Theorem eqgval 18331

Description: Value of the subgroup left coset equivalence relation. (Contributed by Mario Carneiro, 15-Jan-2015.) (Revised by Mario Carneiro, 14-Jun-2015.)

Hypotheses

Ref	Expression
eqgval.x	⊢ 𝑋 = (Base‘𝐺)
eqgval.n	⊢ 𝑁 = (invg‘𝐺)
eqgval.p	⊢ + = (+g‘𝐺)
eqgval.r	⊢ 𝑅 = (𝐺 ~QG 𝑆)

Assertion

Ref	Expression
eqgval	⊢ ((𝐺 ∈ 𝑉 ∧ 𝑆 ⊆ 𝑋) → (𝐴𝑅𝐵 ↔ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ ((𝑁‘𝐴) + 𝐵) ∈ 𝑆)))

Proof of Theorem eqgval

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqgval.x	. . . 4 ⊢ 𝑋 = (Base‘𝐺)
2		eqgval.n	. . . 4 ⊢ 𝑁 = (invg‘𝐺)
3		eqgval.p	. . . 4 ⊢ + = (+g‘𝐺)
4		eqgval.r	. . . 4 ⊢ 𝑅 = (𝐺 ~QG 𝑆)
5	1, 2, 3, 4	eqgfval 18330	. . 3 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑆 ⊆ 𝑋) → 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁‘𝑥) + 𝑦) ∈ 𝑆)})
6	5	breqd 5079	. 2 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑆 ⊆ 𝑋) → (𝐴𝑅𝐵 ↔ 𝐴{⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁‘𝑥) + 𝑦) ∈ 𝑆)}𝐵))
7		brabv 5455	. . . 4 ⊢ (𝐴{⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁‘𝑥) + 𝑦) ∈ 𝑆)}𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
8	7	adantl 484	. . 3 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑆 ⊆ 𝑋) ∧ 𝐴{⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁‘𝑥) + 𝑦) ∈ 𝑆)}𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
9		simpr1 1190	. . . . 5 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑆 ⊆ 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ ((𝑁‘𝐴) + 𝐵) ∈ 𝑆)) → 𝐴 ∈ 𝑋)
10	9	elexd 3516	. . . 4 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑆 ⊆ 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ ((𝑁‘𝐴) + 𝐵) ∈ 𝑆)) → 𝐴 ∈ V)
11		simpr2 1191	. . . . 5 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑆 ⊆ 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ ((𝑁‘𝐴) + 𝐵) ∈ 𝑆)) → 𝐵 ∈ 𝑋)
12	11	elexd 3516	. . . 4 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑆 ⊆ 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ ((𝑁‘𝐴) + 𝐵) ∈ 𝑆)) → 𝐵 ∈ V)
13	10, 12	jca 514	. . 3 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑆 ⊆ 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ ((𝑁‘𝐴) + 𝐵) ∈ 𝑆)) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
14		vex 3499	. . . . . . . 8 ⊢ 𝑥 ∈ V
15		vex 3499	. . . . . . . 8 ⊢ 𝑦 ∈ V
16	14, 15	prss 4755	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ↔ {𝑥, 𝑦} ⊆ 𝑋)
17		eleq1 2902	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝑋 ↔ 𝐴 ∈ 𝑋))
18		eleq1 2902	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ 𝑋 ↔ 𝐵 ∈ 𝑋))
19	17, 18	bi2anan9 637	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ↔ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)))
20	16, 19	syl5bbr 287	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ({𝑥, 𝑦} ⊆ 𝑋 ↔ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)))
21		fveq2 6672	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑁‘𝑥) = (𝑁‘𝐴))
22		id 22	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → 𝑦 = 𝐵)
23	21, 22	oveqan12d 7177	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((𝑁‘𝑥) + 𝑦) = ((𝑁‘𝐴) + 𝐵))
24	23	eleq1d 2899	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (((𝑁‘𝑥) + 𝑦) ∈ 𝑆 ↔ ((𝑁‘𝐴) + 𝐵) ∈ 𝑆))
25	20, 24	anbi12d 632	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁‘𝑥) + 𝑦) ∈ 𝑆) ↔ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑁‘𝐴) + 𝐵) ∈ 𝑆)))
26		df-3an 1085	. . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ ((𝑁‘𝐴) + 𝐵) ∈ 𝑆) ↔ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑁‘𝐴) + 𝐵) ∈ 𝑆))
27	25, 26	syl6bbr 291	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁‘𝑥) + 𝑦) ∈ 𝑆) ↔ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ ((𝑁‘𝐴) + 𝐵) ∈ 𝑆)))
28		eqid 2823	. . . 4 ⊢ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁‘𝑥) + 𝑦) ∈ 𝑆)} = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁‘𝑥) + 𝑦) ∈ 𝑆)}
29	27, 28	brabga 5423	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴{⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁‘𝑥) + 𝑦) ∈ 𝑆)}𝐵 ↔ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ ((𝑁‘𝐴) + 𝐵) ∈ 𝑆)))
30	8, 13, 29	pm5.21nd 800	. 2 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑆 ⊆ 𝑋) → (𝐴{⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁‘𝑥) + 𝑦) ∈ 𝑆)}𝐵 ↔ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ ((𝑁‘𝐴) + 𝐵) ∈ 𝑆)))
31	6, 30	bitrd 281	1 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑆 ⊆ 𝑋) → (𝐴𝑅𝐵 ↔ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ ((𝑁‘𝐴) + 𝐵) ∈ 𝑆)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  Vcvv 3496   ⊆ wss 3938  {cpr 4571   class class class wbr 5068  {copab 5130  ‘cfv 6357  (class class class)co 7158  Basecbs 16485  +_gcplusg 16567  inv_gcminusg 18106   ~_QG cqg 18277

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-iota 6316  df-fun 6359  df-fv 6365  df-ov 7161  df-oprab 7162  df-mpo 7163  df-eqg 18280

This theorem is referenced by:  eqger  18332  eqglact  18333  eqgid  18334  eqgcpbl  18336  gastacos  18442  orbstafun  18443  sylow2blem1  18747  sylow2blem3  18749  eqgabl  18957  tgpconncompeqg  22722  tgpconncomp  22723  qustgpopn  22730  qusker  30920  eqgvscpbl  30921  qusxpid  30930