Theorem efgmnvl 18127

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 5132	. 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 18125	. . . . . . 7 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2𝑜) → (𝑎𝑀𝑏) = ⟨𝑎, (1𝑜 ∖ 𝑏)⟩)
4	3	fveq2d 6195	. . . . . 6 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2𝑜) → (𝑀‘(𝑎𝑀𝑏)) = (𝑀‘⟨𝑎, (1𝑜 ∖ 𝑏)⟩))
5		df-ov 6653	. . . . . 6 ⊢ (𝑎𝑀(1𝑜 ∖ 𝑏)) = (𝑀‘⟨𝑎, (1𝑜 ∖ 𝑏)⟩)
6	4, 5	syl6eqr 2674	. . . . 5 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2𝑜) → (𝑀‘(𝑎𝑀𝑏)) = (𝑎𝑀(1𝑜 ∖ 𝑏)))
7		2oconcl 7583	. . . . . 6 ⊢ (𝑏 ∈ 2𝑜 → (1𝑜 ∖ 𝑏) ∈ 2𝑜)
8	2	efgmval 18125	. . . . . 6 ⊢ ((𝑎 ∈ 𝐼 ∧ (1𝑜 ∖ 𝑏) ∈ 2𝑜) → (𝑎𝑀(1𝑜 ∖ 𝑏)) = ⟨𝑎, (1𝑜 ∖ (1𝑜 ∖ 𝑏))⟩)
9	7, 8	sylan2 491	. . . . 5 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2𝑜) → (𝑎𝑀(1𝑜 ∖ 𝑏)) = ⟨𝑎, (1𝑜 ∖ (1𝑜 ∖ 𝑏))⟩)
10		1on 7567	. . . . . . . . . . 11 ⊢ 1𝑜 ∈ On
11	10	onordi 5832	. . . . . . . . . 10 ⊢ Ord 1𝑜
12		ordtr 5737	. . . . . . . . . 10 ⊢ (Ord 1𝑜 → Tr 1𝑜)
13		trsucss 5811	. . . . . . . . . 10 ⊢ (Tr 1𝑜 → (𝑏 ∈ suc 1𝑜 → 𝑏 ⊆ 1𝑜))
14	11, 12, 13	mp2b 10	. . . . . . . . 9 ⊢ (𝑏 ∈ suc 1𝑜 → 𝑏 ⊆ 1𝑜)
15		df-2o 7561	. . . . . . . . 9 ⊢ 2𝑜 = suc 1𝑜
16	14, 15	eleq2s 2719	. . . . . . . 8 ⊢ (𝑏 ∈ 2𝑜 → 𝑏 ⊆ 1𝑜)
17	16	adantl 482	. . . . . . 7 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2𝑜) → 𝑏 ⊆ 1𝑜)
18		dfss4 3858	. . . . . . 7 ⊢ (𝑏 ⊆ 1𝑜 ↔ (1𝑜 ∖ (1𝑜 ∖ 𝑏)) = 𝑏)
19	17, 18	sylib 208	. . . . . 6 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2𝑜) → (1𝑜 ∖ (1𝑜 ∖ 𝑏)) = 𝑏)
20	19	opeq2d 4409	. . . . 5 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2𝑜) → ⟨𝑎, (1𝑜 ∖ (1𝑜 ∖ 𝑏))⟩ = ⟨𝑎, 𝑏⟩)
21	6, 9, 20	3eqtrd 2660	. . . 4 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2𝑜) → (𝑀‘(𝑎𝑀𝑏)) = ⟨𝑎, 𝑏⟩)
22		fveq2 6191	. . . . . . 7 ⊢ (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑀‘𝐴) = (𝑀‘⟨𝑎, 𝑏⟩))
23		df-ov 6653	. . . . . . 7 ⊢ (𝑎𝑀𝑏) = (𝑀‘⟨𝑎, 𝑏⟩)
24	22, 23	syl6eqr 2674	. . . . . 6 ⊢ (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑀‘𝐴) = (𝑎𝑀𝑏))
25	24	fveq2d 6195	. . . . 5 ⊢ (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑀‘(𝑀‘𝐴)) = (𝑀‘(𝑎𝑀𝑏)))
26		id 22	. . . . 5 ⊢ (𝐴 = ⟨𝑎, 𝑏⟩ → 𝐴 = ⟨𝑎, 𝑏⟩)
27	25, 26	eqeq12d 2637	. . . 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 3036	. 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 384   = wceq 1483   ∈ wcel 1990  ∃wrex 2913   ∖ cdif 3571   ⊆ wss 3574  ⟨cop 4183  Tr wtr 4752   × cxp 5112  Ord word 5722  suc csuc 5725  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652  1_𝑜c1o 7553  2_𝑜c2o 7554

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-ord 5726  df-on 5727  df-suc 5729  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1o 7560  df-2o 7561

This theorem is referenced by:  efginvrel1  18141  efgredlemc  18158