Theorem eqgfval 14002

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 2833	. . . 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 13389	. . . . . . 7 ⊢ Base Fn V
6		funfvex 5707	. . . . . . . 8 ⊢ ((Fun Base ∧ 𝐺 ∈ dom Base) → (Base‘𝐺) ∈ V)
7	6	funfni 5478	. . . . . . 7 ⊢ ((Base Fn V ∧ 𝐺 ∈ V) → (Base‘𝐺) ∈ V)
8	5, 2, 7	sylancr 418	. . . . . 6 ⊢ (𝐺 ∈ 𝑉 → (Base‘𝐺) ∈ V)
9	4, 8	eqeltrid 2325	. . . . 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 4268	. . 3 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑆 ⊆ 𝑋) → 𝑆 ∈ V)
13		xpexg 4884	. . . . 5 ⊢ ((𝑋 ∈ V ∧ 𝑋 ∈ V) → (𝑋 × 𝑋) ∈ V)
14	10, 10, 13	syl2anc 415	. . . 4 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑆 ⊆ 𝑋) → (𝑋 × 𝑋) ∈ V)
15		simpl 109	. . . . . . . 8 ⊢ (({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁‘𝑥) + 𝑦) ∈ 𝑆) → {𝑥, 𝑦} ⊆ 𝑋)
16		vex 2824	. . . . . . . . 9 ⊢ 𝑥 ∈ V
17		vex 2824	. . . . . . . . 9 ⊢ 𝑦 ∈ V
18	16, 17	prss 3866	. . . . . . . 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 4415	. . . . . 6 ⊢ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁‘𝑥) + 𝑦) ∈ 𝑆)} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)}
21		df-xp 4775	. . . . . 6 ⊢ (𝑋 × 𝑋) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)}
22	20, 21	sseqtrri 3283	. . . . 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 4268	. . 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 5694	. . . . . . . 8 ⊢ ((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) → (Base‘𝑔) = (Base‘𝐺))
27	26, 4	eqtr4di 2289	. . . . . . 7 ⊢ ((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) → (Base‘𝑔) = 𝑋)
28	27	sseq2d 3278	. . . . . 6 ⊢ ((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) → ({𝑥, 𝑦} ⊆ (Base‘𝑔) ↔ {𝑥, 𝑦} ⊆ 𝑋))
29	25	fveq2d 5694	. . . . . . . . 9 ⊢ ((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) → (+g‘𝑔) = (+g‘𝐺))
30		eqgval.p	. . . . . . . . 9 ⊢ + = (+g‘𝐺)
31	29, 30	eqtr4di 2289	. . . . . . . 8 ⊢ ((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) → (+g‘𝑔) = + )
32	25	fveq2d 5694	. . . . . . . . . 10 ⊢ ((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) → (invg‘𝑔) = (invg‘𝐺))
33		eqgval.n	. . . . . . . . . 10 ⊢ 𝑁 = (invg‘𝐺)
34	32, 33	eqtr4di 2289	. . . . . . . . 9 ⊢ ((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) → (invg‘𝑔) = 𝑁)
35	34	fveq1d 5692	. . . . . . . 8 ⊢ ((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) → ((invg‘𝑔)‘𝑥) = (𝑁‘𝑥))
36		eqidd 2239	. . . . . . . 8 ⊢ ((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) → 𝑦 = 𝑦)
37	31, 35, 36	oveq123d 6096	. . . . . . 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 2309	. . . . . 6 ⊢ ((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) → ((((invg‘𝑔)‘𝑥)(+g‘𝑔)𝑦) ∈ 𝑠 ↔ ((𝑁‘𝑥) + 𝑦) ∈ 𝑆))
40	28, 39	anbi12d 477	. . . . 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 4192	. . . 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 13952	. . . 4 ⊢ ~QG = (𝑔 ∈ V, 𝑠 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝑔) ∧ (((invg‘𝑔)‘𝑥)(+g‘𝑔)𝑦) ∈ 𝑠)})
43	41, 42	ovmpoga 6208	. . 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 1278	. 2 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑆 ⊆ 𝑋) → (𝐺 ~QG 𝑆) = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁‘𝑥) + 𝑦) ∈ 𝑆)})
45	1, 44	eqtrid 2283	1 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑆 ⊆ 𝑋) → 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁‘𝑥) + 𝑦) ∈ 𝑆)})

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1402   ∈ wcel 2209  Vcvv 2821   ⊆ wss 3220  {cpr 3706  {copab 4186   × cxp 4767   Fn wfn 5367  ‘cfv 5372  (class class class)co 6075  Basecbs 13330  +_gcplusg 13408  inv_gcminusg 13783   ~_QG cqg 13949

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1re 8263  ax-addrcl 8266

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-iota 5332  df-fun 5374  df-fn 5375  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-inn 9284  df-ndx 13333  df-slot 13334  df-base 13336  df-eqg 13952

This theorem is referenced by:  eqgval  14003