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Theorem efgmnvl 19582
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 5701 . 2 (𝐴 ∈ (𝐼 × 2o) ↔ ∃𝑎𝐼𝑏 ∈ 2o 𝐴 = ⟨𝑎, 𝑏⟩)
2 efgmval.m . . . . . . . 8 𝑀 = (𝑦𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o𝑧)⟩)
32efgmval 19580 . . . . . . 7 ((𝑎𝐼𝑏 ∈ 2o) → (𝑎𝑀𝑏) = ⟨𝑎, (1o𝑏)⟩)
43fveq2d 6896 . . . . . 6 ((𝑎𝐼𝑏 ∈ 2o) → (𝑀‘(𝑎𝑀𝑏)) = (𝑀‘⟨𝑎, (1o𝑏)⟩))
5 df-ov 7412 . . . . . 6 (𝑎𝑀(1o𝑏)) = (𝑀‘⟨𝑎, (1o𝑏)⟩)
64, 5eqtr4di 2791 . . . . 5 ((𝑎𝐼𝑏 ∈ 2o) → (𝑀‘(𝑎𝑀𝑏)) = (𝑎𝑀(1o𝑏)))
7 2oconcl 8503 . . . . . 6 (𝑏 ∈ 2o → (1o𝑏) ∈ 2o)
82efgmval 19580 . . . . . 6 ((𝑎𝐼 ∧ (1o𝑏) ∈ 2o) → (𝑎𝑀(1o𝑏)) = ⟨𝑎, (1o ∖ (1o𝑏))⟩)
97, 8sylan2 594 . . . . 5 ((𝑎𝐼𝑏 ∈ 2o) → (𝑎𝑀(1o𝑏)) = ⟨𝑎, (1o ∖ (1o𝑏))⟩)
10 1on 8478 . . . . . . . . . . 11 1o ∈ On
1110onordi 6476 . . . . . . . . . 10 Ord 1o
12 ordtr 6379 . . . . . . . . . 10 (Ord 1o → Tr 1o)
13 trsucss 6453 . . . . . . . . . 10 (Tr 1o → (𝑏 ∈ suc 1o𝑏 ⊆ 1o))
1411, 12, 13mp2b 10 . . . . . . . . 9 (𝑏 ∈ suc 1o𝑏 ⊆ 1o)
15 df-2o 8467 . . . . . . . . 9 2o = suc 1o
1614, 15eleq2s 2852 . . . . . . . 8 (𝑏 ∈ 2o𝑏 ⊆ 1o)
1716adantl 483 . . . . . . 7 ((𝑎𝐼𝑏 ∈ 2o) → 𝑏 ⊆ 1o)
18 dfss4 4259 . . . . . . 7 (𝑏 ⊆ 1o ↔ (1o ∖ (1o𝑏)) = 𝑏)
1917, 18sylib 217 . . . . . 6 ((𝑎𝐼𝑏 ∈ 2o) → (1o ∖ (1o𝑏)) = 𝑏)
2019opeq2d 4881 . . . . 5 ((𝑎𝐼𝑏 ∈ 2o) → ⟨𝑎, (1o ∖ (1o𝑏))⟩ = ⟨𝑎, 𝑏⟩)
216, 9, 203eqtrd 2777 . . . 4 ((𝑎𝐼𝑏 ∈ 2o) → (𝑀‘(𝑎𝑀𝑏)) = ⟨𝑎, 𝑏⟩)
22 fveq2 6892 . . . . . . 7 (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑀𝐴) = (𝑀‘⟨𝑎, 𝑏⟩))
23 df-ov 7412 . . . . . . 7 (𝑎𝑀𝑏) = (𝑀‘⟨𝑎, 𝑏⟩)
2422, 23eqtr4di 2791 . . . . . 6 (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑀𝐴) = (𝑎𝑀𝑏))
2524fveq2d 6896 . . . . 5 (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑀‘(𝑀𝐴)) = (𝑀‘(𝑎𝑀𝑏)))
26 id 22 . . . . 5 (𝐴 = ⟨𝑎, 𝑏⟩ → 𝐴 = ⟨𝑎, 𝑏⟩)
2725, 26eqeq12d 2749 . . . 4 (𝐴 = ⟨𝑎, 𝑏⟩ → ((𝑀‘(𝑀𝐴)) = 𝐴 ↔ (𝑀‘(𝑎𝑀𝑏)) = ⟨𝑎, 𝑏⟩))
2821, 27syl5ibrcom 246 . . 3 ((𝑎𝐼𝑏 ∈ 2o) → (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑀‘(𝑀𝐴)) = 𝐴))
2928rexlimivv 3200 . 2 (∃𝑎𝐼𝑏 ∈ 2o 𝐴 = ⟨𝑎, 𝑏⟩ → (𝑀‘(𝑀𝐴)) = 𝐴)
301, 29sylbi 216 1 (𝐴 ∈ (𝐼 × 2o) → (𝑀‘(𝑀𝐴)) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wrex 3071  cdif 3946  wss 3949  cop 4635  Tr wtr 5266   × cxp 5675  Ord word 6364  suc csuc 6367  cfv 6544  (class class class)co 7409  cmpo 7411  1oc1o 8459  2oc2o 8460
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-ord 6368  df-on 6369  df-suc 6371  df-iota 6496  df-fun 6546  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1o 8466  df-2o 8467
This theorem is referenced by:  efginvrel1  19596  efgredlemc  19613
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