Theorem efgmval 18509

Description: Value of the formal inverse operation for the generating set of a free group. (Contributed by Mario Carneiro, 27-Sep-2015.)

Hypothesis

Ref	Expression
efgmval.m	⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)

Assertion

Ref	Expression
efgmval	⊢ ((𝐴 ∈ 𝐼 ∧ 𝐵 ∈ 2o) → (𝐴𝑀𝐵) = ⟨𝐴, (1o ∖ 𝐵)⟩)

Distinct variable group:   𝑦,𝑧,𝐼

Allowed substitution hints:   𝐴(𝑦,𝑧)   𝐵(𝑦,𝑧)   𝑀(𝑦,𝑧)

Proof of Theorem efgmval

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		opeq1 4636	. 2 ⊢ (𝑎 = 𝐴 → ⟨𝑎, (1o ∖ 𝑏)⟩ = ⟨𝐴, (1o ∖ 𝑏)⟩)
2		difeq2 3945	. . 3 ⊢ (𝑏 = 𝐵 → (1o ∖ 𝑏) = (1o ∖ 𝐵))
3	2	opeq2d 4643	. 2 ⊢ (𝑏 = 𝐵 → ⟨𝐴, (1o ∖ 𝑏)⟩ = ⟨𝐴, (1o ∖ 𝐵)⟩)
4		efgmval.m	. . 3 ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)
5		opeq1 4636	. . . 4 ⊢ (𝑦 = 𝑎 → ⟨𝑦, (1o ∖ 𝑧)⟩ = ⟨𝑎, (1o ∖ 𝑧)⟩)
6		difeq2 3945	. . . . 5 ⊢ (𝑧 = 𝑏 → (1o ∖ 𝑧) = (1o ∖ 𝑏))
7	6	opeq2d 4643	. . . 4 ⊢ (𝑧 = 𝑏 → ⟨𝑎, (1o ∖ 𝑧)⟩ = ⟨𝑎, (1o ∖ 𝑏)⟩)
8	5, 7	cbvmpt2v 7012	. . 3 ⊢ (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩) = (𝑎 ∈ 𝐼, 𝑏 ∈ 2o ↦ ⟨𝑎, (1o ∖ 𝑏)⟩)
9	4, 8	eqtri 2802	. 2 ⊢ 𝑀 = (𝑎 ∈ 𝐼, 𝑏 ∈ 2o ↦ ⟨𝑎, (1o ∖ 𝑏)⟩)
10		opex 5164	. 2 ⊢ ⟨𝐴, (1o ∖ 𝐵)⟩ ∈ V
11	1, 3, 9, 10	ovmpt2 7073	1 ⊢ ((𝐴 ∈ 𝐼 ∧ 𝐵 ∈ 2o) → (𝐴𝑀𝐵) = ⟨𝐴, (1o ∖ 𝐵)⟩)

Syntax hints:   → wi 4   ∧ wa 386   = wceq 1601   ∈ wcel 2107   ∖ cdif 3789  ⟨cop 4404  (class class class)co 6922   ↦ cmpt2 6924  1_oc1o 7836  2_oc2o 7837

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pr 5138

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-iota 6099  df-fun 6137  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927

This theorem is referenced by:  efgmnvl  18511  efgval2  18521  vrgpinv  18568  frgpuptinv  18570  frgpuplem  18571  frgpnabllem1  18662