Theorem eqgfval 19288

Description: Value of the subgroup left coset equivalence relation. (Contributed by Mario Carneiro, 15-Jan-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
eqgfval	⊢ ((𝐺 ∈ 𝑉 ∧ 𝑆 ⊆ 𝑋) → 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁‘𝑥) + 𝑦) ∈ 𝑆)})

Distinct variable groups:   𝑥,𝑦,𝐺   𝑥,𝑁,𝑦   𝑥,𝑆,𝑦   𝑥, + ,𝑦   𝑥,𝑋,𝑦

Allowed substitution hints:   𝑅(𝑥, 𝑦)   𝑉(𝑥, 𝑦)

Proof of Theorem eqgfval

Dummy variables 𝑔 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3478	. 2 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V)
2		eqgval.x	. . . 4 ⊢ 𝑋 = (Base‘𝐺)
3	2	fvexi 6899	. . 3 ⊢ 𝑋 ∈ V
4	3	ssex 5293	. 2 ⊢ (𝑆 ⊆ 𝑋 → 𝑆 ∈ V)
5		eqgval.r	. . 3 ⊢ 𝑅 = (𝐺 ~QG 𝑆)
6		simpl 488	. . . . . . . . 9 ⊢ ((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) → 𝑔 = 𝐺)
7	6	fveq2d 6889	. . . . . . . 8 ⊢ ((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) → (Base‘𝑔) = (Base‘𝐺))
8	7, 2	eqtr4di 2818	. . . . . . 7 ⊢ ((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) → (Base‘𝑔) = 𝑋)
9	8	sseq2d 3970	. . . . . 6 ⊢ ((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) → ({𝑥, 𝑦} ⊆ (Base‘𝑔) ↔ {𝑥, 𝑦} ⊆ 𝑋))
10	6	fveq2d 6889	. . . . . . . . 9 ⊢ ((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) → (+g‘𝑔) = (+g‘𝐺))
11		eqgval.p	. . . . . . . . 9 ⊢ + = (+g‘𝐺)
12	10, 11	eqtr4di 2818	. . . . . . . 8 ⊢ ((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) → (+g‘𝑔) = + )
13	6	fveq2d 6889	. . . . . . . . . 10 ⊢ ((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) → (invg‘𝑔) = (invg‘𝐺))
14		eqgval.n	. . . . . . . . . 10 ⊢ 𝑁 = (invg‘𝐺)
15	13, 14	eqtr4di 2818	. . . . . . . . 9 ⊢ ((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) → (invg‘𝑔) = 𝑁)
16	15	fveq1d 6887	. . . . . . . 8 ⊢ ((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) → ((invg‘𝑔)‘𝑥) = (𝑁‘𝑥))
17		eqidd 2766	. . . . . . . 8 ⊢ ((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) → 𝑦 = 𝑦)
18	12, 16, 17	oveq123d 7440	. . . . . . 7 ⊢ ((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) → (((invg‘𝑔)‘𝑥)(+g‘𝑔)𝑦) = ((𝑁‘𝑥) + 𝑦))
19		simpr 490	. . . . . . 7 ⊢ ((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) → 𝑠 = 𝑆)
20	18, 19	eleq12d 2859	. . . . . 6 ⊢ ((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) → ((((invg‘𝑔)‘𝑥)(+g‘𝑔)𝑦) ∈ 𝑠 ↔ ((𝑁‘𝑥) + 𝑦) ∈ 𝑆))
21	9, 20	anbi12d 644	. . . . 5 ⊢ ((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) → (({𝑥, 𝑦} ⊆ (Base‘𝑔) ∧ (((invg‘𝑔)‘𝑥)(+g‘𝑔)𝑦) ∈ 𝑠) ↔ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁‘𝑥) + 𝑦) ∈ 𝑆)))
22	21	opabbidv 5179	. . . 4 ⊢ ((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) → {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝑔) ∧ (((invg‘𝑔)‘𝑥)(+g‘𝑔)𝑦) ∈ 𝑠)} = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁‘𝑥) + 𝑦) ∈ 𝑆)})
23		df-eqg 19235	. . . 4 ⊢ ~QG = (𝑔 ∈ V, 𝑠 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝑔) ∧ (((invg‘𝑔)‘𝑥)(+g‘𝑔)𝑦) ∈ 𝑠)})
24	3, 3	xpex 7758	. . . . 5 ⊢ (𝑋 × 𝑋) ∈ V
25		vex 3461	. . . . . . . . 9 ⊢ 𝑥 ∈ V
26		vex 3461	. . . . . . . . 9 ⊢ 𝑦 ∈ V
27	25, 26	prss 4788	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ↔ {𝑥, 𝑦} ⊆ 𝑋)
28	27	biranri 511	. . . . . . 7 ⊢ (({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁‘𝑥) + 𝑦) ∈ 𝑆) → (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋))
29	28	ssopab2i 5537	. . . . . 6 ⊢ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁‘𝑥) + 𝑦) ∈ 𝑆)} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)}
30		df-xp 5669	. . . . . 6 ⊢ (𝑋 × 𝑋) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)}
31	29, 30	sseqtrri 3987	. . . . 5 ⊢ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁‘𝑥) + 𝑦) ∈ 𝑆)} ⊆ (𝑋 × 𝑋)
32	24, 31	ssexi 5295	. . . 4 ⊢ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁‘𝑥) + 𝑦) ∈ 𝑆)} ∈ V
33	22, 23, 32	ovmpoa 7574	. . 3 ⊢ ((𝐺 ∈ V ∧ 𝑆 ∈ V) → (𝐺 ~QG 𝑆) = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁‘𝑥) + 𝑦) ∈ 𝑆)})
34	5, 33	eqtrid 2812	. 2 ⊢ ((𝐺 ∈ V ∧ 𝑆 ∈ V) → 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁‘𝑥) + 𝑦) ∈ 𝑆)})
35	1, 4, 34	syl2an 608	1 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑆 ⊆ 𝑋) → 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁‘𝑥) + 𝑦) ∈ 𝑆)})

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  Vcvv 3457   ⊆ wss 3906  {cpr 4593  {copab 5175   × cxp 5661  ‘cfv 6540  (class class class)co 7419  Basecbs 17291  +_gcplusg 17332  inv_gcminusg 19045   ~_QG cqg 19232

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-iota 6496  df-fun 6542  df-fv 6548  df-ov 7422  df-oprab 7423  df-mpo 7424  df-eqg 19235

This theorem is used by:  eqgval  19289  eqg0subg  19311  quslsm  33778  oppreqg  33829