Theorem efgmval 18125

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 4402	. 2 ⊢ (𝑎 = 𝐴 → ⟨𝑎, (1𝑜 ∖ 𝑏)⟩ = ⟨𝐴, (1𝑜 ∖ 𝑏)⟩)
2		difeq2 3722	. . 3 ⊢ (𝑏 = 𝐵 → (1𝑜 ∖ 𝑏) = (1𝑜 ∖ 𝐵))
3	2	opeq2d 4409	. 2 ⊢ (𝑏 = 𝐵 → ⟨𝐴, (1𝑜 ∖ 𝑏)⟩ = ⟨𝐴, (1𝑜 ∖ 𝐵)⟩)
4		efgmval.m	. . 3 ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜 ∖ 𝑧)⟩)
5		opeq1 4402	. . . 4 ⊢ (𝑦 = 𝑎 → ⟨𝑦, (1𝑜 ∖ 𝑧)⟩ = ⟨𝑎, (1𝑜 ∖ 𝑧)⟩)
6		difeq2 3722	. . . . 5 ⊢ (𝑧 = 𝑏 → (1𝑜 ∖ 𝑧) = (1𝑜 ∖ 𝑏))
7	6	opeq2d 4409	. . . 4 ⊢ (𝑧 = 𝑏 → ⟨𝑎, (1𝑜 ∖ 𝑧)⟩ = ⟨𝑎, (1𝑜 ∖ 𝑏)⟩)
8	5, 7	cbvmpt2v 6735	. . 3 ⊢ (𝑦 ∈ 𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜 ∖ 𝑧)⟩) = (𝑎 ∈ 𝐼, 𝑏 ∈ 2𝑜 ↦ ⟨𝑎, (1𝑜 ∖ 𝑏)⟩)
9	4, 8	eqtri 2644	. 2 ⊢ 𝑀 = (𝑎 ∈ 𝐼, 𝑏 ∈ 2𝑜 ↦ ⟨𝑎, (1𝑜 ∖ 𝑏)⟩)
10		opex 4932	. 2 ⊢ ⟨𝐴, (1𝑜 ∖ 𝐵)⟩ ∈ V
11	1, 3, 9, 10	ovmpt2 6796	1 ⊢ ((𝐴 ∈ 𝐼 ∧ 𝐵 ∈ 2𝑜) → (𝐴𝑀𝐵) = ⟨𝐴, (1𝑜 ∖ 𝐵)⟩)

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ∖ cdif 3571  ⟨cop 4183  (class class class)co 6650   ↦ cmpt2 6652  1_𝑜c1o 7553  2_𝑜c2o 7554

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655

This theorem is referenced by:  efgmnvl  18127  efgval2  18137  vrgpinv  18182  frgpuptinv  18184  frgpuplem  18185  frgpnabllem1  18276