Theorem eqgex 13291

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 2771	. . . 4 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V)
2	1	adantr 276	. . 3 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → 𝐺 ∈ V)
3		elex 2771	. . . 4 ⊢ (𝑆 ∈ 𝑊 → 𝑆 ∈ V)
4	3	adantl 277	. . 3 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → 𝑆 ∈ V)
5		vex 2763	. . . . . . 7 ⊢ 𝑥 ∈ V
6		vex 2763	. . . . . . 7 ⊢ 𝑦 ∈ V
7	5, 6	prss 3774	. . . . . 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 4096	. . . 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 12676	. . . . . . 7 ⊢ Base Fn V
11		funfvex 5571	. . . . . . . 8 ⊢ ((Fun Base ∧ 𝐺 ∈ dom Base) → (Base‘𝐺) ∈ V)
12	11	funfni 5354	. . . . . . 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 4773	. . . . . 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 4733	. . . . . 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 4169	. . . 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 2281	. . 3 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝐺) ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦) ∈ 𝑆)} ∈ V)
20		fveq2 5554	. . . . . . 7 ⊢ (𝑟 = 𝐺 → (Base‘𝑟) = (Base‘𝐺))
21	20	sseq2d 3209	. . . . . 6 ⊢ (𝑟 = 𝐺 → ({𝑥, 𝑦} ⊆ (Base‘𝑟) ↔ {𝑥, 𝑦} ⊆ (Base‘𝐺)))
22		fveq2 5554	. . . . . . . 8 ⊢ (𝑟 = 𝐺 → (+g‘𝑟) = (+g‘𝐺))
23		fveq2 5554	. . . . . . . . 9 ⊢ (𝑟 = 𝐺 → (invg‘𝑟) = (invg‘𝐺))
24	23	fveq1d 5556	. . . . . . . 8 ⊢ (𝑟 = 𝐺 → ((invg‘𝑟)‘𝑥) = ((invg‘𝐺)‘𝑥))
25		eqidd 2194	. . . . . . . 8 ⊢ (𝑟 = 𝐺 → 𝑦 = 𝑦)
26	22, 24, 25	oveq123d 5939	. . . . . . 7 ⊢ (𝑟 = 𝐺 → (((invg‘𝑟)‘𝑥)(+g‘𝑟)𝑦) = (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦))
27	26	eleq1d 2262	. . . . . 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 4095	. . . 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 2257	. . . . . 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 4095	. . . 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 13242	. . . 4 ⊢ ~QG = (𝑟 ∈ V, 𝑖 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝑟) ∧ (((invg‘𝑟)‘𝑥)(+g‘𝑟)𝑦) ∈ 𝑖)})
34	29, 32, 33	ovmpog 6053	. . 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 1249	. 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 2270	1 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝐺 ~QG 𝑆) ∈ V)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2164  Vcvv 2760   ⊆ wss 3153  {cpr 3619  {copab 4089   × cxp 4657   Fn wfn 5249  ‘cfv 5254  (class class class)co 5918  Basecbs 12618  +_gcplusg 12695  inv_gcminusg 13073   ~_QG cqg 13239

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1re 7966  ax-addrcl 7969

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-iota 5215  df-fun 5256  df-fn 5257  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-inn 8983  df-ndx 12621  df-slot 12622  df-base 12624  df-eqg 13242

This theorem is referenced by:  quselbasg  13300  quseccl0g  13301  qusghm  13352  quscrng  14029  znval  14124  znle  14125  znbaslemnn  14127  znbas  14132  znzrhval  14135  znzrhfo  14136