Theorem eqgval 19246

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 19245	. . 3 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑆 ⊆ 𝑋) → 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁‘𝑥) + 𝑦) ∈ 𝑆)})
6	5	breqd 5121	. 2 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑆 ⊆ 𝑋) → (𝐴𝑅𝐵 ↔ 𝐴{⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁‘𝑥) + 𝑦) ∈ 𝑆)}𝐵))
7		brabv 5553	. . . 4 ⊢ (𝐴{⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁‘𝑥) + 𝑦) ∈ 𝑆)}𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
8	7	adantl 486	. . 3 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑆 ⊆ 𝑋) ∧ 𝐴{⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁‘𝑥) + 𝑦) ∈ 𝑆)}𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
9		simpr1 1213	. . . . 5 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑆 ⊆ 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ ((𝑁‘𝐴) + 𝐵) ∈ 𝑆)) → 𝐴 ∈ 𝑋)
10	9	elexd 3478	. . . 4 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑆 ⊆ 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ ((𝑁‘𝐴) + 𝐵) ∈ 𝑆)) → 𝐴 ∈ V)
11		simpr2 1214	. . . . 5 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑆 ⊆ 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ ((𝑁‘𝐴) + 𝐵) ∈ 𝑆)) → 𝐵 ∈ 𝑋)
12	11	elexd 3478	. . . 4 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑆 ⊆ 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ ((𝑁‘𝐴) + 𝐵) ∈ 𝑆)) → 𝐵 ∈ V)
13	10, 12	jca 520	. . 3 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑆 ⊆ 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ ((𝑁‘𝐴) + 𝐵) ∈ 𝑆)) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
14		vex 3459	. . . . . . . 8 ⊢ 𝑥 ∈ V
15		vex 3459	. . . . . . . 8 ⊢ 𝑦 ∈ V
16	14, 15	prss 4787	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ↔ {𝑥, 𝑦} ⊆ 𝑋)
17		eleq1 2851	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝑋 ↔ 𝐴 ∈ 𝑋))
18		eleq1 2851	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ 𝑋 ↔ 𝐵 ∈ 𝑋))
19	17, 18	bi2anan9 649	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ↔ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)))
20	16, 19	bitr3id 288	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ({𝑥, 𝑦} ⊆ 𝑋 ↔ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)))
21		fveq2 6883	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑁‘𝑥) = (𝑁‘𝐴))
22		id 23	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → 𝑦 = 𝐵)
23	21, 22	oveqan12d 7431	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((𝑁‘𝑥) + 𝑦) = ((𝑁‘𝐴) + 𝐵))
24	23	eleq1d 2848	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (((𝑁‘𝑥) + 𝑦) ∈ 𝑆 ↔ ((𝑁‘𝐴) + 𝐵) ∈ 𝑆))
25	20, 24	anbi12d 643	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁‘𝑥) + 𝑦) ∈ 𝑆) ↔ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑁‘𝐴) + 𝐵) ∈ 𝑆)))
26		df-3an 1105	. . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ ((𝑁‘𝐴) + 𝐵) ∈ 𝑆) ↔ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑁‘𝐴) + 𝐵) ∈ 𝑆))
27	25, 26	bitr4di 292	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁‘𝑥) + 𝑦) ∈ 𝑆) ↔ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ ((𝑁‘𝐴) + 𝐵) ∈ 𝑆)))
28		eqid 2763	. . . 4 ⊢ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁‘𝑥) + 𝑦) ∈ 𝑆)} = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁‘𝑥) + 𝑦) ∈ 𝑆)}
29	27, 28	brabga 5520	. . 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 813	. 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 282	1 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑆 ⊆ 𝑋) → (𝐴𝑅𝐵 ↔ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ ((𝑁‘𝐴) + 𝐵) ∈ 𝑆)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  Vcvv 3455   ⊆ wss 3906  {cpr 4592   class class class wbr 5110  {copab 5174  ‘cfv 6538  (class class class)co 7412  Basecbs 17270  +_gcplusg 17311  inv_gcminusg 19002   ~_QG cqg 19189

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3746  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-iota 6494  df-fun 6540  df-fv 6546  df-ov 7415  df-oprab 7416  df-mpo 7417  df-eqg 19192

This theorem is referenced by:  eqger  19247  eqglact  19248  eqgid  19249  eqgcpbl  19251  qusxpid  19252  ghmqusnsglem1  19351  ghmquskerlem1  19354  gastacos  19381  orbstafun  19382  sylow2blem1  19691  sylow2blem3  19693  eqgabl  19905  qsnzr  21464  tgpconncompeqg  24250  tgpconncomp  24251  qustgpopn  24258  qusker  33650  eqgvscpbl  33651  nsgqusf1olem3  33705  qsdrngilem  33757  qsdrnglem2  33759