Theorem releqgg 13011

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 4752	. 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 2748	. . . . . 6 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V)
4	3	adantr 276	. . . . 5 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → 𝐺 ∈ V)
5		elex 2748	. . . . . 6 ⊢ (𝑆 ∈ 𝑊 → 𝑆 ∈ V)
6	5	adantl 277	. . . . 5 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → 𝑆 ∈ V)
7		vex 2740	. . . . . . . . 9 ⊢ 𝑥 ∈ V
8		vex 2740	. . . . . . . . 9 ⊢ 𝑦 ∈ V
9	7, 8	prss 3748	. . . . . . . 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 4069	. . . . . 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 12512	. . . . . . . . 9 ⊢ Base Fn V
13		funfvex 5531	. . . . . . . . . 10 ⊢ ((Fun Base ∧ 𝐺 ∈ dom Base) → (Base‘𝐺) ∈ V)
14	13	funfni 5315	. . . . . . . . 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 4739	. . . . . . . 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 4699	. . . . . . . 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 4142	. . . . . 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 2265	. . . . 5 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝐺) ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦) ∈ 𝑆)} ∈ V)
22		fveq2 5514	. . . . . . . . 9 ⊢ (𝑟 = 𝐺 → (Base‘𝑟) = (Base‘𝐺))
23	22	sseq2d 3185	. . . . . . . 8 ⊢ (𝑟 = 𝐺 → ({𝑥, 𝑦} ⊆ (Base‘𝑟) ↔ {𝑥, 𝑦} ⊆ (Base‘𝐺)))
24		fveq2 5514	. . . . . . . . . 10 ⊢ (𝑟 = 𝐺 → (+g‘𝑟) = (+g‘𝐺))
25		fveq2 5514	. . . . . . . . . . 11 ⊢ (𝑟 = 𝐺 → (invg‘𝑟) = (invg‘𝐺))
26	25	fveq1d 5516	. . . . . . . . . 10 ⊢ (𝑟 = 𝐺 → ((invg‘𝑟)‘𝑥) = ((invg‘𝐺)‘𝑥))
27		eqidd 2178	. . . . . . . . . 10 ⊢ (𝑟 = 𝐺 → 𝑦 = 𝑦)
28	24, 26, 27	oveq123d 5893	. . . . . . . . 9 ⊢ (𝑟 = 𝐺 → (((invg‘𝑟)‘𝑥)(+g‘𝑟)𝑦) = (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦))
29	28	eleq1d 2246	. . . . . . . 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 4068	. . . . . 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 2241	. . . . . . . 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 4068	. . . . . 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 12963	. . . . . 6 ⊢ ~QG = (𝑟 ∈ V, 𝑖 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝑟) ∧ (((invg‘𝑟)‘𝑥)(+g‘𝑟)𝑦) ∈ 𝑖)})
36	31, 34, 35	ovmpog 6006	. . . . 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 1238	. . . 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 2222	. . 3 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝐺) ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦) ∈ 𝑆)})
39	38	releqd 4709	. 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 1353   ∈ wcel 2148  Vcvv 2737   ⊆ wss 3129  {cpr 3593  {copab 4062   × cxp 4623  Rel wrel 4630   Fn wfn 5210  ‘cfv 5215  (class class class)co 5872  Basecbs 12454  +_gcplusg 12528  inv_gcminusg 12810   ~_QG cqg 12960

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535  ax-cnex 7899  ax-resscn 7900  ax-1re 7902  ax-addrcl 7905

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-iota 5177  df-fun 5217  df-fn 5218  df-fv 5223  df-ov 5875  df-oprab 5876  df-mpo 5877  df-inn 8916  df-ndx 12457  df-slot 12458  df-base 12460  df-eqg 12963

This theorem is referenced by:  eqger  13014  eqgid  13016