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Theorem efgmval 19745
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 4878 . 2 (𝑎 = 𝐴 → ⟨𝑎, (1o𝑏)⟩ = ⟨𝐴, (1o𝑏)⟩)
2 difeq2 4130 . . 3 (𝑏 = 𝐵 → (1o𝑏) = (1o𝐵))
32opeq2d 4885 . 2 (𝑏 = 𝐵 → ⟨𝐴, (1o𝑏)⟩ = ⟨𝐴, (1o𝐵)⟩)
4 efgmval.m . . 3 𝑀 = (𝑦𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o𝑧)⟩)
5 opeq1 4878 . . . 4 (𝑦 = 𝑎 → ⟨𝑦, (1o𝑧)⟩ = ⟨𝑎, (1o𝑧)⟩)
6 difeq2 4130 . . . . 5 (𝑧 = 𝑏 → (1o𝑧) = (1o𝑏))
76opeq2d 4885 . . . 4 (𝑧 = 𝑏 → ⟨𝑎, (1o𝑧)⟩ = ⟨𝑎, (1o𝑏)⟩)
85, 7cbvmpov 7528 . . 3 (𝑦𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o𝑧)⟩) = (𝑎𝐼, 𝑏 ∈ 2o ↦ ⟨𝑎, (1o𝑏)⟩)
94, 8eqtri 2763 . 2 𝑀 = (𝑎𝐼, 𝑏 ∈ 2o ↦ ⟨𝑎, (1o𝑏)⟩)
10 opex 5475 . 2 𝐴, (1o𝐵)⟩ ∈ V
111, 3, 9, 10ovmpo 7593 1 ((𝐴𝐼𝐵 ∈ 2o) → (𝐴𝑀𝐵) = ⟨𝐴, (1o𝐵)⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  cdif 3960  cop 4637  (class class class)co 7431  cmpo 7433  1oc1o 8498  2oc2o 8499
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436
This theorem is referenced by:  efgmnvl  19747  efgval2  19757  vrgpinv  19802  frgpuptinv  19804  frgpuplem  19805  frgpnabllem1  19906
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