Theorem efgmnvl 17950

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 5056	. 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 17948	. . . . . . 7 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2𝑜) → (𝑎𝑀𝑏) = ⟨𝑎, (1𝑜 ∖ 𝑏)⟩)
4	3	fveq2d 6107	. . . . . 6 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2𝑜) → (𝑀‘(𝑎𝑀𝑏)) = (𝑀‘⟨𝑎, (1𝑜 ∖ 𝑏)⟩))
5		df-ov 6552	. . . . . 6 ⊢ (𝑎𝑀(1𝑜 ∖ 𝑏)) = (𝑀‘⟨𝑎, (1𝑜 ∖ 𝑏)⟩)
6	4, 5	syl6eqr 2662	. . . . 5 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2𝑜) → (𝑀‘(𝑎𝑀𝑏)) = (𝑎𝑀(1𝑜 ∖ 𝑏)))
7		2oconcl 7470	. . . . . 6 ⊢ (𝑏 ∈ 2𝑜 → (1𝑜 ∖ 𝑏) ∈ 2𝑜)
8	2	efgmval 17948	. . . . . 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 7454	. . . . . . . . . . 11 ⊢ 1𝑜 ∈ On
11	10	onordi 5749	. . . . . . . . . 10 ⊢ Ord 1𝑜
12		ordtr 5654	. . . . . . . . . 10 ⊢ (Ord 1𝑜 → Tr 1𝑜)
13		trsucss 5728	. . . . . . . . . 10 ⊢ (Tr 1𝑜 → (𝑏 ∈ suc 1𝑜 → 𝑏 ⊆ 1𝑜))
14	11, 12, 13	mp2b 10	. . . . . . . . 9 ⊢ (𝑏 ∈ suc 1𝑜 → 𝑏 ⊆ 1𝑜)
15		df-2o 7448	. . . . . . . . 9 ⊢ 2𝑜 = suc 1𝑜
16	14, 15	eleq2s 2706	. . . . . . . 8 ⊢ (𝑏 ∈ 2𝑜 → 𝑏 ⊆ 1𝑜)
17	16	adantl 481	. . . . . . 7 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2𝑜) → 𝑏 ⊆ 1𝑜)
18		dfss4 3820	. . . . . . 7 ⊢ (𝑏 ⊆ 1𝑜 ↔ (1𝑜 ∖ (1𝑜 ∖ 𝑏)) = 𝑏)
19	17, 18	sylib 207	. . . . . 6 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2𝑜) → (1𝑜 ∖ (1𝑜 ∖ 𝑏)) = 𝑏)
20	19	opeq2d 4347	. . . . 5 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2𝑜) → ⟨𝑎, (1𝑜 ∖ (1𝑜 ∖ 𝑏))⟩ = ⟨𝑎, 𝑏⟩)
21	6, 9, 20	3eqtrd 2648	. . . 4 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2𝑜) → (𝑀‘(𝑎𝑀𝑏)) = ⟨𝑎, 𝑏⟩)
22		fveq2 6103	. . . . . . 7 ⊢ (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑀‘𝐴) = (𝑀‘⟨𝑎, 𝑏⟩))
23		df-ov 6552	. . . . . . 7 ⊢ (𝑎𝑀𝑏) = (𝑀‘⟨𝑎, 𝑏⟩)
24	22, 23	syl6eqr 2662	. . . . . 6 ⊢ (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑀‘𝐴) = (𝑎𝑀𝑏))
25	24	fveq2d 6107	. . . . 5 ⊢ (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑀‘(𝑀‘𝐴)) = (𝑀‘(𝑎𝑀𝑏)))
26		id 22	. . . . 5 ⊢ (𝐴 = ⟨𝑎, 𝑏⟩ → 𝐴 = ⟨𝑎, 𝑏⟩)
27	25, 26	eqeq12d 2625	. . . 4 ⊢ (𝐴 = ⟨𝑎, 𝑏⟩ → ((𝑀‘(𝑀‘𝐴)) = 𝐴 ↔ (𝑀‘(𝑎𝑀𝑏)) = ⟨𝑎, 𝑏⟩))
28	21, 27	syl5ibrcom 236	. . 3 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2𝑜) → (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑀‘(𝑀‘𝐴)) = 𝐴))
29	28	rexlimivv 3018	. 2 ⊢ (∃𝑎 ∈ 𝐼 ∃𝑏 ∈ 2𝑜 𝐴 = ⟨𝑎, 𝑏⟩ → (𝑀‘(𝑀‘𝐴)) = 𝐴)
30	1, 29	sylbi 206	1 ⊢ (𝐴 ∈ (𝐼 × 2𝑜) → (𝑀‘(𝑀‘𝐴)) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃wrex 2897   ∖ cdif 3537   ⊆ wss 3540  ⟨cop 4131  Tr wtr 4680   × cxp 5036  Ord word 5639  suc csuc 5642  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  1_𝑜c1o 7440  2_𝑜c2o 7441

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-ord 5643  df-on 5644  df-suc 5646  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1o 7447  df-2o 7448

This theorem is referenced by:  efginvrel1  17964  efgredlemc  17981