Theorem eqgval 19208

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 19207	. . 3 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑆 ⊆ 𝑋) → 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁‘𝑥) + 𝑦) ∈ 𝑆)})
6	5	breqd 5108	. 2 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑆 ⊆ 𝑋) → (𝐴𝑅𝐵 ↔ 𝐴{⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁‘𝑥) + 𝑦) ∈ 𝑆)}𝐵))
7		brabv 5533	. . . 4 ⊢ (𝐴{⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁‘𝑥) + 𝑦) ∈ 𝑆)}𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
8	7	adantl 485	. . 3 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑆 ⊆ 𝑋) ∧ 𝐴{⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁‘𝑥) + 𝑦) ∈ 𝑆)}𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
9		simpr1 1207	. . . . 5 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑆 ⊆ 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ ((𝑁‘𝐴) + 𝐵) ∈ 𝑆)) → 𝐴 ∈ 𝑋)
10	9	elexd 3476	. . . 4 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑆 ⊆ 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ ((𝑁‘𝐴) + 𝐵) ∈ 𝑆)) → 𝐴 ∈ V)
11		simpr2 1208	. . . . 5 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑆 ⊆ 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ ((𝑁‘𝐴) + 𝐵) ∈ 𝑆)) → 𝐵 ∈ 𝑋)
12	11	elexd 3476	. . . 4 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑆 ⊆ 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ ((𝑁‘𝐴) + 𝐵) ∈ 𝑆)) → 𝐵 ∈ V)
13	10, 12	jca 519	. . 3 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑆 ⊆ 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ ((𝑁‘𝐴) + 𝐵) ∈ 𝑆)) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
14		vex 3457	. . . . . . . 8 ⊢ 𝑥 ∈ V
15		vex 3457	. . . . . . . 8 ⊢ 𝑦 ∈ V
16	14, 15	prss 4775	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ↔ {𝑥, 𝑦} ⊆ 𝑋)
17		eleq1 2849	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝑋 ↔ 𝐴 ∈ 𝑋))
18		eleq1 2849	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ 𝑋 ↔ 𝐵 ∈ 𝑋))
19	17, 18	bi2anan9 647	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ↔ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)))
20	16, 19	bitr3id 287	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ({𝑥, 𝑦} ⊆ 𝑋 ↔ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)))
21		fveq2 6861	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑁‘𝑥) = (𝑁‘𝐴))
22		id 22	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → 𝑦 = 𝐵)
23	21, 22	oveqan12d 7409	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((𝑁‘𝑥) + 𝑦) = ((𝑁‘𝐴) + 𝐵))
24	23	eleq1d 2846	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (((𝑁‘𝑥) + 𝑦) ∈ 𝑆 ↔ ((𝑁‘𝐴) + 𝐵) ∈ 𝑆))
25	20, 24	anbi12d 641	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁‘𝑥) + 𝑦) ∈ 𝑆) ↔ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑁‘𝐴) + 𝐵) ∈ 𝑆)))
26		df-3an 1099	. . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ ((𝑁‘𝐴) + 𝐵) ∈ 𝑆) ↔ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑁‘𝐴) + 𝐵) ∈ 𝑆))
27	25, 26	bitr4di 291	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁‘𝑥) + 𝑦) ∈ 𝑆) ↔ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ ((𝑁‘𝐴) + 𝐵) ∈ 𝑆)))
28		eqid 2761	. . . 4 ⊢ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁‘𝑥) + 𝑦) ∈ 𝑆)} = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁‘𝑥) + 𝑦) ∈ 𝑆)}
29	27, 28	brabga 5501	. . 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 811	. 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 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141  Vcvv 3453   ⊆ wss 3902  {cpr 4581   class class class wbr 5097  {copab 5159  ‘cfv 6515  (class class class)co 7390  Basecbs 17235  +_gcplusg 17276  inv_gcminusg 18966   ~_QG cqg 19154

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7712

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-sbc 3743  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-iota 6471  df-fun 6517  df-fv 6523  df-ov 7393  df-oprab 7394  df-mpo 7395  df-eqg 19157

This theorem is referenced by:  eqger  19209  eqglact  19210  eqgid  19211  eqgcpbl  19213  ghmqusnsglem1  19310  ghmquskerlem1  19313  gastacos  19340  orbstafun  19341  sylow2blem1  19650  sylow2blem3  19652  eqgabl  19864  tgpconncompeqg  24159  tgpconncomp  24160  qustgpopn  24167  qusker  33495  eqgvscpbl  33496  qusxpid  33509  nsgqusf1olem3  33561  qsnzr  33602  qsdrngilem  33642  qsdrnglem2  33644