Theorem releqgg 13752

Description: The left coset equivalence relation is a relation. (Contributed by Mario Carneiro, 14-Jun-2015.)

Hypothesis

Ref	Expression
releqg.r	⊢ 𝑅 = (𝐺 ~QG 𝑆)

Assertion

Ref	Expression
releqgg	⊢ ((𝐺 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → Rel 𝑅)

Proof of Theorem releqgg

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

Step	Hyp	Ref	Expression
1		relopab 4847	. 2 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝐺) ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦) ∈ 𝑆)}
2		releqg.r	. . . 4 ⊢ 𝑅 = (𝐺 ~QG 𝑆)
3		elex 2811	. . . . . 6 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V)
4	3	adantr 276	. . . . 5 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → 𝐺 ∈ V)
5		elex 2811	. . . . . 6 ⊢ (𝑆 ∈ 𝑊 → 𝑆 ∈ V)
6	5	adantl 277	. . . . 5 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → 𝑆 ∈ V)
7		vex 2802	. . . . . . . . 9 ⊢ 𝑥 ∈ V
8		vex 2802	. . . . . . . . 9 ⊢ 𝑦 ∈ V
9	7, 8	prss 3823	. . . . . . . 8 ⊢ ((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) ↔ {𝑥, 𝑦} ⊆ (Base‘𝐺))
10	9	anbi1i 458	. . . . . . 7 ⊢ (((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦) ∈ 𝑆) ↔ ({𝑥, 𝑦} ⊆ (Base‘𝐺) ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦) ∈ 𝑆))
11	10	opabbii 4150	. . . . . 6 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦) ∈ 𝑆)} = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝐺) ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦) ∈ 𝑆)}
12		basfn 13086	. . . . . . . . 9 ⊢ Base Fn V
13		funfvex 5643	. . . . . . . . . 10 ⊢ ((Fun Base ∧ 𝐺 ∈ dom Base) → (Base‘𝐺) ∈ V)
14	13	funfni 5422	. . . . . . . . 9 ⊢ ((Base Fn V ∧ 𝐺 ∈ V) → (Base‘𝐺) ∈ V)
15	12, 4, 14	sylancr 414	. . . . . . . 8 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (Base‘𝐺) ∈ V)
16		xpexg 4832	. . . . . . . 8 ⊢ (((Base‘𝐺) ∈ V ∧ (Base‘𝐺) ∈ V) → ((Base‘𝐺) × (Base‘𝐺)) ∈ V)
17	15, 15, 16	syl2anc 411	. . . . . . 7 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → ((Base‘𝐺) × (Base‘𝐺)) ∈ V)
18		opabssxp 4792	. . . . . . . 8 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦) ∈ 𝑆)} ⊆ ((Base‘𝐺) × (Base‘𝐺))
19	18	a1i 9	. . . . . . 7 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦) ∈ 𝑆)} ⊆ ((Base‘𝐺) × (Base‘𝐺)))
20	17, 19	ssexd 4223	. . . . . 6 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦) ∈ 𝑆)} ∈ V)
21	11, 20	eqeltrrid 2317	. . . . 5 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝐺) ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦) ∈ 𝑆)} ∈ V)
22		fveq2 5626	. . . . . . . . 9 ⊢ (𝑟 = 𝐺 → (Base‘𝑟) = (Base‘𝐺))
23	22	sseq2d 3254	. . . . . . . 8 ⊢ (𝑟 = 𝐺 → ({𝑥, 𝑦} ⊆ (Base‘𝑟) ↔ {𝑥, 𝑦} ⊆ (Base‘𝐺)))
24		fveq2 5626	. . . . . . . . . 10 ⊢ (𝑟 = 𝐺 → (+g‘𝑟) = (+g‘𝐺))
25		fveq2 5626	. . . . . . . . . . 11 ⊢ (𝑟 = 𝐺 → (invg‘𝑟) = (invg‘𝐺))
26	25	fveq1d 5628	. . . . . . . . . 10 ⊢ (𝑟 = 𝐺 → ((invg‘𝑟)‘𝑥) = ((invg‘𝐺)‘𝑥))
27		eqidd 2230	. . . . . . . . . 10 ⊢ (𝑟 = 𝐺 → 𝑦 = 𝑦)
28	24, 26, 27	oveq123d 6021	. . . . . . . . 9 ⊢ (𝑟 = 𝐺 → (((invg‘𝑟)‘𝑥)(+g‘𝑟)𝑦) = (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦))
29	28	eleq1d 2298	. . . . . . . 8 ⊢ (𝑟 = 𝐺 → ((((invg‘𝑟)‘𝑥)(+g‘𝑟)𝑦) ∈ 𝑖 ↔ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦) ∈ 𝑖))
30	23, 29	anbi12d 473	. . . . . . 7 ⊢ (𝑟 = 𝐺 → (({𝑥, 𝑦} ⊆ (Base‘𝑟) ∧ (((invg‘𝑟)‘𝑥)(+g‘𝑟)𝑦) ∈ 𝑖) ↔ ({𝑥, 𝑦} ⊆ (Base‘𝐺) ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦) ∈ 𝑖)))
31	30	opabbidv 4149	. . . . . 6 ⊢ (𝑟 = 𝐺 → {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝑟) ∧ (((invg‘𝑟)‘𝑥)(+g‘𝑟)𝑦) ∈ 𝑖)} = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝐺) ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦) ∈ 𝑖)})
32		eleq2 2293	. . . . . . . 8 ⊢ (𝑖 = 𝑆 → ((((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦) ∈ 𝑖 ↔ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦) ∈ 𝑆))
33	32	anbi2d 464	. . . . . . 7 ⊢ (𝑖 = 𝑆 → (({𝑥, 𝑦} ⊆ (Base‘𝐺) ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦) ∈ 𝑖) ↔ ({𝑥, 𝑦} ⊆ (Base‘𝐺) ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦) ∈ 𝑆)))
34	33	opabbidv 4149	. . . . . 6 ⊢ (𝑖 = 𝑆 → {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝐺) ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦) ∈ 𝑖)} = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝐺) ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦) ∈ 𝑆)})
35		df-eqg 13704	. . . . . 6 ⊢ ~QG = (𝑟 ∈ V, 𝑖 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝑟) ∧ (((invg‘𝑟)‘𝑥)(+g‘𝑟)𝑦) ∈ 𝑖)})
36	31, 34, 35	ovmpog 6138	. . . . 5 ⊢ ((𝐺 ∈ V ∧ 𝑆 ∈ V ∧ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝐺) ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦) ∈ 𝑆)} ∈ V) → (𝐺 ~QG 𝑆) = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝐺) ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦) ∈ 𝑆)})
37	4, 6, 21, 36	syl3anc 1271	. . . 4 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝐺 ~QG 𝑆) = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝐺) ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦) ∈ 𝑆)})
38	2, 37	eqtrid 2274	. . 3 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝐺) ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦) ∈ 𝑆)})
39	38	releqd 4802	. 2 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (Rel 𝑅 ↔ Rel {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝐺) ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦) ∈ 𝑆)}))
40	1, 39	mpbiri 168	1 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → Rel 𝑅)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1395   ∈ wcel 2200  Vcvv 2799   ⊆ wss 3197  {cpr 3667  {copab 4143   × cxp 4716  Rel wrel 4723   Fn wfn 5312  ‘cfv 5317  (class class class)co 6000  Basecbs 13027  +_gcplusg 13105  inv_gcminusg 13529   ~_QG cqg 13701

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-cnex 8086  ax-resscn 8087  ax-1re 8089  ax-addrcl 8092

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-iota 5277  df-fun 5319  df-fn 5320  df-fv 5325  df-ov 6003  df-oprab 6004  df-mpo 6005  df-inn 9107  df-ndx 13030  df-slot 13031  df-base 13033  df-eqg 13704

This theorem is referenced by:  eqger  13756  eqgid  13758