Theorem eqgfval 18320

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		elex 3459	. 2 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V)
2		eqgval.x	. . . 4 ⊢ 𝑋 = (Base‘𝐺)
3	2	fvexi 6659	. . 3 ⊢ 𝑋 ∈ V
4	3	ssex 5189	. 2 ⊢ (𝑆 ⊆ 𝑋 → 𝑆 ∈ V)
5		eqgval.r	. . 3 ⊢ 𝑅 = (𝐺 ~QG 𝑆)
6		simpl 486	. . . . . . . . 9 ⊢ ((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) → 𝑔 = 𝐺)
7	6	fveq2d 6649	. . . . . . . 8 ⊢ ((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) → (Base‘𝑔) = (Base‘𝐺))
8	7, 2	eqtr4di 2851	. . . . . . 7 ⊢ ((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) → (Base‘𝑔) = 𝑋)
9	8	sseq2d 3947	. . . . . 6 ⊢ ((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) → ({𝑥, 𝑦} ⊆ (Base‘𝑔) ↔ {𝑥, 𝑦} ⊆ 𝑋))
10	6	fveq2d 6649	. . . . . . . . 9 ⊢ ((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) → (+g‘𝑔) = (+g‘𝐺))
11		eqgval.p	. . . . . . . . 9 ⊢ + = (+g‘𝐺)
12	10, 11	eqtr4di 2851	. . . . . . . 8 ⊢ ((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) → (+g‘𝑔) = + )
13	6	fveq2d 6649	. . . . . . . . . 10 ⊢ ((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) → (invg‘𝑔) = (invg‘𝐺))
14		eqgval.n	. . . . . . . . . 10 ⊢ 𝑁 = (invg‘𝐺)
15	13, 14	eqtr4di 2851	. . . . . . . . 9 ⊢ ((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) → (invg‘𝑔) = 𝑁)
16	15	fveq1d 6647	. . . . . . . 8 ⊢ ((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) → ((invg‘𝑔)‘𝑥) = (𝑁‘𝑥))
17		eqidd 2799	. . . . . . . 8 ⊢ ((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) → 𝑦 = 𝑦)
18	12, 16, 17	oveq123d 7156	. . . . . . 7 ⊢ ((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) → (((invg‘𝑔)‘𝑥)(+g‘𝑔)𝑦) = ((𝑁‘𝑥) + 𝑦))
19		simpr 488	. . . . . . 7 ⊢ ((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) → 𝑠 = 𝑆)
20	18, 19	eleq12d 2884	. . . . . 6 ⊢ ((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) → ((((invg‘𝑔)‘𝑥)(+g‘𝑔)𝑦) ∈ 𝑠 ↔ ((𝑁‘𝑥) + 𝑦) ∈ 𝑆))
21	9, 20	anbi12d 633	. . . . 5 ⊢ ((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) → (({𝑥, 𝑦} ⊆ (Base‘𝑔) ∧ (((invg‘𝑔)‘𝑥)(+g‘𝑔)𝑦) ∈ 𝑠) ↔ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁‘𝑥) + 𝑦) ∈ 𝑆)))
22	21	opabbidv 5096	. . . 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)})
23		df-eqg 18270	. . . 4 ⊢ ~QG = (𝑔 ∈ V, 𝑠 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝑔) ∧ (((invg‘𝑔)‘𝑥)(+g‘𝑔)𝑦) ∈ 𝑠)})
24	3, 3	xpex 7456	. . . . 5 ⊢ (𝑋 × 𝑋) ∈ V
25		simpl 486	. . . . . . . 8 ⊢ (({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁‘𝑥) + 𝑦) ∈ 𝑆) → {𝑥, 𝑦} ⊆ 𝑋)
26		vex 3444	. . . . . . . . 9 ⊢ 𝑥 ∈ V
27		vex 3444	. . . . . . . . 9 ⊢ 𝑦 ∈ V
28	26, 27	prss 4713	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ↔ {𝑥, 𝑦} ⊆ 𝑋)
29	25, 28	sylibr 237	. . . . . . 7 ⊢ (({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁‘𝑥) + 𝑦) ∈ 𝑆) → (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋))
30	29	ssopab2i 5402	. . . . . 6 ⊢ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁‘𝑥) + 𝑦) ∈ 𝑆)} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)}
31		df-xp 5525	. . . . . 6 ⊢ (𝑋 × 𝑋) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)}
32	30, 31	sseqtrri 3952	. . . . 5 ⊢ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁‘𝑥) + 𝑦) ∈ 𝑆)} ⊆ (𝑋 × 𝑋)
33	24, 32	ssexi 5190	. . . 4 ⊢ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁‘𝑥) + 𝑦) ∈ 𝑆)} ∈ V
34	22, 23, 33	ovmpoa 7284	. . 3 ⊢ ((𝐺 ∈ V ∧ 𝑆 ∈ V) → (𝐺 ~QG 𝑆) = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁‘𝑥) + 𝑦) ∈ 𝑆)})
35	5, 34	syl5eq 2845	. 2 ⊢ ((𝐺 ∈ V ∧ 𝑆 ∈ V) → 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁‘𝑥) + 𝑦) ∈ 𝑆)})
36	1, 4, 35	syl2an 598	1 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑆 ⊆ 𝑋) → 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁‘𝑥) + 𝑦) ∈ 𝑆)})

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  Vcvv 3441   ⊆ wss 3881  {cpr 4527  {copab 5092   × cxp 5517  ‘cfv 6324  (class class class)co 7135  Basecbs 16475  +_gcplusg 16557  inv_gcminusg 18096   ~_QG cqg 18267

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6283  df-fun 6326  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-eqg 18270

This theorem is referenced by:  eqgval  18321