Theorem eqgfval 13832

Description: Value of the subgroup left coset equivalence relation. (Contributed by Mario Carneiro, 15-Jan-2015.)

Hypotheses

Ref	Expression
eqgval.x	⊢ 𝑋 = (Base‘𝐺)
eqgval.n	⊢ 𝑁 = (invg‘𝐺)
eqgval.p	⊢ + = (+g‘𝐺)
eqgval.r	⊢ 𝑅 = (𝐺 ~QG 𝑆)

Assertion

Ref	Expression
eqgfval	⊢ ((𝐺 ∈ 𝑉 ∧ 𝑆 ⊆ 𝑋) → 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁‘𝑥) + 𝑦) ∈ 𝑆)})

Distinct variable groups:   𝑥,𝑦,𝐺   𝑥,𝑁,𝑦   𝑥,𝑆,𝑦   𝑥, + ,𝑦   𝑥,𝑋,𝑦

Allowed substitution hints:   𝑅(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem eqgfval

Dummy variables 𝑔 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqgval.r	. 2 ⊢ 𝑅 = (𝐺 ~QG 𝑆)
2		elex 2813	. . . 4 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V)
3	2	adantr 276	. . 3 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑆 ⊆ 𝑋) → 𝐺 ∈ V)
4		eqgval.x	. . . . . 6 ⊢ 𝑋 = (Base‘𝐺)
5		basfn 13164	. . . . . . 7 ⊢ Base Fn V
6		funfvex 5659	. . . . . . . 8 ⊢ ((Fun Base ∧ 𝐺 ∈ dom Base) → (Base‘𝐺) ∈ V)
7	6	funfni 5434	. . . . . . 7 ⊢ ((Base Fn V ∧ 𝐺 ∈ V) → (Base‘𝐺) ∈ V)
8	5, 2, 7	sylancr 414	. . . . . 6 ⊢ (𝐺 ∈ 𝑉 → (Base‘𝐺) ∈ V)
9	4, 8	eqeltrid 2317	. . . . 5 ⊢ (𝐺 ∈ 𝑉 → 𝑋 ∈ V)
10	9	adantr 276	. . . 4 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑆 ⊆ 𝑋) → 𝑋 ∈ V)
11		simpr 110	. . . 4 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ 𝑋)
12	10, 11	ssexd 4230	. . 3 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑆 ⊆ 𝑋) → 𝑆 ∈ V)
13		xpexg 4842	. . . . 5 ⊢ ((𝑋 ∈ V ∧ 𝑋 ∈ V) → (𝑋 × 𝑋) ∈ V)
14	10, 10, 13	syl2anc 411	. . . 4 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑆 ⊆ 𝑋) → (𝑋 × 𝑋) ∈ V)
15		simpl 109	. . . . . . . 8 ⊢ (({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁‘𝑥) + 𝑦) ∈ 𝑆) → {𝑥, 𝑦} ⊆ 𝑋)
16		vex 2804	. . . . . . . . 9 ⊢ 𝑥 ∈ V
17		vex 2804	. . . . . . . . 9 ⊢ 𝑦 ∈ V
18	16, 17	prss 3830	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ↔ {𝑥, 𝑦} ⊆ 𝑋)
19	15, 18	sylibr 134	. . . . . . 7 ⊢ (({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁‘𝑥) + 𝑦) ∈ 𝑆) → (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋))
20	19	ssopab2i 4374	. . . . . 6 ⊢ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁‘𝑥) + 𝑦) ∈ 𝑆)} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)}
21		df-xp 4733	. . . . . 6 ⊢ (𝑋 × 𝑋) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)}
22	20, 21	sseqtrri 3261	. . . . 5 ⊢ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁‘𝑥) + 𝑦) ∈ 𝑆)} ⊆ (𝑋 × 𝑋)
23	22	a1i 9	. . . 4 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑆 ⊆ 𝑋) → {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁‘𝑥) + 𝑦) ∈ 𝑆)} ⊆ (𝑋 × 𝑋))
24	14, 23	ssexd 4230	. . 3 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑆 ⊆ 𝑋) → {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁‘𝑥) + 𝑦) ∈ 𝑆)} ∈ V)
25		simpl 109	. . . . . . . . 9 ⊢ ((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) → 𝑔 = 𝐺)
26	25	fveq2d 5646	. . . . . . . 8 ⊢ ((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) → (Base‘𝑔) = (Base‘𝐺))
27	26, 4	eqtr4di 2281	. . . . . . 7 ⊢ ((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) → (Base‘𝑔) = 𝑋)
28	27	sseq2d 3256	. . . . . 6 ⊢ ((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) → ({𝑥, 𝑦} ⊆ (Base‘𝑔) ↔ {𝑥, 𝑦} ⊆ 𝑋))
29	25	fveq2d 5646	. . . . . . . . 9 ⊢ ((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) → (+g‘𝑔) = (+g‘𝐺))
30		eqgval.p	. . . . . . . . 9 ⊢ + = (+g‘𝐺)
31	29, 30	eqtr4di 2281	. . . . . . . 8 ⊢ ((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) → (+g‘𝑔) = + )
32	25	fveq2d 5646	. . . . . . . . . 10 ⊢ ((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) → (invg‘𝑔) = (invg‘𝐺))
33		eqgval.n	. . . . . . . . . 10 ⊢ 𝑁 = (invg‘𝐺)
34	32, 33	eqtr4di 2281	. . . . . . . . 9 ⊢ ((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) → (invg‘𝑔) = 𝑁)
35	34	fveq1d 5644	. . . . . . . 8 ⊢ ((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) → ((invg‘𝑔)‘𝑥) = (𝑁‘𝑥))
36		eqidd 2231	. . . . . . . 8 ⊢ ((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) → 𝑦 = 𝑦)
37	31, 35, 36	oveq123d 6044	. . . . . . 7 ⊢ ((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) → (((invg‘𝑔)‘𝑥)(+g‘𝑔)𝑦) = ((𝑁‘𝑥) + 𝑦))
38		simpr 110	. . . . . . 7 ⊢ ((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) → 𝑠 = 𝑆)
39	37, 38	eleq12d 2301	. . . . . 6 ⊢ ((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) → ((((invg‘𝑔)‘𝑥)(+g‘𝑔)𝑦) ∈ 𝑠 ↔ ((𝑁‘𝑥) + 𝑦) ∈ 𝑆))
40	28, 39	anbi12d 473	. . . . 5 ⊢ ((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) → (({𝑥, 𝑦} ⊆ (Base‘𝑔) ∧ (((invg‘𝑔)‘𝑥)(+g‘𝑔)𝑦) ∈ 𝑠) ↔ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁‘𝑥) + 𝑦) ∈ 𝑆)))
41	40	opabbidv 4156	. . . 4 ⊢ ((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) → {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝑔) ∧ (((invg‘𝑔)‘𝑥)(+g‘𝑔)𝑦) ∈ 𝑠)} = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁‘𝑥) + 𝑦) ∈ 𝑆)})
42		df-eqg 13782	. . . 4 ⊢ ~QG = (𝑔 ∈ V, 𝑠 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝑔) ∧ (((invg‘𝑔)‘𝑥)(+g‘𝑔)𝑦) ∈ 𝑠)})
43	41, 42	ovmpoga 6156	. . 3 ⊢ ((𝐺 ∈ V ∧ 𝑆 ∈ V ∧ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁‘𝑥) + 𝑦) ∈ 𝑆)} ∈ V) → (𝐺 ~QG 𝑆) = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁‘𝑥) + 𝑦) ∈ 𝑆)})
44	3, 12, 24, 43	syl3anc 1273	. 2 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑆 ⊆ 𝑋) → (𝐺 ~QG 𝑆) = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁‘𝑥) + 𝑦) ∈ 𝑆)})
45	1, 44	eqtrid 2275	1 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑆 ⊆ 𝑋) → 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁‘𝑥) + 𝑦) ∈ 𝑆)})

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1397   ∈ wcel 2201  Vcvv 2801   ⊆ wss 3199  {cpr 3671  {copab 4150   × cxp 4725   Fn wfn 5323  ‘cfv 5328  (class class class)co 6023  Basecbs 13105  +_gcplusg 13183  inv_gcminusg 13607   ~_QG cqg 13779

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-sep 4208  ax-pow 4266  ax-pr 4301  ax-un 4532  ax-setind 4637  ax-cnex 8128  ax-resscn 8129  ax-1re 8131  ax-addrcl 8134

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-ral 2514  df-rex 2515  df-v 2803  df-sbc 3031  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-int 3930  df-br 4090  df-opab 4152  df-mpt 4153  df-id 4392  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-res 4739  df-iota 5288  df-fun 5330  df-fn 5331  df-fv 5336  df-ov 6026  df-oprab 6027  df-mpo 6028  df-inn 9149  df-ndx 13108  df-slot 13109  df-base 13111  df-eqg 13782

This theorem is referenced by:  eqgval  13833