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 = (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 5166	. 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 18171	. . . . . . 7 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2𝑜) → (𝑎𝑀𝑏) = ⟨𝑎, (1𝑜 ∖ 𝑏)⟩)
4	3	fveq2d 6233	. . . . . 6 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2𝑜) → (𝑀‘(𝑎𝑀𝑏)) = (𝑀‘⟨𝑎, (1𝑜 ∖ 𝑏)⟩))
5		df-ov 6693	. . . . . 6 ⊢ (𝑎𝑀(1𝑜 ∖ 𝑏)) = (𝑀‘⟨𝑎, (1𝑜 ∖ 𝑏)⟩)
6	4, 5	syl6eqr 2703	. . . . 5 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2𝑜) → (𝑀‘(𝑎𝑀𝑏)) = (𝑎𝑀(1𝑜 ∖ 𝑏)))
7		2oconcl 7628	. . . . . 6 ⊢ (𝑏 ∈ 2𝑜 → (1𝑜 ∖ 𝑏) ∈ 2𝑜)
8	2	efgmval 18171	. . . . . 6 ⊢ ((𝑎 ∈ 𝐼 ∧ (1𝑜 ∖ 𝑏) ∈ 2𝑜) → (𝑎𝑀(1𝑜 ∖ 𝑏)) = ⟨𝑎, (1𝑜 ∖ (1𝑜 ∖ 𝑏))⟩)
9	7, 8	sylan2 490	. . . . 5 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2𝑜) → (𝑎𝑀(1𝑜 ∖ 𝑏)) = ⟨𝑎, (1𝑜 ∖ (1𝑜 ∖ 𝑏))⟩)
10		1on 7612	. . . . . . . . . . 11 ⊢ 1𝑜 ∈ On
11	10	onordi 5870	. . . . . . . . . 10 ⊢ Ord 1𝑜
12		ordtr 5775	. . . . . . . . . 10 ⊢ (Ord 1𝑜 → Tr 1𝑜)
13		trsucss 5849	. . . . . . . . . 10 ⊢ (Tr 1𝑜 → (𝑏 ∈ suc 1𝑜 → 𝑏 ⊆ 1𝑜))
14	11, 12, 13	mp2b 10	. . . . . . . . 9 ⊢ (𝑏 ∈ suc 1𝑜 → 𝑏 ⊆ 1𝑜)
15		df-2o 7606	. . . . . . . . 9 ⊢ 2𝑜 = suc 1𝑜
16	14, 15	eleq2s 2748	. . . . . . . 8 ⊢ (𝑏 ∈ 2𝑜 → 𝑏 ⊆ 1𝑜)
17	16	adantl 481	. . . . . . 7 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2𝑜) → 𝑏 ⊆ 1𝑜)
18		dfss4 3891	. . . . . . 7 ⊢ (𝑏 ⊆ 1𝑜 ↔ (1𝑜 ∖ (1𝑜 ∖ 𝑏)) = 𝑏)
19	17, 18	sylib 208	. . . . . 6 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2𝑜) → (1𝑜 ∖ (1𝑜 ∖ 𝑏)) = 𝑏)
20	19	opeq2d 4440	. . . . 5 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2𝑜) → ⟨𝑎, (1𝑜 ∖ (1𝑜 ∖ 𝑏))⟩ = ⟨𝑎, 𝑏⟩)
21	6, 9, 20	3eqtrd 2689	. . . 4 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2𝑜) → (𝑀‘(𝑎𝑀𝑏)) = ⟨𝑎, 𝑏⟩)
22		fveq2 6229	. . . . . . 7 ⊢ (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑀‘𝐴) = (𝑀‘⟨𝑎, 𝑏⟩))
23		df-ov 6693	. . . . . . 7 ⊢ (𝑎𝑀𝑏) = (𝑀‘⟨𝑎, 𝑏⟩)
24	22, 23	syl6eqr 2703	. . . . . 6 ⊢ (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑀‘𝐴) = (𝑎𝑀𝑏))
25	24	fveq2d 6233	. . . . 5 ⊢ (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑀‘(𝑀‘𝐴)) = (𝑀‘(𝑎𝑀𝑏)))
26		id 22	. . . . 5 ⊢ (𝐴 = ⟨𝑎, 𝑏⟩ → 𝐴 = ⟨𝑎, 𝑏⟩)
27	25, 26	eqeq12d 2666	. . . 4 ⊢ (𝐴 = ⟨𝑎, 𝑏⟩ → ((𝑀‘(𝑀‘𝐴)) = 𝐴 ↔ (𝑀‘(𝑎𝑀𝑏)) = ⟨𝑎, 𝑏⟩))
28	21, 27	syl5ibrcom 237	. . 3 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2𝑜) → (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑀‘(𝑀‘𝐴)) = 𝐴))
29	28	rexlimivv 3065	. 2 ⊢ (∃𝑎 ∈ 𝐼 ∃𝑏 ∈ 2𝑜 𝐴 = ⟨𝑎, 𝑏⟩ → (𝑀‘(𝑀‘𝐴)) = 𝐴)
30	1, 29	sylbi 207	1 ⊢ (𝐴 ∈ (𝐼 × 2𝑜) → (𝑀‘(𝑀‘𝐴)) = 𝐴)

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  1_𝑜c1o 7598  2_𝑜c2o 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