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Theorem efgmnvl 19747
Description: The inversion function on the generators is an involution. (Contributed by Mario Carneiro, 1-Oct-2015.)
Hypothesis
Ref Expression
efgmval.m 𝑀 = (𝑦𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o𝑧)⟩)
Assertion
Ref Expression
efgmnvl (𝐴 ∈ (𝐼 × 2o) → (𝑀‘(𝑀𝐴)) = 𝐴)
Distinct variable group:   𝑦,𝑧,𝐼
Allowed substitution hints:   𝐴(𝑦,𝑧)   𝑀(𝑦,𝑧)

Proof of Theorem efgmnvl
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elxp2 5713 . 2 (𝐴 ∈ (𝐼 × 2o) ↔ ∃𝑎𝐼𝑏 ∈ 2o 𝐴 = ⟨𝑎, 𝑏⟩)
2 efgmval.m . . . . . . . 8 𝑀 = (𝑦𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o𝑧)⟩)
32efgmval 19745 . . . . . . 7 ((𝑎𝐼𝑏 ∈ 2o) → (𝑎𝑀𝑏) = ⟨𝑎, (1o𝑏)⟩)
43fveq2d 6911 . . . . . 6 ((𝑎𝐼𝑏 ∈ 2o) → (𝑀‘(𝑎𝑀𝑏)) = (𝑀‘⟨𝑎, (1o𝑏)⟩))
5 df-ov 7434 . . . . . 6 (𝑎𝑀(1o𝑏)) = (𝑀‘⟨𝑎, (1o𝑏)⟩)
64, 5eqtr4di 2793 . . . . 5 ((𝑎𝐼𝑏 ∈ 2o) → (𝑀‘(𝑎𝑀𝑏)) = (𝑎𝑀(1o𝑏)))
7 2oconcl 8540 . . . . . 6 (𝑏 ∈ 2o → (1o𝑏) ∈ 2o)
82efgmval 19745 . . . . . 6 ((𝑎𝐼 ∧ (1o𝑏) ∈ 2o) → (𝑎𝑀(1o𝑏)) = ⟨𝑎, (1o ∖ (1o𝑏))⟩)
97, 8sylan2 593 . . . . 5 ((𝑎𝐼𝑏 ∈ 2o) → (𝑎𝑀(1o𝑏)) = ⟨𝑎, (1o ∖ (1o𝑏))⟩)
10 1on 8517 . . . . . . . . . . 11 1o ∈ On
1110onordi 6497 . . . . . . . . . 10 Ord 1o
12 ordtr 6400 . . . . . . . . . 10 (Ord 1o → Tr 1o)
13 trsucss 6474 . . . . . . . . . 10 (Tr 1o → (𝑏 ∈ suc 1o𝑏 ⊆ 1o))
1411, 12, 13mp2b 10 . . . . . . . . 9 (𝑏 ∈ suc 1o𝑏 ⊆ 1o)
15 df-2o 8506 . . . . . . . . 9 2o = suc 1o
1614, 15eleq2s 2857 . . . . . . . 8 (𝑏 ∈ 2o𝑏 ⊆ 1o)
1716adantl 481 . . . . . . 7 ((𝑎𝐼𝑏 ∈ 2o) → 𝑏 ⊆ 1o)
18 dfss4 4275 . . . . . . 7 (𝑏 ⊆ 1o ↔ (1o ∖ (1o𝑏)) = 𝑏)
1917, 18sylib 218 . . . . . 6 ((𝑎𝐼𝑏 ∈ 2o) → (1o ∖ (1o𝑏)) = 𝑏)
2019opeq2d 4885 . . . . 5 ((𝑎𝐼𝑏 ∈ 2o) → ⟨𝑎, (1o ∖ (1o𝑏))⟩ = ⟨𝑎, 𝑏⟩)
216, 9, 203eqtrd 2779 . . . 4 ((𝑎𝐼𝑏 ∈ 2o) → (𝑀‘(𝑎𝑀𝑏)) = ⟨𝑎, 𝑏⟩)
22 fveq2 6907 . . . . . . 7 (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑀𝐴) = (𝑀‘⟨𝑎, 𝑏⟩))
23 df-ov 7434 . . . . . . 7 (𝑎𝑀𝑏) = (𝑀‘⟨𝑎, 𝑏⟩)
2422, 23eqtr4di 2793 . . . . . 6 (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑀𝐴) = (𝑎𝑀𝑏))
2524fveq2d 6911 . . . . 5 (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑀‘(𝑀𝐴)) = (𝑀‘(𝑎𝑀𝑏)))
26 id 22 . . . . 5 (𝐴 = ⟨𝑎, 𝑏⟩ → 𝐴 = ⟨𝑎, 𝑏⟩)
2725, 26eqeq12d 2751 . . . 4 (𝐴 = ⟨𝑎, 𝑏⟩ → ((𝑀‘(𝑀𝐴)) = 𝐴 ↔ (𝑀‘(𝑎𝑀𝑏)) = ⟨𝑎, 𝑏⟩))
2821, 27syl5ibrcom 247 . . 3 ((𝑎𝐼𝑏 ∈ 2o) → (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑀‘(𝑀𝐴)) = 𝐴))
2928rexlimivv 3199 . 2 (∃𝑎𝐼𝑏 ∈ 2o 𝐴 = ⟨𝑎, 𝑏⟩ → (𝑀‘(𝑀𝐴)) = 𝐴)
301, 29sylbi 217 1 (𝐴 ∈ (𝐼 × 2o) → (𝑀‘(𝑀𝐴)) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wrex 3068  cdif 3960  wss 3963  cop 4637  Tr wtr 5265   × cxp 5687  Ord word 6385  suc csuc 6388  cfv 6563  (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-3or 1087  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-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-ord 6389  df-on 6390  df-suc 6392  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1o 8505  df-2o 8506
This theorem is referenced by:  efginvrel1  19761  efgredlemc  19778
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