Theorem efgmnvl 19732

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 5709	. 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 19730	. . . . . . 7 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o) → (𝑎𝑀𝑏) = ⟨𝑎, (1o ∖ 𝑏)⟩)
4	3	fveq2d 6910	. . . . . 6 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o) → (𝑀‘(𝑎𝑀𝑏)) = (𝑀‘⟨𝑎, (1o ∖ 𝑏)⟩))
5		df-ov 7434	. . . . . 6 ⊢ (𝑎𝑀(1o ∖ 𝑏)) = (𝑀‘⟨𝑎, (1o ∖ 𝑏)⟩)
6	4, 5	eqtr4di 2795	. . . . 5 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o) → (𝑀‘(𝑎𝑀𝑏)) = (𝑎𝑀(1o ∖ 𝑏)))
7		2oconcl 8541	. . . . . 6 ⊢ (𝑏 ∈ 2o → (1o ∖ 𝑏) ∈ 2o)
8	2	efgmval 19730	. . . . . 6 ⊢ ((𝑎 ∈ 𝐼 ∧ (1o ∖ 𝑏) ∈ 2o) → (𝑎𝑀(1o ∖ 𝑏)) = ⟨𝑎, (1o ∖ (1o ∖ 𝑏))⟩)
9	7, 8	sylan2 593	. . . . 5 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o) → (𝑎𝑀(1o ∖ 𝑏)) = ⟨𝑎, (1o ∖ (1o ∖ 𝑏))⟩)
10		1on 8518	. . . . . . . . . . 11 ⊢ 1o ∈ On
11	10	onordi 6495	. . . . . . . . . 10 ⊢ Ord 1o
12		ordtr 6398	. . . . . . . . . 10 ⊢ (Ord 1o → Tr 1o)
13		trsucss 6472	. . . . . . . . . 10 ⊢ (Tr 1o → (𝑏 ∈ suc 1o → 𝑏 ⊆ 1o))
14	11, 12, 13	mp2b 10	. . . . . . . . 9 ⊢ (𝑏 ∈ suc 1o → 𝑏 ⊆ 1o)
15		df-2o 8507	. . . . . . . . 9 ⊢ 2o = suc 1o
16	14, 15	eleq2s 2859	. . . . . . . 8 ⊢ (𝑏 ∈ 2o → 𝑏 ⊆ 1o)
17	16	adantl 481	. . . . . . 7 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o) → 𝑏 ⊆ 1o)
18		dfss4 4269	. . . . . . 7 ⊢ (𝑏 ⊆ 1o ↔ (1o ∖ (1o ∖ 𝑏)) = 𝑏)
19	17, 18	sylib 218	. . . . . 6 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o) → (1o ∖ (1o ∖ 𝑏)) = 𝑏)
20	19	opeq2d 4880	. . . . 5 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o) → ⟨𝑎, (1o ∖ (1o ∖ 𝑏))⟩ = ⟨𝑎, 𝑏⟩)
21	6, 9, 20	3eqtrd 2781	. . . 4 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o) → (𝑀‘(𝑎𝑀𝑏)) = ⟨𝑎, 𝑏⟩)
22		fveq2 6906	. . . . . . 7 ⊢ (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑀‘𝐴) = (𝑀‘⟨𝑎, 𝑏⟩))
23		df-ov 7434	. . . . . . 7 ⊢ (𝑎𝑀𝑏) = (𝑀‘⟨𝑎, 𝑏⟩)
24	22, 23	eqtr4di 2795	. . . . . 6 ⊢ (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑀‘𝐴) = (𝑎𝑀𝑏))
25	24	fveq2d 6910	. . . . 5 ⊢ (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑀‘(𝑀‘𝐴)) = (𝑀‘(𝑎𝑀𝑏)))
26		id 22	. . . . 5 ⊢ (𝐴 = ⟨𝑎, 𝑏⟩ → 𝐴 = ⟨𝑎, 𝑏⟩)
27	25, 26	eqeq12d 2753	. . . 4 ⊢ (𝐴 = ⟨𝑎, 𝑏⟩ → ((𝑀‘(𝑀‘𝐴)) = 𝐴 ↔ (𝑀‘(𝑎𝑀𝑏)) = ⟨𝑎, 𝑏⟩))
28	21, 27	syl5ibrcom 247	. . 3 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o) → (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑀‘(𝑀‘𝐴)) = 𝐴))
29	28	rexlimivv 3201	. 2 ⊢ (∃𝑎 ∈ 𝐼 ∃𝑏 ∈ 2o 𝐴 = ⟨𝑎, 𝑏⟩ → (𝑀‘(𝑀‘𝐴)) = 𝐴)
30	1, 29	sylbi 217	1 ⊢ (𝐴 ∈ (𝐼 × 2o) → (𝑀‘(𝑀‘𝐴)) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∃wrex 3070   ∖ cdif 3948   ⊆ wss 3951  ⟨cop 4632  Tr wtr 5259   × cxp 5683  Ord word 6383  suc csuc 6386  ‘cfv 6561  (class class class)co 7431   ∈ cmpo 7433  1_oc1o 8499  2_oc2o 8500

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-ord 6387  df-on 6388  df-suc 6390  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1o 8506  df-2o 8507

This theorem is referenced by:  efginvrel1  19746  efgredlemc  19763