Theorem efgmnvl 19584

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 5700	. 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 19582	. . . . . . 7 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o) → (𝑎𝑀𝑏) = ⟨𝑎, (1o ∖ 𝑏)⟩)
4	3	fveq2d 6895	. . . . . 6 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o) → (𝑀‘(𝑎𝑀𝑏)) = (𝑀‘⟨𝑎, (1o ∖ 𝑏)⟩))
5		df-ov 7414	. . . . . 6 ⊢ (𝑎𝑀(1o ∖ 𝑏)) = (𝑀‘⟨𝑎, (1o ∖ 𝑏)⟩)
6	4, 5	eqtr4di 2790	. . . . 5 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o) → (𝑀‘(𝑎𝑀𝑏)) = (𝑎𝑀(1o ∖ 𝑏)))
7		2oconcl 8505	. . . . . 6 ⊢ (𝑏 ∈ 2o → (1o ∖ 𝑏) ∈ 2o)
8	2	efgmval 19582	. . . . . 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 8480	. . . . . . . . . . 11 ⊢ 1o ∈ On
11	10	onordi 6475	. . . . . . . . . 10 ⊢ Ord 1o
12		ordtr 6378	. . . . . . . . . 10 ⊢ (Ord 1o → Tr 1o)
13		trsucss 6452	. . . . . . . . . 10 ⊢ (Tr 1o → (𝑏 ∈ suc 1o → 𝑏 ⊆ 1o))
14	11, 12, 13	mp2b 10	. . . . . . . . 9 ⊢ (𝑏 ∈ suc 1o → 𝑏 ⊆ 1o)
15		df-2o 8469	. . . . . . . . 9 ⊢ 2o = suc 1o
16	14, 15	eleq2s 2851	. . . . . . . 8 ⊢ (𝑏 ∈ 2o → 𝑏 ⊆ 1o)
17	16	adantl 482	. . . . . . 7 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o) → 𝑏 ⊆ 1o)
18		dfss4 4258	. . . . . . 7 ⊢ (𝑏 ⊆ 1o ↔ (1o ∖ (1o ∖ 𝑏)) = 𝑏)
19	17, 18	sylib 217	. . . . . 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 2776	. . . 4 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o) → (𝑀‘(𝑎𝑀𝑏)) = ⟨𝑎, 𝑏⟩)
22		fveq2 6891	. . . . . . 7 ⊢ (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑀‘𝐴) = (𝑀‘⟨𝑎, 𝑏⟩))
23		df-ov 7414	. . . . . . 7 ⊢ (𝑎𝑀𝑏) = (𝑀‘⟨𝑎, 𝑏⟩)
24	22, 23	eqtr4di 2790	. . . . . 6 ⊢ (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑀‘𝐴) = (𝑎𝑀𝑏))
25	24	fveq2d 6895	. . . . 5 ⊢ (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑀‘(𝑀‘𝐴)) = (𝑀‘(𝑎𝑀𝑏)))
26		id 22	. . . . 5 ⊢ (𝐴 = ⟨𝑎, 𝑏⟩ → 𝐴 = ⟨𝑎, 𝑏⟩)
27	25, 26	eqeq12d 2748	. . . 4 ⊢ (𝐴 = ⟨𝑎, 𝑏⟩ → ((𝑀‘(𝑀‘𝐴)) = 𝐴 ↔ (𝑀‘(𝑎𝑀𝑏)) = ⟨𝑎, 𝑏⟩))
28	21, 27	syl5ibrcom 246	. . 3 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o) → (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑀‘(𝑀‘𝐴)) = 𝐴))
29	28	rexlimivv 3199	. 2 ⊢ (∃𝑎 ∈ 𝐼 ∃𝑏 ∈ 2o 𝐴 = ⟨𝑎, 𝑏⟩ → (𝑀‘(𝑀‘𝐴)) = 𝐴)
30	1, 29	sylbi 216	1 ⊢ (𝐴 ∈ (𝐼 × 2o) → (𝑀‘(𝑀‘𝐴)) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∃wrex 3070   ∖ cdif 3945   ⊆ wss 3948  ⟨cop 4634  Tr wtr 5265   × cxp 5674  Ord word 6363  suc csuc 6366  ‘cfv 6543  (class class class)co 7411   ∈ cmpo 7413  1_oc1o 8461  2_oc2o 8462

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-ord 6367  df-on 6368  df-suc 6370  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1o 8468  df-2o 8469

This theorem is referenced by:  efginvrel1  19598  efgredlemc  19615