Theorem efgmval 19685

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 4811	. 2 ⊢ (𝑎 = 𝐴 → ⟨𝑎, (1o ∖ 𝑏)⟩ = ⟨𝐴, (1o ∖ 𝑏)⟩)
2		difeq2 4058	. . 3 ⊢ (𝑏 = 𝐵 → (1o ∖ 𝑏) = (1o ∖ 𝐵))
3	2	opeq2d 4818	. 2 ⊢ (𝑏 = 𝐵 → ⟨𝐴, (1o ∖ 𝑏)⟩ = ⟨𝐴, (1o ∖ 𝐵)⟩)
4		efgmval.m	. . 3 ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)
5		opeq1 4811	. . . 4 ⊢ (𝑦 = 𝑎 → ⟨𝑦, (1o ∖ 𝑧)⟩ = ⟨𝑎, (1o ∖ 𝑧)⟩)
6		difeq2 4058	. . . . 5 ⊢ (𝑧 = 𝑏 → (1o ∖ 𝑧) = (1o ∖ 𝑏))
7	6	opeq2d 4818	. . . 4 ⊢ (𝑧 = 𝑏 → ⟨𝑎, (1o ∖ 𝑧)⟩ = ⟨𝑎, (1o ∖ 𝑏)⟩)
8	5, 7	cbvmpov 7458	. . 3 ⊢ (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩) = (𝑎 ∈ 𝐼, 𝑏 ∈ 2o ↦ ⟨𝑎, (1o ∖ 𝑏)⟩)
9	4, 8	eqtri 2763	. 2 ⊢ 𝑀 = (𝑎 ∈ 𝐼, 𝑏 ∈ 2o ↦ ⟨𝑎, (1o ∖ 𝑏)⟩)
10		opex 5410	. 2 ⊢ ⟨𝐴, (1o ∖ 𝐵)⟩ ∈ V
11	1, 3, 9, 10	ovmpo 7523	1 ⊢ ((𝐴 ∈ 𝐼 ∧ 𝐵 ∈ 2o) → (𝐴𝑀𝐵) = ⟨𝐴, (1o ∖ 𝐵)⟩)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119   ∖ cdif 3887  ⟨cop 4568  (class class class)co 7363   ∈ cmpo 7365  1_oc1o 8395  2_oc2o 8396

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-sbc 3731  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-iota 6448  df-fun 6494  df-fv 6500  df-ov 7366  df-oprab 7367  df-mpo 7368

This theorem is referenced by:  efgmnvl  19687  efgval2  19697  vrgpinv  19742  frgpuptinv  19744  frgpuplem  19745  frgpnabllem1  19846