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Theorem efgmnvl 18059
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 5097 . 2 (𝐴 ∈ (𝐼 × 2𝑜) ↔ ∃𝑎𝐼𝑏 ∈ 2𝑜 𝐴 = ⟨𝑎, 𝑏⟩)
2 efgmval.m . . . . . . . 8 𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)
32efgmval 18057 . . . . . . 7 ((𝑎𝐼𝑏 ∈ 2𝑜) → (𝑎𝑀𝑏) = ⟨𝑎, (1𝑜𝑏)⟩)
43fveq2d 6157 . . . . . 6 ((𝑎𝐼𝑏 ∈ 2𝑜) → (𝑀‘(𝑎𝑀𝑏)) = (𝑀‘⟨𝑎, (1𝑜𝑏)⟩))
5 df-ov 6613 . . . . . 6 (𝑎𝑀(1𝑜𝑏)) = (𝑀‘⟨𝑎, (1𝑜𝑏)⟩)
64, 5syl6eqr 2673 . . . . 5 ((𝑎𝐼𝑏 ∈ 2𝑜) → (𝑀‘(𝑎𝑀𝑏)) = (𝑎𝑀(1𝑜𝑏)))
7 2oconcl 7535 . . . . . 6 (𝑏 ∈ 2𝑜 → (1𝑜𝑏) ∈ 2𝑜)
82efgmval 18057 . . . . . 6 ((𝑎𝐼 ∧ (1𝑜𝑏) ∈ 2𝑜) → (𝑎𝑀(1𝑜𝑏)) = ⟨𝑎, (1𝑜 ∖ (1𝑜𝑏))⟩)
97, 8sylan2 491 . . . . 5 ((𝑎𝐼𝑏 ∈ 2𝑜) → (𝑎𝑀(1𝑜𝑏)) = ⟨𝑎, (1𝑜 ∖ (1𝑜𝑏))⟩)
10 1on 7519 . . . . . . . . . . 11 1𝑜 ∈ On
1110onordi 5796 . . . . . . . . . 10 Ord 1𝑜
12 ordtr 5701 . . . . . . . . . 10 (Ord 1𝑜 → Tr 1𝑜)
13 trsucss 5775 . . . . . . . . . 10 (Tr 1𝑜 → (𝑏 ∈ suc 1𝑜𝑏 ⊆ 1𝑜))
1411, 12, 13mp2b 10 . . . . . . . . 9 (𝑏 ∈ suc 1𝑜𝑏 ⊆ 1𝑜)
15 df-2o 7513 . . . . . . . . 9 2𝑜 = suc 1𝑜
1614, 15eleq2s 2716 . . . . . . . 8 (𝑏 ∈ 2𝑜𝑏 ⊆ 1𝑜)
1716adantl 482 . . . . . . 7 ((𝑎𝐼𝑏 ∈ 2𝑜) → 𝑏 ⊆ 1𝑜)
18 dfss4 3841 . . . . . . 7 (𝑏 ⊆ 1𝑜 ↔ (1𝑜 ∖ (1𝑜𝑏)) = 𝑏)
1917, 18sylib 208 . . . . . 6 ((𝑎𝐼𝑏 ∈ 2𝑜) → (1𝑜 ∖ (1𝑜𝑏)) = 𝑏)
2019opeq2d 4382 . . . . 5 ((𝑎𝐼𝑏 ∈ 2𝑜) → ⟨𝑎, (1𝑜 ∖ (1𝑜𝑏))⟩ = ⟨𝑎, 𝑏⟩)
216, 9, 203eqtrd 2659 . . . 4 ((𝑎𝐼𝑏 ∈ 2𝑜) → (𝑀‘(𝑎𝑀𝑏)) = ⟨𝑎, 𝑏⟩)
22 fveq2 6153 . . . . . . 7 (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑀𝐴) = (𝑀‘⟨𝑎, 𝑏⟩))
23 df-ov 6613 . . . . . . 7 (𝑎𝑀𝑏) = (𝑀‘⟨𝑎, 𝑏⟩)
2422, 23syl6eqr 2673 . . . . . 6 (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑀𝐴) = (𝑎𝑀𝑏))
2524fveq2d 6157 . . . . 5 (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑀‘(𝑀𝐴)) = (𝑀‘(𝑎𝑀𝑏)))
26 id 22 . . . . 5 (𝐴 = ⟨𝑎, 𝑏⟩ → 𝐴 = ⟨𝑎, 𝑏⟩)
2725, 26eqeq12d 2636 . . . 4 (𝐴 = ⟨𝑎, 𝑏⟩ → ((𝑀‘(𝑀𝐴)) = 𝐴 ↔ (𝑀‘(𝑎𝑀𝑏)) = ⟨𝑎, 𝑏⟩))
2821, 27syl5ibrcom 237 . . 3 ((𝑎𝐼𝑏 ∈ 2𝑜) → (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑀‘(𝑀𝐴)) = 𝐴))
2928rexlimivv 3030 . 2 (∃𝑎𝐼𝑏 ∈ 2𝑜 𝐴 = ⟨𝑎, 𝑏⟩ → (𝑀‘(𝑀𝐴)) = 𝐴)
301, 29sylbi 207 1 (𝐴 ∈ (𝐼 × 2𝑜) → (𝑀‘(𝑀𝐴)) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wrex 2908  cdif 3556  wss 3559  cop 4159  Tr wtr 4717   × cxp 5077  Ord word 5686  suc csuc 5689  cfv 5852  (class class class)co 6610  cmpt2 6612  1𝑜c1o 7505  2𝑜c2o 7506
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-ord 5690  df-on 5691  df-suc 5693  df-iota 5815  df-fun 5854  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-1o 7512  df-2o 7513
This theorem is referenced by:  efginvrel1  18073  efgredlemc  18090
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