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Theorem efgmnvl 18842
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 5567 . 2 (𝐴 ∈ (𝐼 × 2o) ↔ ∃𝑎𝐼𝑏 ∈ 2o 𝐴 = ⟨𝑎, 𝑏⟩)
2 efgmval.m . . . . . . . 8 𝑀 = (𝑦𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o𝑧)⟩)
32efgmval 18840 . . . . . . 7 ((𝑎𝐼𝑏 ∈ 2o) → (𝑎𝑀𝑏) = ⟨𝑎, (1o𝑏)⟩)
43fveq2d 6667 . . . . . 6 ((𝑎𝐼𝑏 ∈ 2o) → (𝑀‘(𝑎𝑀𝑏)) = (𝑀‘⟨𝑎, (1o𝑏)⟩))
5 df-ov 7154 . . . . . 6 (𝑎𝑀(1o𝑏)) = (𝑀‘⟨𝑎, (1o𝑏)⟩)
64, 5eqtr4di 2877 . . . . 5 ((𝑎𝐼𝑏 ∈ 2o) → (𝑀‘(𝑎𝑀𝑏)) = (𝑎𝑀(1o𝑏)))
7 2oconcl 8126 . . . . . 6 (𝑏 ∈ 2o → (1o𝑏) ∈ 2o)
82efgmval 18840 . . . . . 6 ((𝑎𝐼 ∧ (1o𝑏) ∈ 2o) → (𝑎𝑀(1o𝑏)) = ⟨𝑎, (1o ∖ (1o𝑏))⟩)
97, 8sylan2 595 . . . . 5 ((𝑎𝐼𝑏 ∈ 2o) → (𝑎𝑀(1o𝑏)) = ⟨𝑎, (1o ∖ (1o𝑏))⟩)
10 1on 8107 . . . . . . . . . . 11 1o ∈ On
1110onordi 6284 . . . . . . . . . 10 Ord 1o
12 ordtr 6194 . . . . . . . . . 10 (Ord 1o → Tr 1o)
13 trsucss 6265 . . . . . . . . . 10 (Tr 1o → (𝑏 ∈ suc 1o𝑏 ⊆ 1o))
1411, 12, 13mp2b 10 . . . . . . . . 9 (𝑏 ∈ suc 1o𝑏 ⊆ 1o)
15 df-2o 8101 . . . . . . . . 9 2o = suc 1o
1614, 15eleq2s 2934 . . . . . . . 8 (𝑏 ∈ 2o𝑏 ⊆ 1o)
1716adantl 485 . . . . . . 7 ((𝑎𝐼𝑏 ∈ 2o) → 𝑏 ⊆ 1o)
18 dfss4 4220 . . . . . . 7 (𝑏 ⊆ 1o ↔ (1o ∖ (1o𝑏)) = 𝑏)
1917, 18sylib 221 . . . . . 6 ((𝑎𝐼𝑏 ∈ 2o) → (1o ∖ (1o𝑏)) = 𝑏)
2019opeq2d 4796 . . . . 5 ((𝑎𝐼𝑏 ∈ 2o) → ⟨𝑎, (1o ∖ (1o𝑏))⟩ = ⟨𝑎, 𝑏⟩)
216, 9, 203eqtrd 2863 . . . 4 ((𝑎𝐼𝑏 ∈ 2o) → (𝑀‘(𝑎𝑀𝑏)) = ⟨𝑎, 𝑏⟩)
22 fveq2 6663 . . . . . . 7 (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑀𝐴) = (𝑀‘⟨𝑎, 𝑏⟩))
23 df-ov 7154 . . . . . . 7 (𝑎𝑀𝑏) = (𝑀‘⟨𝑎, 𝑏⟩)
2422, 23eqtr4di 2877 . . . . . 6 (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑀𝐴) = (𝑎𝑀𝑏))
2524fveq2d 6667 . . . . 5 (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑀‘(𝑀𝐴)) = (𝑀‘(𝑎𝑀𝑏)))
26 id 22 . . . . 5 (𝐴 = ⟨𝑎, 𝑏⟩ → 𝐴 = ⟨𝑎, 𝑏⟩)
2725, 26eqeq12d 2840 . . . 4 (𝐴 = ⟨𝑎, 𝑏⟩ → ((𝑀‘(𝑀𝐴)) = 𝐴 ↔ (𝑀‘(𝑎𝑀𝑏)) = ⟨𝑎, 𝑏⟩))
2821, 27syl5ibrcom 250 . . 3 ((𝑎𝐼𝑏 ∈ 2o) → (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑀‘(𝑀𝐴)) = 𝐴))
2928rexlimivv 3284 . 2 (∃𝑎𝐼𝑏 ∈ 2o 𝐴 = ⟨𝑎, 𝑏⟩ → (𝑀‘(𝑀𝐴)) = 𝐴)
301, 29sylbi 220 1 (𝐴 ∈ (𝐼 × 2o) → (𝑀‘(𝑀𝐴)) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  wrex 3134  cdif 3916  wss 3919  cop 4556  Tr wtr 5159   × cxp 5541  Ord word 6179  suc csuc 6182  cfv 6345  (class class class)co 7151  cmpo 7153  1oc1o 8093  2oc2o 8094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pr 5318  ax-un 7457
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-br 5054  df-opab 5116  df-tr 5160  df-id 5448  df-eprel 5453  df-po 5462  df-so 5463  df-fr 5502  df-we 5504  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-ord 6183  df-on 6184  df-suc 6186  df-iota 6304  df-fun 6347  df-fv 6353  df-ov 7154  df-oprab 7155  df-mpo 7156  df-1o 8100  df-2o 8101
This theorem is referenced by:  efginvrel1  18856  efgredlemc  18873
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