Theorem releqgg 13350

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 4792	. 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 2774	. . . . . 6 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V)
4	3	adantr 276	. . . . 5 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → 𝐺 ∈ V)
5		elex 2774	. . . . . 6 ⊢ (𝑆 ∈ 𝑊 → 𝑆 ∈ V)
6	5	adantl 277	. . . . 5 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → 𝑆 ∈ V)
7		vex 2766	. . . . . . . . 9 ⊢ 𝑥 ∈ V
8		vex 2766	. . . . . . . . 9 ⊢ 𝑦 ∈ V
9	7, 8	prss 3778	. . . . . . . 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 4100	. . . . . 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 12736	. . . . . . . . 9 ⊢ Base Fn V
13		funfvex 5575	. . . . . . . . . 10 ⊢ ((Fun Base ∧ 𝐺 ∈ dom Base) → (Base‘𝐺) ∈ V)
14	13	funfni 5358	. . . . . . . . 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 4777	. . . . . . . 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 4737	. . . . . . . 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 4173	. . . . . 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 2284	. . . . 5 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝐺) ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦) ∈ 𝑆)} ∈ V)
22		fveq2 5558	. . . . . . . . 9 ⊢ (𝑟 = 𝐺 → (Base‘𝑟) = (Base‘𝐺))
23	22	sseq2d 3213	. . . . . . . 8 ⊢ (𝑟 = 𝐺 → ({𝑥, 𝑦} ⊆ (Base‘𝑟) ↔ {𝑥, 𝑦} ⊆ (Base‘𝐺)))
24		fveq2 5558	. . . . . . . . . 10 ⊢ (𝑟 = 𝐺 → (+g‘𝑟) = (+g‘𝐺))
25		fveq2 5558	. . . . . . . . . . 11 ⊢ (𝑟 = 𝐺 → (invg‘𝑟) = (invg‘𝐺))
26	25	fveq1d 5560	. . . . . . . . . 10 ⊢ (𝑟 = 𝐺 → ((invg‘𝑟)‘𝑥) = ((invg‘𝐺)‘𝑥))
27		eqidd 2197	. . . . . . . . . 10 ⊢ (𝑟 = 𝐺 → 𝑦 = 𝑦)
28	24, 26, 27	oveq123d 5943	. . . . . . . . 9 ⊢ (𝑟 = 𝐺 → (((invg‘𝑟)‘𝑥)(+g‘𝑟)𝑦) = (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦))
29	28	eleq1d 2265	. . . . . . . 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 4099	. . . . . 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 2260	. . . . . . . 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 4099	. . . . . 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 13302	. . . . . 6 ⊢ ~QG = (𝑟 ∈ V, 𝑖 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝑟) ∧ (((invg‘𝑟)‘𝑥)(+g‘𝑟)𝑦) ∈ 𝑖)})
36	31, 34, 35	ovmpog 6057	. . . . 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 1249	. . . 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 2241	. . 3 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝐺) ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦) ∈ 𝑆)})
39	38	releqd 4747	. 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 1364   ∈ wcel 2167  Vcvv 2763   ⊆ wss 3157  {cpr 3623  {copab 4093   × cxp 4661  Rel wrel 4668   Fn wfn 5253  ‘cfv 5258  (class class class)co 5922  Basecbs 12678  +_gcplusg 12755  inv_gcminusg 13133   ~_QG cqg 13299

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1re 7973  ax-addrcl 7976

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-iota 5219  df-fun 5260  df-fn 5261  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-inn 8991  df-ndx 12681  df-slot 12682  df-base 12684  df-eqg 13302

This theorem is referenced by:  eqger  13354  eqgid  13356