Theorem eqgval 19143

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 19142	. . 3 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑆 ⊆ 𝑋) → 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁‘𝑥) + 𝑦) ∈ 𝑆)})
6	5	breqd 5097	. 2 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑆 ⊆ 𝑋) → (𝐴𝑅𝐵 ↔ 𝐴{⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁‘𝑥) + 𝑦) ∈ 𝑆)}𝐵))
7		brabv 5514	. . . 4 ⊢ (𝐴{⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁‘𝑥) + 𝑦) ∈ 𝑆)}𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
8	7	adantl 481	. . 3 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑆 ⊆ 𝑋) ∧ 𝐴{⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁‘𝑥) + 𝑦) ∈ 𝑆)}𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
9		simpr1 1196	. . . . 5 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑆 ⊆ 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ ((𝑁‘𝐴) + 𝐵) ∈ 𝑆)) → 𝐴 ∈ 𝑋)
10	9	elexd 3454	. . . 4 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑆 ⊆ 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ ((𝑁‘𝐴) + 𝐵) ∈ 𝑆)) → 𝐴 ∈ V)
11		simpr2 1197	. . . . 5 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑆 ⊆ 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ ((𝑁‘𝐴) + 𝐵) ∈ 𝑆)) → 𝐵 ∈ 𝑋)
12	11	elexd 3454	. . . 4 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑆 ⊆ 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ ((𝑁‘𝐴) + 𝐵) ∈ 𝑆)) → 𝐵 ∈ V)
13	10, 12	jca 511	. . 3 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑆 ⊆ 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ ((𝑁‘𝐴) + 𝐵) ∈ 𝑆)) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
14		vex 3434	. . . . . . . 8 ⊢ 𝑥 ∈ V
15		vex 3434	. . . . . . . 8 ⊢ 𝑦 ∈ V
16	14, 15	prss 4764	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ↔ {𝑥, 𝑦} ⊆ 𝑋)
17		eleq1 2825	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝑋 ↔ 𝐴 ∈ 𝑋))
18		eleq1 2825	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ 𝑋 ↔ 𝐵 ∈ 𝑋))
19	17, 18	bi2anan9 639	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ↔ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)))
20	16, 19	bitr3id 285	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ({𝑥, 𝑦} ⊆ 𝑋 ↔ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)))
21		fveq2 6834	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑁‘𝑥) = (𝑁‘𝐴))
22		id 22	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → 𝑦 = 𝐵)
23	21, 22	oveqan12d 7379	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((𝑁‘𝑥) + 𝑦) = ((𝑁‘𝐴) + 𝐵))
24	23	eleq1d 2822	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (((𝑁‘𝑥) + 𝑦) ∈ 𝑆 ↔ ((𝑁‘𝐴) + 𝐵) ∈ 𝑆))
25	20, 24	anbi12d 633	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁‘𝑥) + 𝑦) ∈ 𝑆) ↔ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑁‘𝐴) + 𝐵) ∈ 𝑆)))
26		df-3an 1089	. . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ ((𝑁‘𝐴) + 𝐵) ∈ 𝑆) ↔ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑁‘𝐴) + 𝐵) ∈ 𝑆))
27	25, 26	bitr4di 289	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁‘𝑥) + 𝑦) ∈ 𝑆) ↔ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ ((𝑁‘𝐴) + 𝐵) ∈ 𝑆)))
28		eqid 2737	. . . 4 ⊢ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁‘𝑥) + 𝑦) ∈ 𝑆)} = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁‘𝑥) + 𝑦) ∈ 𝑆)}
29	27, 28	brabga 5482	. . 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 802	. 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 279	1 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑆 ⊆ 𝑋) → (𝐴𝑅𝐵 ↔ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ ((𝑁‘𝐴) + 𝐵) ∈ 𝑆)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  Vcvv 3430   ⊆ wss 3890  {cpr 4570   class class class wbr 5086  {copab 5148  ‘cfv 6492  (class class class)co 7360  Basecbs 17170  +_gcplusg 17211  inv_gcminusg 18901   ~_QG cqg 19089

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-iota 6448  df-fun 6494  df-fv 6500  df-ov 7363  df-oprab 7364  df-mpo 7365  df-eqg 19092

This theorem is referenced by:  eqger  19144  eqglact  19145  eqgid  19146  eqgcpbl  19148  ghmqusnsglem1  19246  ghmquskerlem1  19249  gastacos  19276  orbstafun  19277  sylow2blem1  19586  sylow2blem3  19588  eqgabl  19800  tgpconncompeqg  24087  tgpconncomp  24088  qustgpopn  24095  qusker  33424  eqgvscpbl  33425  qusxpid  33438  nsgqusf1olem3  33490  qsnzr  33530  qsdrngilem  33569  qsdrnglem2  33571