Theorem efgmnvl 18832

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 5543	. 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 18830	. . . . . . 7 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o) → (𝑎𝑀𝑏) = ⟨𝑎, (1o ∖ 𝑏)⟩)
4	3	fveq2d 6649	. . . . . 6 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o) → (𝑀‘(𝑎𝑀𝑏)) = (𝑀‘⟨𝑎, (1o ∖ 𝑏)⟩))
5		df-ov 7138	. . . . . 6 ⊢ (𝑎𝑀(1o ∖ 𝑏)) = (𝑀‘⟨𝑎, (1o ∖ 𝑏)⟩)
6	4, 5	eqtr4di 2851	. . . . 5 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o) → (𝑀‘(𝑎𝑀𝑏)) = (𝑎𝑀(1o ∖ 𝑏)))
7		2oconcl 8111	. . . . . 6 ⊢ (𝑏 ∈ 2o → (1o ∖ 𝑏) ∈ 2o)
8	2	efgmval 18830	. . . . . 6 ⊢ ((𝑎 ∈ 𝐼 ∧ (1o ∖ 𝑏) ∈ 2o) → (𝑎𝑀(1o ∖ 𝑏)) = ⟨𝑎, (1o ∖ (1o ∖ 𝑏))⟩)
9	7, 8	sylan2 595	. . . . 5 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o) → (𝑎𝑀(1o ∖ 𝑏)) = ⟨𝑎, (1o ∖ (1o ∖ 𝑏))⟩)
10		1on 8092	. . . . . . . . . . 11 ⊢ 1o ∈ On
11	10	onordi 6263	. . . . . . . . . 10 ⊢ Ord 1o
12		ordtr 6173	. . . . . . . . . 10 ⊢ (Ord 1o → Tr 1o)
13		trsucss 6244	. . . . . . . . . 10 ⊢ (Tr 1o → (𝑏 ∈ suc 1o → 𝑏 ⊆ 1o))
14	11, 12, 13	mp2b 10	. . . . . . . . 9 ⊢ (𝑏 ∈ suc 1o → 𝑏 ⊆ 1o)
15		df-2o 8086	. . . . . . . . 9 ⊢ 2o = suc 1o
16	14, 15	eleq2s 2908	. . . . . . . 8 ⊢ (𝑏 ∈ 2o → 𝑏 ⊆ 1o)
17	16	adantl 485	. . . . . . 7 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o) → 𝑏 ⊆ 1o)
18		dfss4 4185	. . . . . . 7 ⊢ (𝑏 ⊆ 1o ↔ (1o ∖ (1o ∖ 𝑏)) = 𝑏)
19	17, 18	sylib 221	. . . . . 6 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o) → (1o ∖ (1o ∖ 𝑏)) = 𝑏)
20	19	opeq2d 4772	. . . . 5 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o) → ⟨𝑎, (1o ∖ (1o ∖ 𝑏))⟩ = ⟨𝑎, 𝑏⟩)
21	6, 9, 20	3eqtrd 2837	. . . 4 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o) → (𝑀‘(𝑎𝑀𝑏)) = ⟨𝑎, 𝑏⟩)
22		fveq2 6645	. . . . . . 7 ⊢ (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑀‘𝐴) = (𝑀‘⟨𝑎, 𝑏⟩))
23		df-ov 7138	. . . . . . 7 ⊢ (𝑎𝑀𝑏) = (𝑀‘⟨𝑎, 𝑏⟩)
24	22, 23	eqtr4di 2851	. . . . . 6 ⊢ (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑀‘𝐴) = (𝑎𝑀𝑏))
25	24	fveq2d 6649	. . . . 5 ⊢ (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑀‘(𝑀‘𝐴)) = (𝑀‘(𝑎𝑀𝑏)))
26		id 22	. . . . 5 ⊢ (𝐴 = ⟨𝑎, 𝑏⟩ → 𝐴 = ⟨𝑎, 𝑏⟩)
27	25, 26	eqeq12d 2814	. . . 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 3251	. 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)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∃wrex 3107   ∖ cdif 3878   ⊆ wss 3881  ⟨cop 4531  Tr wtr 5136   × cxp 5517  Ord word 6158  suc csuc 6161  ‘cfv 6324  (class class class)co 7135   ∈ cmpo 7137  1_oc1o 8078  2_oc2o 8079

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-ord 6162  df-on 6163  df-suc 6165  df-iota 6283  df-fun 6326  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-1o 8085  df-2o 8086

This theorem is referenced by:  efginvrel1  18846  efgredlemc  18863