Theorem efgmval 17948

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 4340	. 2 ⊢ (𝑎 = 𝐴 → ⟨𝑎, (1𝑜 ∖ 𝑏)⟩ = ⟨𝐴, (1𝑜 ∖ 𝑏)⟩)
2		difeq2 3684	. . 3 ⊢ (𝑏 = 𝐵 → (1𝑜 ∖ 𝑏) = (1𝑜 ∖ 𝐵))
3	2	opeq2d 4347	. 2 ⊢ (𝑏 = 𝐵 → ⟨𝐴, (1𝑜 ∖ 𝑏)⟩ = ⟨𝐴, (1𝑜 ∖ 𝐵)⟩)
4		efgmval.m	. . 3 ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜 ∖ 𝑧)⟩)
5		opeq1 4340	. . . 4 ⊢ (𝑦 = 𝑎 → ⟨𝑦, (1𝑜 ∖ 𝑧)⟩ = ⟨𝑎, (1𝑜 ∖ 𝑧)⟩)
6		difeq2 3684	. . . . 5 ⊢ (𝑧 = 𝑏 → (1𝑜 ∖ 𝑧) = (1𝑜 ∖ 𝑏))
7	6	opeq2d 4347	. . . 4 ⊢ (𝑧 = 𝑏 → ⟨𝑎, (1𝑜 ∖ 𝑧)⟩ = ⟨𝑎, (1𝑜 ∖ 𝑏)⟩)
8	5, 7	cbvmpt2v 6633	. . 3 ⊢ (𝑦 ∈ 𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜 ∖ 𝑧)⟩) = (𝑎 ∈ 𝐼, 𝑏 ∈ 2𝑜 ↦ ⟨𝑎, (1𝑜 ∖ 𝑏)⟩)
9	4, 8	eqtri 2632	. 2 ⊢ 𝑀 = (𝑎 ∈ 𝐼, 𝑏 ∈ 2𝑜 ↦ ⟨𝑎, (1𝑜 ∖ 𝑏)⟩)
10		opex 4859	. 2 ⊢ ⟨𝐴, (1𝑜 ∖ 𝐵)⟩ ∈ V
11	1, 3, 9, 10	ovmpt2 6694	1 ⊢ ((𝐴 ∈ 𝐼 ∧ 𝐵 ∈ 2𝑜) → (𝐴𝑀𝐵) = ⟨𝐴, (1𝑜 ∖ 𝐵)⟩)

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ∖ cdif 3537  ⟨cop 4131  (class class class)co 6549   ↦ cmpt2 6551  1_𝑜c1o 7440  2_𝑜c2o 7441

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554

This theorem is referenced by:  efgmnvl  17950  efgval2  17960  vrgpinv  18005  frgpuptinv  18007  frgpuplem  18008  frgpnabllem1  18099