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Theorem efgmnvl 18173
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 5166 . 2 (𝐴 ∈ (𝐼 × 2𝑜) ↔ ∃𝑎𝐼𝑏 ∈ 2𝑜 𝐴 = ⟨𝑎, 𝑏⟩)
2 efgmval.m . . . . . . . 8 𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)
32efgmval 18171 . . . . . . 7 ((𝑎𝐼𝑏 ∈ 2𝑜) → (𝑎𝑀𝑏) = ⟨𝑎, (1𝑜𝑏)⟩)
43fveq2d 6233 . . . . . 6 ((𝑎𝐼𝑏 ∈ 2𝑜) → (𝑀‘(𝑎𝑀𝑏)) = (𝑀‘⟨𝑎, (1𝑜𝑏)⟩))
5 df-ov 6693 . . . . . 6 (𝑎𝑀(1𝑜𝑏)) = (𝑀‘⟨𝑎, (1𝑜𝑏)⟩)
64, 5syl6eqr 2703 . . . . 5 ((𝑎𝐼𝑏 ∈ 2𝑜) → (𝑀‘(𝑎𝑀𝑏)) = (𝑎𝑀(1𝑜𝑏)))
7 2oconcl 7628 . . . . . 6 (𝑏 ∈ 2𝑜 → (1𝑜𝑏) ∈ 2𝑜)
82efgmval 18171 . . . . . 6 ((𝑎𝐼 ∧ (1𝑜𝑏) ∈ 2𝑜) → (𝑎𝑀(1𝑜𝑏)) = ⟨𝑎, (1𝑜 ∖ (1𝑜𝑏))⟩)
97, 8sylan2 490 . . . . 5 ((𝑎𝐼𝑏 ∈ 2𝑜) → (𝑎𝑀(1𝑜𝑏)) = ⟨𝑎, (1𝑜 ∖ (1𝑜𝑏))⟩)
10 1on 7612 . . . . . . . . . . 11 1𝑜 ∈ On
1110onordi 5870 . . . . . . . . . 10 Ord 1𝑜
12 ordtr 5775 . . . . . . . . . 10 (Ord 1𝑜 → Tr 1𝑜)
13 trsucss 5849 . . . . . . . . . 10 (Tr 1𝑜 → (𝑏 ∈ suc 1𝑜𝑏 ⊆ 1𝑜))
1411, 12, 13mp2b 10 . . . . . . . . 9 (𝑏 ∈ suc 1𝑜𝑏 ⊆ 1𝑜)
15 df-2o 7606 . . . . . . . . 9 2𝑜 = suc 1𝑜
1614, 15eleq2s 2748 . . . . . . . 8 (𝑏 ∈ 2𝑜𝑏 ⊆ 1𝑜)
1716adantl 481 . . . . . . 7 ((𝑎𝐼𝑏 ∈ 2𝑜) → 𝑏 ⊆ 1𝑜)
18 dfss4 3891 . . . . . . 7 (𝑏 ⊆ 1𝑜 ↔ (1𝑜 ∖ (1𝑜𝑏)) = 𝑏)
1917, 18sylib 208 . . . . . 6 ((𝑎𝐼𝑏 ∈ 2𝑜) → (1𝑜 ∖ (1𝑜𝑏)) = 𝑏)
2019opeq2d 4440 . . . . 5 ((𝑎𝐼𝑏 ∈ 2𝑜) → ⟨𝑎, (1𝑜 ∖ (1𝑜𝑏))⟩ = ⟨𝑎, 𝑏⟩)
216, 9, 203eqtrd 2689 . . . 4 ((𝑎𝐼𝑏 ∈ 2𝑜) → (𝑀‘(𝑎𝑀𝑏)) = ⟨𝑎, 𝑏⟩)
22 fveq2 6229 . . . . . . 7 (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑀𝐴) = (𝑀‘⟨𝑎, 𝑏⟩))
23 df-ov 6693 . . . . . . 7 (𝑎𝑀𝑏) = (𝑀‘⟨𝑎, 𝑏⟩)
2422, 23syl6eqr 2703 . . . . . 6 (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑀𝐴) = (𝑎𝑀𝑏))
2524fveq2d 6233 . . . . 5 (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑀‘(𝑀𝐴)) = (𝑀‘(𝑎𝑀𝑏)))
26 id 22 . . . . 5 (𝐴 = ⟨𝑎, 𝑏⟩ → 𝐴 = ⟨𝑎, 𝑏⟩)
2725, 26eqeq12d 2666 . . . 4 (𝐴 = ⟨𝑎, 𝑏⟩ → ((𝑀‘(𝑀𝐴)) = 𝐴 ↔ (𝑀‘(𝑎𝑀𝑏)) = ⟨𝑎, 𝑏⟩))
2821, 27syl5ibrcom 237 . . 3 ((𝑎𝐼𝑏 ∈ 2𝑜) → (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑀‘(𝑀𝐴)) = 𝐴))
2928rexlimivv 3065 . 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 383   = wceq 1523  wcel 2030  wrex 2942  cdif 3604  wss 3607  cop 4216  Tr wtr 4785   × cxp 5141  Ord word 5760  suc csuc 5763  cfv 5926  (class class class)co 6690  cmpt2 6692  1𝑜c1o 7598  2𝑜c2o 7599
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-ord 5764  df-on 5765  df-suc 5767  df-iota 5889  df-fun 5928  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1o 7605  df-2o 7606
This theorem is referenced by:  efginvrel1  18187  efgredlemc  18204
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