Theorem efgmnvl 19929

Description: The inversion function on the generators is an involution. (Contributed by Mario Carneiro, 1-Oct-2015.)

Hypothesis

Ref	Expression
efgmval.m	⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)

Assertion

Ref	Expression
efgmnvl	⊢ (𝐴 ∈ (𝐼 × 2o) → (𝑀‘(𝑀‘𝐴)) = 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.

Step	Hyp	Ref	Expression
1		elxp2 5674	. 2 ⊢ (𝐴 ∈ (𝐼 × 2o) ↔ ∃𝑎 ∈ 𝐼 ∃𝑏 ∈ 2o 𝐴 = ⟨𝑎, 𝑏⟩)
2		efgmval.m	. . . . . . . 8 ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)
3	2	efgmval 19927	. . . . . . 7 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o) → (𝑎𝑀𝑏) = ⟨𝑎, (1o ∖ 𝑏)⟩)
4	3	fveq2d 6889	. . . . . 6 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o) → (𝑀‘(𝑎𝑀𝑏)) = (𝑀‘⟨𝑎, (1o ∖ 𝑏)⟩))
5		df-ov 7423	. . . . . 6 ⊢ (𝑎𝑀(1o ∖ 𝑏)) = (𝑀‘⟨𝑎, (1o ∖ 𝑏)⟩)
6	4, 5	eqtr4di 2813	. . . . 5 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o) → (𝑀‘(𝑎𝑀𝑏)) = (𝑎𝑀(1o ∖ 𝑏)))
7		2oconcl 8512	. . . . . 6 ⊢ (𝑏 ∈ 2o → (1o ∖ 𝑏) ∈ 2o)
8	2	efgmval 19927	. . . . . 6 ⊢ ((𝑎 ∈ 𝐼 ∧ (1o ∖ 𝑏) ∈ 2o) → (𝑎𝑀(1o ∖ 𝑏)) = ⟨𝑎, (1o ∖ (1o ∖ 𝑏))⟩)
9	7, 8	sylan2 605	. . . . 5 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o) → (𝑎𝑀(1o ∖ 𝑏)) = ⟨𝑎, (1o ∖ (1o ∖ 𝑏))⟩)
10		1on 8490	. . . . . . . . . . 11 ⊢ 1o ∈ On
11	10	onordi 6476	. . . . . . . . . 10 ⊢ Ord 1o
12		ordtr 6376	. . . . . . . . . 10 ⊢ (Ord 1o → Tr 1o)
13		trsucss 6453	. . . . . . . . . 10 ⊢ (Tr 1o → (𝑏 ∈ suc 1o → 𝑏 ⊆ 1o))
14	11, 12, 13	mp2b 10	. . . . . . . . 9 ⊢ (𝑏 ∈ suc 1o → 𝑏 ⊆ 1o)
15		df-2o 8478	. . . . . . . . 9 ⊢ 2o = suc 1o
16	14, 15	eleq2s 2878	. . . . . . . 8 ⊢ (𝑏 ∈ 2o → 𝑏 ⊆ 1o)
17	16	adantl 487	. . . . . . 7 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o) → 𝑏 ⊆ 1o)
18		dfss4 4214	. . . . . . 7 ⊢ (𝑏 ⊆ 1o ↔ (1o ∖ (1o ∖ 𝑏)) = 𝑏)
19	17, 18	sylib 221	. . . . . 6 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o) → (1o ∖ (1o ∖ 𝑏)) = 𝑏)
20	19	opeq2d 4839	. . . . 5 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o) → ⟨𝑎, (1o ∖ (1o ∖ 𝑏))⟩ = ⟨𝑎, 𝑏⟩)
21	6, 9, 20	3eqtrd 2799	. . . 4 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o) → (𝑀‘(𝑎𝑀𝑏)) = ⟨𝑎, 𝑏⟩)
22		fveq2 6885	. . . . . . 7 ⊢ (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑀‘𝐴) = (𝑀‘⟨𝑎, 𝑏⟩))
23		df-ov 7423	. . . . . . 7 ⊢ (𝑎𝑀𝑏) = (𝑀‘⟨𝑎, 𝑏⟩)
24	22, 23	eqtr4di 2813	. . . . . 6 ⊢ (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑀‘𝐴) = (𝑎𝑀𝑏))
25	24	fveq2d 6889	. . . . 5 ⊢ (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑀‘(𝑀‘𝐴)) = (𝑀‘(𝑎𝑀𝑏)))
26		id 23	. . . . 5 ⊢ (𝐴 = ⟨𝑎, 𝑏⟩ → 𝐴 = ⟨𝑎, 𝑏⟩)
27	25, 26	eqeq12d 2776	. . . 4 ⊢ (𝐴 = ⟨𝑎, 𝑏⟩ → ((𝑀‘(𝑀‘𝐴)) = 𝐴 ↔ (𝑀‘(𝑎𝑀𝑏)) = ⟨𝑎, 𝑏⟩))
28	21, 27	syl5ibrcom 250	. . 3 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o) → (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑀‘(𝑀‘𝐴)) = 𝐴))
29	28	rexlimivv 3204	. 2 ⊢ (∃𝑎 ∈ 𝐼 ∃𝑏 ∈ 2o 𝐴 = ⟨𝑎, 𝑏⟩ → (𝑀‘(𝑀‘𝐴)) = 𝐴)
30	1, 29	sylbi 220	1 ⊢ (𝐴 ∈ (𝐼 × 2o) → (𝑀‘(𝑀‘𝐴)) = 𝐴)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∃wrex 3086   ⊆ wss 3895   ∖ cdif 4045  ⟨cop 4589  Tr wtr 5211   × cxp 5648  Ord word 6361  suc csuc 6364  ‘cfv 6538  (class class class)co 7420   ∈ cmpo 7422  1_oc1o 8470  2_oc2o 8471

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-ord 6365  df-on 6366  df-suc 6368  df-iota 6494  df-fun 6540  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1o 8477  df-2o 8478

This theorem is used by:  efginvrel1  19943  efgredlemc  19960