Theorem efgmval 19641

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 4829	. 2 ⊢ (𝑎 = 𝐴 → ⟨𝑎, (1o ∖ 𝑏)⟩ = ⟨𝐴, (1o ∖ 𝑏)⟩)
2		difeq2 4072	. . 3 ⊢ (𝑏 = 𝐵 → (1o ∖ 𝑏) = (1o ∖ 𝐵))
3	2	opeq2d 4836	. 2 ⊢ (𝑏 = 𝐵 → ⟨𝐴, (1o ∖ 𝑏)⟩ = ⟨𝐴, (1o ∖ 𝐵)⟩)
4		efgmval.m	. . 3 ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)
5		opeq1 4829	. . . 4 ⊢ (𝑦 = 𝑎 → ⟨𝑦, (1o ∖ 𝑧)⟩ = ⟨𝑎, (1o ∖ 𝑧)⟩)
6		difeq2 4072	. . . . 5 ⊢ (𝑧 = 𝑏 → (1o ∖ 𝑧) = (1o ∖ 𝑏))
7	6	opeq2d 4836	. . . 4 ⊢ (𝑧 = 𝑏 → ⟨𝑎, (1o ∖ 𝑧)⟩ = ⟨𝑎, (1o ∖ 𝑏)⟩)
8	5, 7	cbvmpov 7453	. . 3 ⊢ (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩) = (𝑎 ∈ 𝐼, 𝑏 ∈ 2o ↦ ⟨𝑎, (1o ∖ 𝑏)⟩)
9	4, 8	eqtri 2759	. 2 ⊢ 𝑀 = (𝑎 ∈ 𝐼, 𝑏 ∈ 2o ↦ ⟨𝑎, (1o ∖ 𝑏)⟩)
10		opex 5412	. 2 ⊢ ⟨𝐴, (1o ∖ 𝐵)⟩ ∈ V
11	1, 3, 9, 10	ovmpo 7518	1 ⊢ ((𝐴 ∈ 𝐼 ∧ 𝐵 ∈ 2o) → (𝐴𝑀𝐵) = ⟨𝐴, (1o ∖ 𝐵)⟩)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113   ∖ cdif 3898  ⟨cop 4586  (class class class)co 7358   ∈ cmpo 7360  1_oc1o 8390  2_oc2o 8391

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-iota 6448  df-fun 6494  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363

This theorem is referenced by:  efgmnvl  19643  efgval2  19653  vrgpinv  19698  frgpuptinv  19700  frgpuplem  19701  frgpnabllem1  19802