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Theorem efgmnvl 19581
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 5700 . 2 (𝐴 ∈ (𝐼 × 2o) ↔ ∃𝑎𝐼𝑏 ∈ 2o 𝐴 = ⟨𝑎, 𝑏⟩)
2 efgmval.m . . . . . . . 8 𝑀 = (𝑦𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o𝑧)⟩)
32efgmval 19579 . . . . . . 7 ((𝑎𝐼𝑏 ∈ 2o) → (𝑎𝑀𝑏) = ⟨𝑎, (1o𝑏)⟩)
43fveq2d 6895 . . . . . 6 ((𝑎𝐼𝑏 ∈ 2o) → (𝑀‘(𝑎𝑀𝑏)) = (𝑀‘⟨𝑎, (1o𝑏)⟩))
5 df-ov 7411 . . . . . 6 (𝑎𝑀(1o𝑏)) = (𝑀‘⟨𝑎, (1o𝑏)⟩)
64, 5eqtr4di 2790 . . . . 5 ((𝑎𝐼𝑏 ∈ 2o) → (𝑀‘(𝑎𝑀𝑏)) = (𝑎𝑀(1o𝑏)))
7 2oconcl 8502 . . . . . 6 (𝑏 ∈ 2o → (1o𝑏) ∈ 2o)
82efgmval 19579 . . . . . 6 ((𝑎𝐼 ∧ (1o𝑏) ∈ 2o) → (𝑎𝑀(1o𝑏)) = ⟨𝑎, (1o ∖ (1o𝑏))⟩)
97, 8sylan2 593 . . . . 5 ((𝑎𝐼𝑏 ∈ 2o) → (𝑎𝑀(1o𝑏)) = ⟨𝑎, (1o ∖ (1o𝑏))⟩)
10 1on 8477 . . . . . . . . . . 11 1o ∈ On
1110onordi 6475 . . . . . . . . . 10 Ord 1o
12 ordtr 6378 . . . . . . . . . 10 (Ord 1o → Tr 1o)
13 trsucss 6452 . . . . . . . . . 10 (Tr 1o → (𝑏 ∈ suc 1o𝑏 ⊆ 1o))
1411, 12, 13mp2b 10 . . . . . . . . 9 (𝑏 ∈ suc 1o𝑏 ⊆ 1o)
15 df-2o 8466 . . . . . . . . 9 2o = suc 1o
1614, 15eleq2s 2851 . . . . . . . 8 (𝑏 ∈ 2o𝑏 ⊆ 1o)
1716adantl 482 . . . . . . 7 ((𝑎𝐼𝑏 ∈ 2o) → 𝑏 ⊆ 1o)
18 dfss4 4258 . . . . . . 7 (𝑏 ⊆ 1o ↔ (1o ∖ (1o𝑏)) = 𝑏)
1917, 18sylib 217 . . . . . 6 ((𝑎𝐼𝑏 ∈ 2o) → (1o ∖ (1o𝑏)) = 𝑏)
2019opeq2d 4880 . . . . 5 ((𝑎𝐼𝑏 ∈ 2o) → ⟨𝑎, (1o ∖ (1o𝑏))⟩ = ⟨𝑎, 𝑏⟩)
216, 9, 203eqtrd 2776 . . . 4 ((𝑎𝐼𝑏 ∈ 2o) → (𝑀‘(𝑎𝑀𝑏)) = ⟨𝑎, 𝑏⟩)
22 fveq2 6891 . . . . . . 7 (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑀𝐴) = (𝑀‘⟨𝑎, 𝑏⟩))
23 df-ov 7411 . . . . . . 7 (𝑎𝑀𝑏) = (𝑀‘⟨𝑎, 𝑏⟩)
2422, 23eqtr4di 2790 . . . . . 6 (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑀𝐴) = (𝑎𝑀𝑏))
2524fveq2d 6895 . . . . 5 (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑀‘(𝑀𝐴)) = (𝑀‘(𝑎𝑀𝑏)))
26 id 22 . . . . 5 (𝐴 = ⟨𝑎, 𝑏⟩ → 𝐴 = ⟨𝑎, 𝑏⟩)
2725, 26eqeq12d 2748 . . . 4 (𝐴 = ⟨𝑎, 𝑏⟩ → ((𝑀‘(𝑀𝐴)) = 𝐴 ↔ (𝑀‘(𝑎𝑀𝑏)) = ⟨𝑎, 𝑏⟩))
2821, 27syl5ibrcom 246 . . 3 ((𝑎𝐼𝑏 ∈ 2o) → (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑀‘(𝑀𝐴)) = 𝐴))
2928rexlimivv 3199 . 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 396   = wceq 1541  wcel 2106  wrex 3070  cdif 3945  wss 3948  cop 4634  Tr wtr 5265   × cxp 5674  Ord word 6363  suc csuc 6366  cfv 6543  (class class class)co 7408  cmpo 7410  1oc1o 8458  2oc2o 8459
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-ord 6367  df-on 6368  df-suc 6370  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-1o 8465  df-2o 8466
This theorem is referenced by:  efginvrel1  19595  efgredlemc  19612
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