Theorem efgmval 19609

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 4827	. 2 ⊢ (𝑎 = 𝐴 → ⟨𝑎, (1o ∖ 𝑏)⟩ = ⟨𝐴, (1o ∖ 𝑏)⟩)
2		difeq2 4073	. . 3 ⊢ (𝑏 = 𝐵 → (1o ∖ 𝑏) = (1o ∖ 𝐵))
3	2	opeq2d 4834	. 2 ⊢ (𝑏 = 𝐵 → ⟨𝐴, (1o ∖ 𝑏)⟩ = ⟨𝐴, (1o ∖ 𝐵)⟩)
4		efgmval.m	. . 3 ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)
5		opeq1 4827	. . . 4 ⊢ (𝑦 = 𝑎 → ⟨𝑦, (1o ∖ 𝑧)⟩ = ⟨𝑎, (1o ∖ 𝑧)⟩)
6		difeq2 4073	. . . . 5 ⊢ (𝑧 = 𝑏 → (1o ∖ 𝑧) = (1o ∖ 𝑏))
7	6	opeq2d 4834	. . . 4 ⊢ (𝑧 = 𝑏 → ⟨𝑎, (1o ∖ 𝑧)⟩ = ⟨𝑎, (1o ∖ 𝑏)⟩)
8	5, 7	cbvmpov 7448	. . 3 ⊢ (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩) = (𝑎 ∈ 𝐼, 𝑏 ∈ 2o ↦ ⟨𝑎, (1o ∖ 𝑏)⟩)
9	4, 8	eqtri 2752	. 2 ⊢ 𝑀 = (𝑎 ∈ 𝐼, 𝑏 ∈ 2o ↦ ⟨𝑎, (1o ∖ 𝑏)⟩)
10		opex 5411	. 2 ⊢ ⟨𝐴, (1o ∖ 𝐵)⟩ ∈ V
11	1, 3, 9, 10	ovmpo 7513	1 ⊢ ((𝐴 ∈ 𝐼 ∧ 𝐵 ∈ 2o) → (𝐴𝑀𝐵) = ⟨𝐴, (1o ∖ 𝐵)⟩)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ∖ cdif 3902  ⟨cop 4585  (class class class)co 7353   ∈ cmpo 7355  1_oc1o 8388  2_oc2o 8389

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-sbc 3745  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-iota 6442  df-fun 6488  df-fv 6494  df-ov 7356  df-oprab 7357  df-mpo 7358

This theorem is referenced by:  efgmnvl  19611  efgval2  19621  vrgpinv  19666  frgpuptinv  19668  frgpuplem  19669  frgpnabllem1  19770