Step
*
of Lemma
funinv-ap-equals
∀[n:ℕ]. ∀[f:ℕn →⟶ ℕn]. ∀[a,b:ℕn]. (inv(f) b) = a ∈ ℤ supposing (f a) = b ∈ ℤ
BY
{ (Auto
THEN (Assert inv(f) ∈ ℕn →⟶ ℕn BY
Auto)
THEN StrengthenEquation (-2)
THEN (ApFunToHypEquands `Z' ⌜inv(f) Z⌝ ⌜ℤ⌝ (-1)⋅ THENA Auto)
THEN NthHypEqTrans (-1)) }
1
.....assertion.....
1. n : ℕ
2. f : ℕn →⟶ ℕn
3. a : ℕn
4. b : ℕn
5. (f a) = b ∈ ℤ
6. inv(f) ∈ ℕn →⟶ ℕn
7. (f a) = b ∈ {z:ℤ| (z = (f a) ∈ ℤ) ∧ (z = b ∈ ℤ)}
8. (inv(f) (f a)) = (inv(f) b) ∈ ℤ
⊢ (inv(f) (f a)) = a ∈ ℤ
Latex:
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[f:\mBbbN{}n \mrightarrow{}{}\mrightarrow{} \mBbbN{}n]. \mforall{}[a,b:\mBbbN{}n]. (inv(f) b) = a supposing (f a) = b
By
Latex:
(Auto
THEN (Assert inv(f) \mmember{} \mBbbN{}n \mrightarrow{}{}\mrightarrow{} \mBbbN{}n BY
Auto)
THEN StrengthenEquation (-2)
THEN (ApFunToHypEquands `Z' \mkleeneopen{}inv(f) Z\mkleeneclose{} \mkleeneopen{}\mBbbZ{}\mkleeneclose{} (-1)\mcdot{} THENA Auto)
THEN NthHypEqTrans (-1))
Home
Index