MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  efgmnvl Structured version   Visualization version   GIF version

Theorem efgmnvl 18515
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 5381 . 2 (𝐴 ∈ (𝐼 × 2o) ↔ ∃𝑎𝐼𝑏 ∈ 2o 𝐴 = ⟨𝑎, 𝑏⟩)
2 efgmval.m . . . . . . . 8 𝑀 = (𝑦𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o𝑧)⟩)
32efgmval 18513 . . . . . . 7 ((𝑎𝐼𝑏 ∈ 2o) → (𝑎𝑀𝑏) = ⟨𝑎, (1o𝑏)⟩)
43fveq2d 6452 . . . . . 6 ((𝑎𝐼𝑏 ∈ 2o) → (𝑀‘(𝑎𝑀𝑏)) = (𝑀‘⟨𝑎, (1o𝑏)⟩))
5 df-ov 6927 . . . . . 6 (𝑎𝑀(1o𝑏)) = (𝑀‘⟨𝑎, (1o𝑏)⟩)
64, 5syl6eqr 2832 . . . . 5 ((𝑎𝐼𝑏 ∈ 2o) → (𝑀‘(𝑎𝑀𝑏)) = (𝑎𝑀(1o𝑏)))
7 2oconcl 7869 . . . . . 6 (𝑏 ∈ 2o → (1o𝑏) ∈ 2o)
82efgmval 18513 . . . . . 6 ((𝑎𝐼 ∧ (1o𝑏) ∈ 2o) → (𝑎𝑀(1o𝑏)) = ⟨𝑎, (1o ∖ (1o𝑏))⟩)
97, 8sylan2 586 . . . . 5 ((𝑎𝐼𝑏 ∈ 2o) → (𝑎𝑀(1o𝑏)) = ⟨𝑎, (1o ∖ (1o𝑏))⟩)
10 1on 7852 . . . . . . . . . . 11 1o ∈ On
1110onordi 6082 . . . . . . . . . 10 Ord 1o
12 ordtr 5992 . . . . . . . . . 10 (Ord 1o → Tr 1o)
13 trsucss 6063 . . . . . . . . . 10 (Tr 1o → (𝑏 ∈ suc 1o𝑏 ⊆ 1o))
1411, 12, 13mp2b 10 . . . . . . . . 9 (𝑏 ∈ suc 1o𝑏 ⊆ 1o)
15 df-2o 7846 . . . . . . . . 9 2o = suc 1o
1614, 15eleq2s 2877 . . . . . . . 8 (𝑏 ∈ 2o𝑏 ⊆ 1o)
1716adantl 475 . . . . . . 7 ((𝑎𝐼𝑏 ∈ 2o) → 𝑏 ⊆ 1o)
18 dfss4 4085 . . . . . . 7 (𝑏 ⊆ 1o ↔ (1o ∖ (1o𝑏)) = 𝑏)
1917, 18sylib 210 . . . . . 6 ((𝑎𝐼𝑏 ∈ 2o) → (1o ∖ (1o𝑏)) = 𝑏)
2019opeq2d 4645 . . . . 5 ((𝑎𝐼𝑏 ∈ 2o) → ⟨𝑎, (1o ∖ (1o𝑏))⟩ = ⟨𝑎, 𝑏⟩)
216, 9, 203eqtrd 2818 . . . 4 ((𝑎𝐼𝑏 ∈ 2o) → (𝑀‘(𝑎𝑀𝑏)) = ⟨𝑎, 𝑏⟩)
22 fveq2 6448 . . . . . . 7 (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑀𝐴) = (𝑀‘⟨𝑎, 𝑏⟩))
23 df-ov 6927 . . . . . . 7 (𝑎𝑀𝑏) = (𝑀‘⟨𝑎, 𝑏⟩)
2422, 23syl6eqr 2832 . . . . . 6 (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑀𝐴) = (𝑎𝑀𝑏))
2524fveq2d 6452 . . . . 5 (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑀‘(𝑀𝐴)) = (𝑀‘(𝑎𝑀𝑏)))
26 id 22 . . . . 5 (𝐴 = ⟨𝑎, 𝑏⟩ → 𝐴 = ⟨𝑎, 𝑏⟩)
2725, 26eqeq12d 2793 . . . 4 (𝐴 = ⟨𝑎, 𝑏⟩ → ((𝑀‘(𝑀𝐴)) = 𝐴 ↔ (𝑀‘(𝑎𝑀𝑏)) = ⟨𝑎, 𝑏⟩))
2821, 27syl5ibrcom 239 . . 3 ((𝑎𝐼𝑏 ∈ 2o) → (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑀‘(𝑀𝐴)) = 𝐴))
2928rexlimivv 3219 . 2 (∃𝑎𝐼𝑏 ∈ 2o 𝐴 = ⟨𝑎, 𝑏⟩ → (𝑀‘(𝑀𝐴)) = 𝐴)
301, 29sylbi 209 1 (𝐴 ∈ (𝐼 × 2o) → (𝑀‘(𝑀𝐴)) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1601  wcel 2107  wrex 3091  cdif 3789  wss 3792  cop 4404  Tr wtr 4989   × cxp 5355  Ord word 5977  suc csuc 5980  cfv 6137  (class class class)co 6924  cmpt2 6926  1oc1o 7838  2oc2o 7839
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pr 5140  ax-un 7228
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4674  df-br 4889  df-opab 4951  df-tr 4990  df-id 5263  df-eprel 5268  df-po 5276  df-so 5277  df-fr 5316  df-we 5318  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-ord 5981  df-on 5982  df-suc 5984  df-iota 6101  df-fun 6139  df-fv 6145  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-1o 7845  df-2o 7846
This theorem is referenced by:  efginvrel1  18529  efgredlemc  18547
  Copyright terms: Public domain W3C validator