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Theorem efgmval 18817
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 4779 . 2 (𝑎 = 𝐴 → ⟨𝑎, (1o𝑏)⟩ = ⟨𝐴, (1o𝑏)⟩)
2 difeq2 4072 . . 3 (𝑏 = 𝐵 → (1o𝑏) = (1o𝐵))
32opeq2d 4786 . 2 (𝑏 = 𝐵 → ⟨𝐴, (1o𝑏)⟩ = ⟨𝐴, (1o𝐵)⟩)
4 efgmval.m . . 3 𝑀 = (𝑦𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o𝑧)⟩)
5 opeq1 4779 . . . 4 (𝑦 = 𝑎 → ⟨𝑦, (1o𝑧)⟩ = ⟨𝑎, (1o𝑧)⟩)
6 difeq2 4072 . . . . 5 (𝑧 = 𝑏 → (1o𝑧) = (1o𝑏))
76opeq2d 4786 . . . 4 (𝑧 = 𝑏 → ⟨𝑎, (1o𝑧)⟩ = ⟨𝑎, (1o𝑏)⟩)
85, 7cbvmpov 7226 . . 3 (𝑦𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o𝑧)⟩) = (𝑎𝐼, 𝑏 ∈ 2o ↦ ⟨𝑎, (1o𝑏)⟩)
94, 8eqtri 2843 . 2 𝑀 = (𝑎𝐼, 𝑏 ∈ 2o ↦ ⟨𝑎, (1o𝑏)⟩)
10 opex 5332 . 2 𝐴, (1o𝐵)⟩ ∈ V
111, 3, 9, 10ovmpo 7287 1 ((𝐴𝐼𝐵 ∈ 2o) → (𝐴𝑀𝐵) = ⟨𝐴, (1o𝐵)⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1537  wcel 2114  cdif 3910  cop 4549  (class class class)co 7133  cmpo 7135  1oc1o 8073  2oc2o 8074
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179  ax-nul 5186  ax-pr 5306
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3475  df-sbc 3753  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-sn 4544  df-pr 4546  df-op 4550  df-uni 4815  df-br 5043  df-opab 5105  df-id 5436  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-iota 6290  df-fun 6333  df-fv 6339  df-ov 7136  df-oprab 7137  df-mpo 7138
This theorem is referenced by:  efgmnvl  18819  efgval2  18829  vrgpinv  18874  frgpuptinv  18876  frgpuplem  18877  frgpnabllem1  18972
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