Theorem eqgex 13113

Description: The left coset equivalence relation exists. (Contributed by Jim Kingdon, 25-Apr-2025.)

Assertion

Ref	Expression
eqgex	⊢ ((𝐺 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝐺 ~QG 𝑆) ∈ V)

Proof of Theorem eqgex

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

Step	Hyp	Ref	Expression
1		elex 2760	. . . 4 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V)
2	1	adantr 276	. . 3 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → 𝐺 ∈ V)
3		elex 2760	. . . 4 ⊢ (𝑆 ∈ 𝑊 → 𝑆 ∈ V)
4	3	adantl 277	. . 3 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → 𝑆 ∈ V)
5		vex 2752	. . . . . . 7 ⊢ 𝑥 ∈ V
6		vex 2752	. . . . . . 7 ⊢ 𝑦 ∈ V
7	5, 6	prss 3760	. . . . . 6 ⊢ ((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) ↔ {𝑥, 𝑦} ⊆ (Base‘𝐺))
8	7	anbi1i 458	. . . . 5 ⊢ (((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦) ∈ 𝑆) ↔ ({𝑥, 𝑦} ⊆ (Base‘𝐺) ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦) ∈ 𝑆))
9	8	opabbii 4082	. . . 4 ⊢ {⟨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)}
10		basfn 12534	. . . . . . 7 ⊢ Base Fn V
11		funfvex 5544	. . . . . . . 8 ⊢ ((Fun Base ∧ 𝐺 ∈ dom Base) → (Base‘𝐺) ∈ V)
12	11	funfni 5328	. . . . . . 7 ⊢ ((Base Fn V ∧ 𝐺 ∈ V) → (Base‘𝐺) ∈ V)
13	10, 2, 12	sylancr 414	. . . . . 6 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (Base‘𝐺) ∈ V)
14		xpexg 4752	. . . . . 6 ⊢ (((Base‘𝐺) ∈ V ∧ (Base‘𝐺) ∈ V) → ((Base‘𝐺) × (Base‘𝐺)) ∈ V)
15	13, 13, 14	syl2anc 411	. . . . 5 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → ((Base‘𝐺) × (Base‘𝐺)) ∈ V)
16		opabssxp 4712	. . . . . 6 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦) ∈ 𝑆)} ⊆ ((Base‘𝐺) × (Base‘𝐺))
17	16	a1i 9	. . . . 5 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦) ∈ 𝑆)} ⊆ ((Base‘𝐺) × (Base‘𝐺)))
18	15, 17	ssexd 4155	. . . 4 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦) ∈ 𝑆)} ∈ V)
19	9, 18	eqeltrrid 2275	. . 3 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝐺) ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦) ∈ 𝑆)} ∈ V)
20		fveq2 5527	. . . . . . 7 ⊢ (𝑟 = 𝐺 → (Base‘𝑟) = (Base‘𝐺))
21	20	sseq2d 3197	. . . . . 6 ⊢ (𝑟 = 𝐺 → ({𝑥, 𝑦} ⊆ (Base‘𝑟) ↔ {𝑥, 𝑦} ⊆ (Base‘𝐺)))
22		fveq2 5527	. . . . . . . 8 ⊢ (𝑟 = 𝐺 → (+g‘𝑟) = (+g‘𝐺))
23		fveq2 5527	. . . . . . . . 9 ⊢ (𝑟 = 𝐺 → (invg‘𝑟) = (invg‘𝐺))
24	23	fveq1d 5529	. . . . . . . 8 ⊢ (𝑟 = 𝐺 → ((invg‘𝑟)‘𝑥) = ((invg‘𝐺)‘𝑥))
25		eqidd 2188	. . . . . . . 8 ⊢ (𝑟 = 𝐺 → 𝑦 = 𝑦)
26	22, 24, 25	oveq123d 5909	. . . . . . 7 ⊢ (𝑟 = 𝐺 → (((invg‘𝑟)‘𝑥)(+g‘𝑟)𝑦) = (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦))
27	26	eleq1d 2256	. . . . . 6 ⊢ (𝑟 = 𝐺 → ((((invg‘𝑟)‘𝑥)(+g‘𝑟)𝑦) ∈ 𝑖 ↔ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦) ∈ 𝑖))
28	21, 27	anbi12d 473	. . . . 5 ⊢ (𝑟 = 𝐺 → (({𝑥, 𝑦} ⊆ (Base‘𝑟) ∧ (((invg‘𝑟)‘𝑥)(+g‘𝑟)𝑦) ∈ 𝑖) ↔ ({𝑥, 𝑦} ⊆ (Base‘𝐺) ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦) ∈ 𝑖)))
29	28	opabbidv 4081	. . . 4 ⊢ (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)})
30		eleq2 2251	. . . . . 6 ⊢ (𝑖 = 𝑆 → ((((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦) ∈ 𝑖 ↔ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦) ∈ 𝑆))
31	30	anbi2d 464	. . . . 5 ⊢ (𝑖 = 𝑆 → (({𝑥, 𝑦} ⊆ (Base‘𝐺) ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦) ∈ 𝑖) ↔ ({𝑥, 𝑦} ⊆ (Base‘𝐺) ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦) ∈ 𝑆)))
32	31	opabbidv 4081	. . . 4 ⊢ (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)})
33		df-eqg 13064	. . . 4 ⊢ ~QG = (𝑟 ∈ V, 𝑖 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝑟) ∧ (((invg‘𝑟)‘𝑥)(+g‘𝑟)𝑦) ∈ 𝑖)})
34	29, 32, 33	ovmpog 6023	. . 3 ⊢ ((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)})
35	2, 4, 19, 34	syl3anc 1248	. 2 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝐺 ~QG 𝑆) = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝐺) ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦) ∈ 𝑆)})
36	35, 19	eqeltrd 2264	1 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝐺 ~QG 𝑆) ∈ V)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1363   ∈ wcel 2158  Vcvv 2749   ⊆ wss 3141  {cpr 3605  {copab 4075   × cxp 4636   Fn wfn 5223  ‘cfv 5228  (class class class)co 5888  Basecbs 12476  +_gcplusg 12551  inv_gcminusg 12900   ~_QG cqg 13061

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7916  ax-resscn 7917  ax-1re 7919  ax-addrcl 7922

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-v 2751  df-sbc 2975  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-iota 5190  df-fun 5230  df-fn 5231  df-fv 5236  df-ov 5891  df-oprab 5892  df-mpo 5893  df-inn 8934  df-ndx 12479  df-slot 12480  df-base 12482  df-eqg 13064

This theorem is referenced by:  quscrng  13720