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Theorem efgmval 18106
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 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)
Assertion
Ref Expression
efgmval ((𝐴𝐼𝐵 ∈ 2𝑜) → (𝐴𝑀𝐵) = ⟨𝐴, (1𝑜𝐵)⟩)
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.
StepHypRef Expression
1 opeq1 4393 . 2 (𝑎 = 𝐴 → ⟨𝑎, (1𝑜𝑏)⟩ = ⟨𝐴, (1𝑜𝑏)⟩)
2 difeq2 3714 . . 3 (𝑏 = 𝐵 → (1𝑜𝑏) = (1𝑜𝐵))
32opeq2d 4400 . 2 (𝑏 = 𝐵 → ⟨𝐴, (1𝑜𝑏)⟩ = ⟨𝐴, (1𝑜𝐵)⟩)
4 efgmval.m . . 3 𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)
5 opeq1 4393 . . . 4 (𝑦 = 𝑎 → ⟨𝑦, (1𝑜𝑧)⟩ = ⟨𝑎, (1𝑜𝑧)⟩)
6 difeq2 3714 . . . . 5 (𝑧 = 𝑏 → (1𝑜𝑧) = (1𝑜𝑏))
76opeq2d 4400 . . . 4 (𝑧 = 𝑏 → ⟨𝑎, (1𝑜𝑧)⟩ = ⟨𝑎, (1𝑜𝑏)⟩)
85, 7cbvmpt2v 6720 . . 3 (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩) = (𝑎𝐼, 𝑏 ∈ 2𝑜 ↦ ⟨𝑎, (1𝑜𝑏)⟩)
94, 8eqtri 2642 . 2 𝑀 = (𝑎𝐼, 𝑏 ∈ 2𝑜 ↦ ⟨𝑎, (1𝑜𝑏)⟩)
10 opex 4923 . 2 𝐴, (1𝑜𝐵)⟩ ∈ V
111, 3, 9, 10ovmpt2 6781 1 ((𝐴𝐼𝐵 ∈ 2𝑜) → (𝐴𝑀𝐵) = ⟨𝐴, (1𝑜𝐵)⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1481  wcel 1988  cdif 3564  cop 4174  (class class class)co 6635  cmpt2 6637  1𝑜c1o 7538  2𝑜c2o 7539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-iota 5839  df-fun 5878  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640
This theorem is referenced by:  efgmnvl  18108  efgval2  18118  vrgpinv  18163  frgpuptinv  18165  frgpuplem  18166  frgpnabllem1  18257
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