Theorem brecop 6889

Description: Binary relation on a quotient set. Lemma for real number construction. (Contributed by NM, 29-Jan-1996.)

Hypotheses

Ref	Expression
brecop.1	|- .~ e. _V
brecop.2	|- .~ Er ( G X. G )
brecop.4	|- H = ( ( G X. G ) /. .~ )
brecop.5	|- .<_ = { <. x , y >. | ( ( x e. H /\ y e. H ) /\ E. z E. w E. v E. u ( ( x = [ <. z , w >. ] .~ /\ y = [ <. v , u >. ] .~ ) /\ ph ) ) }
brecop.6	|- ( ( ( ( z e. G /\ w e. G ) /\ ( A e. G /\ B e. G ) ) /\ ( ( v e. G /\ u e. G ) /\ ( C e. G /\ D e. G ) ) ) -> ( ( [ <. z , w >. ] .~ = [ <. A , B >. ] .~ /\ [ <. v , u >. ] .~ = [ <. C , D >. ] .~ ) -> ( ph <-> ps ) ) )

Assertion

Ref	Expression
brecop	|- ( ( ( A e. G /\ B e. G ) /\ ( C e. G /\ D e. G ) ) -> ( [ <. A , B >. ] .~ .<_ [ <. C , D >. ] .~ <-> ps ) )

Distinct variable groups:   x, y, z, w, v, u, A   x, B, y, z, w, v, u   x, C, y, z, w, v, u   x, D, y, z, w, v, u   x, .~ , y, z, w, v, u   x, H, y   z, G, w, v, u   ph, x, y   ps, z, w, v, u

Allowed substitution hints:   ph( z, w, v, u)   ps( x, y)   G( x, y)   H( z, w, v, u)   .<_ ( x, y, z, w, v, u)

Proof of Theorem brecop

Step	Hyp	Ref	Expression
1		brecop.1	. . . 4 |- .~ e. _V
2		brecop.4	. . . 4 |- H = ( ( G X. G ) /. .~ )
3	1, 2	ecopqsi 6854	. . 3 |- ( ( A e. G /\ B e. G ) -> [ <. A , B >. ] .~ e. H )
4	1, 2	ecopqsi 6854	. . 3 |- ( ( C e. G /\ D e. G ) -> [ <. C , D >. ] .~ e. H )
5		df-br 4126	. . . . 5 |- ( [ <. A , B >. ] .~ .<_ [ <. C , D >. ] .~ <-> <. [ <. A , B >. ] .~ , [ <. C , D >. ] .~ >. e. .<_ )
6		brecop.5	. . . . . 6 |- .<_ = { <. x , y >. | ( ( x e. H /\ y e. H ) /\ E. z E. w E. v E. u ( ( x = [ <. z , w >. ] .~ /\ y = [ <. v , u >. ] .~ ) /\ ph ) ) }
7	6	eleq2i 2305	. . . . 5 |- ( <. [ <. A , B >. ] .~ , [ <. C , D >. ] .~ >. e. .<_ <-> <. [ <. A , B >. ] .~ , [ <. C , D >. ] .~ >. e. { <. x , y >. | ( ( x e. H /\ y e. H ) /\ E. z E. w E. v E. u ( ( x = [ <. z , w >. ] .~ /\ y = [ <. v , u >. ] .~ ) /\ ph ) ) } )
8	5, 7	bitri 184	. . . 4 |- ( [ <. A , B >. ] .~ .<_ [ <. C , D >. ] .~ <-> <. [ <. A , B >. ] .~ , [ <. C , D >. ] .~ >. e. { <. x , y >. | ( ( x e. H /\ y e. H ) /\ E. z E. w E. v E. u ( ( x = [ <. z , w >. ] .~ /\ y = [ <. v , u >. ] .~ ) /\ ph ) ) } )
9		eqeq1 2245	. . . . . . . 8 |- ( x = [ <. A , B >. ] .~ -> ( x = [ <. z , w >. ] .~ <-> [ <. A , B >. ] .~ = [ <. z , w >. ] .~ ) )
10	9	anbi1d 469	. . . . . . 7 |- ( x = [ <. A , B >. ] .~ -> ( ( x = [ <. z , w >. ] .~ /\ y = [ <. v , u >. ] .~ ) <-> ( [ <. A , B >. ] .~ = [ <. z , w >. ] .~ /\ y = [ <. v , u >. ] .~ ) ) )
11	10	anbi1d 469	. . . . . 6 |- ( x = [ <. A , B >. ] .~ -> ( ( ( x = [ <. z , w >. ] .~ /\ y = [ <. v , u >. ] .~ ) /\ ph ) <-> ( ( [ <. A , B >. ] .~ = [ <. z , w >. ] .~ /\ y = [ <. v , u >. ] .~ ) /\ ph ) ) )
12	11	4exbidv 1923	. . . . 5 |- ( x = [ <. A , B >. ] .~ -> ( E. z E. w E. v E. u ( ( x = [ <. z , w >. ] .~ /\ y = [ <. v , u >. ] .~ ) /\ ph ) <-> E. z E. w E. v E. u ( ( [ <. A , B >. ] .~ = [ <. z , w >. ] .~ /\ y = [ <. v , u >. ] .~ ) /\ ph ) ) )
13		eqeq1 2245	. . . . . . . 8 |- ( y = [ <. C , D >. ] .~ -> ( y = [ <. v , u >. ] .~ <-> [ <. C , D >. ] .~ = [ <. v , u >. ] .~ ) )
14	13	anbi2d 468	. . . . . . 7 |- ( y = [ <. C , D >. ] .~ -> ( ( [ <. A , B >. ] .~ = [ <. z , w >. ] .~ /\ y = [ <. v , u >. ] .~ ) <-> ( [ <. A , B >. ] .~ = [ <. z , w >. ] .~ /\ [ <. C , D >. ] .~ = [ <. v , u >. ] .~ ) ) )
15	14	anbi1d 469	. . . . . 6 |- ( y = [ <. C , D >. ] .~ -> ( ( ( [ <. A , B >. ] .~ = [ <. z , w >. ] .~ /\ y = [ <. v , u >. ] .~ ) /\ ph ) <-> ( ( [ <. A , B >. ] .~ = [ <. z , w >. ] .~ /\ [ <. C , D >. ] .~ = [ <. v , u >. ] .~ ) /\ ph ) ) )
16	15	4exbidv 1923	. . . . 5 |- ( y = [ <. C , D >. ] .~ -> ( E. z E. w E. v E. u ( ( [ <. A , B >. ] .~ = [ <. z , w >. ] .~ /\ y = [ <. v , u >. ] .~ ) /\ ph ) <-> E. z E. w E. v E. u ( ( [ <. A , B >. ] .~ = [ <. z , w >. ] .~ /\ [ <. C , D >. ] .~ = [ <. v , u >. ] .~ ) /\ ph ) ) )
17	12, 16	opelopab2 4408	. . . 4 |- ( ( [ <. A , B >. ] .~ e. H /\ [ <. C , D >. ] .~ e. H ) -> ( <. [ <. A , B >. ] .~ , [ <. C , D >. ] .~ >. e. { <. x , y >. | ( ( x e. H /\ y e. H ) /\ E. z E. w E. v E. u ( ( x = [ <. z , w >. ] .~ /\ y = [ <. v , u >. ] .~ ) /\ ph ) ) } <-> E. z E. w E. v E. u ( ( [ <. A , B >. ] .~ = [ <. z , w >. ] .~ /\ [ <. C , D >. ] .~ = [ <. v , u >. ] .~ ) /\ ph ) ) )
18	8, 17	bitrid 192	. . 3 |- ( ( [ <. A , B >. ] .~ e. H /\ [ <. C , D >. ] .~ e. H ) -> ( [ <. A , B >. ] .~ .<_ [ <. C , D >. ] .~ <-> E. z E. w E. v E. u ( ( [ <. A , B >. ] .~ = [ <. z , w >. ] .~ /\ [ <. C , D >. ] .~ = [ <. v , u >. ] .~ ) /\ ph ) ) )
19	3, 4, 18	syl2an 289	. 2 |- ( ( ( A e. G /\ B e. G ) /\ ( C e. G /\ D e. G ) ) -> ( [ <. A , B >. ] .~ .<_ [ <. C , D >. ] .~ <-> E. z E. w E. v E. u ( ( [ <. A , B >. ] .~ = [ <. z , w >. ] .~ /\ [ <. C , D >. ] .~ = [ <. v , u >. ] .~ ) /\ ph ) ) )
20		opeq12 3901	. . . . . 6 |- ( ( z = A /\ w = B ) -> <. z , w >. = <. A , B >. )
21	20	eceq1d 6833	. . . . 5 |- ( ( z = A /\ w = B ) -> [ <. z , w >. ] .~ = [ <. A , B >. ] .~ )
22		opeq12 3901	. . . . . 6 |- ( ( v = C /\ u = D ) -> <. v , u >. = <. C , D >. )
23	22	eceq1d 6833	. . . . 5 |- ( ( v = C /\ u = D ) -> [ <. v , u >. ] .~ = [ <. C , D >. ] .~ )
24	21, 23	anim12i 338	. . . 4 |- ( ( ( z = A /\ w = B ) /\ ( v = C /\ u = D ) ) -> ( [ <. z , w >. ] .~ = [ <. A , B >. ] .~ /\ [ <. v , u >. ] .~ = [ <. C , D >. ] .~ ) )
25		opelxpi 4801	. . . . . . . 8 |- ( ( A e. G /\ B e. G ) -> <. A , B >. e. ( G X. G ) )
26		opelxp 4799	. . . . . . . . 9 |- ( <. z , w >. e. ( G X. G ) <-> ( z e. G /\ w e. G ) )
27		brecop.2	. . . . . . . . . . 11 |- .~ Er ( G X. G )
28	27	a1i 9	. . . . . . . . . 10 |- ( [ <. z , w >. ] .~ = [ <. A , B >. ] .~ -> .~ Er ( G X. G ) )
29		id 19	. . . . . . . . . 10 |- ( [ <. z , w >. ] .~ = [ <. A , B >. ] .~ -> [ <. z , w >. ] .~ = [ <. A , B >. ] .~ )
30	28, 29	ereldm 6842	. . . . . . . . 9 |- ( [ <. z , w >. ] .~ = [ <. A , B >. ] .~ -> ( <. z , w >. e. ( G X. G ) <-> <. A , B >. e. ( G X. G ) ) )
31	26, 30	bitr3id 194	. . . . . . . 8 |- ( [ <. z , w >. ] .~ = [ <. A , B >. ] .~ -> ( ( z e. G /\ w e. G ) <-> <. A , B >. e. ( G X. G ) ) )
32	25, 31	imbitrrid 156	. . . . . . 7 |- ( [ <. z , w >. ] .~ = [ <. A , B >. ] .~ -> ( ( A e. G /\ B e. G ) -> ( z e. G /\ w e. G ) ) )
33		opelxpi 4801	. . . . . . . 8 |- ( ( C e. G /\ D e. G ) -> <. C , D >. e. ( G X. G ) )
34		opelxp 4799	. . . . . . . . 9 |- ( <. v , u >. e. ( G X. G ) <-> ( v e. G /\ u e. G ) )
35	27	a1i 9	. . . . . . . . . 10 |- ( [ <. v , u >. ] .~ = [ <. C , D >. ] .~ -> .~ Er ( G X. G ) )
36		id 19	. . . . . . . . . 10 |- ( [ <. v , u >. ] .~ = [ <. C , D >. ] .~ -> [ <. v , u >. ] .~ = [ <. C , D >. ] .~ )
37	35, 36	ereldm 6842	. . . . . . . . 9 |- ( [ <. v , u >. ] .~ = [ <. C , D >. ] .~ -> ( <. v , u >. e. ( G X. G ) <-> <. C , D >. e. ( G X. G ) ) )
38	34, 37	bitr3id 194	. . . . . . . 8 |- ( [ <. v , u >. ] .~ = [ <. C , D >. ] .~ -> ( ( v e. G /\ u e. G ) <-> <. C , D >. e. ( G X. G ) ) )
39	33, 38	imbitrrid 156	. . . . . . 7 |- ( [ <. v , u >. ] .~ = [ <. C , D >. ] .~ -> ( ( C e. G /\ D e. G ) -> ( v e. G /\ u e. G ) ) )
40	32, 39	im2anan9 606	. . . . . 6 |- ( ( [ <. z , w >. ] .~ = [ <. A , B >. ] .~ /\ [ <. v , u >. ] .~ = [ <. C , D >. ] .~ ) -> ( ( ( A e. G /\ B e. G ) /\ ( C e. G /\ D e. G ) ) -> ( ( z e. G /\ w e. G ) /\ ( v e. G /\ u e. G ) ) ) )
41		brecop.6	. . . . . . . . 9 |- ( ( ( ( z e. G /\ w e. G ) /\ ( A e. G /\ B e. G ) ) /\ ( ( v e. G /\ u e. G ) /\ ( C e. G /\ D e. G ) ) ) -> ( ( [ <. z , w >. ] .~ = [ <. A , B >. ] .~ /\ [ <. v , u >. ] .~ = [ <. C , D >. ] .~ ) -> ( ph <-> ps ) ) )
42	41	an4s 596	. . . . . . . 8 |- ( ( ( ( z e. G /\ w e. G ) /\ ( v e. G /\ u e. G ) ) /\ ( ( A e. G /\ B e. G ) /\ ( C e. G /\ D e. G ) ) ) -> ( ( [ <. z , w >. ] .~ = [ <. A , B >. ] .~ /\ [ <. v , u >. ] .~ = [ <. C , D >. ] .~ ) -> ( ph <-> ps ) ) )
43	42	ex 115	. . . . . . 7 |- ( ( ( z e. G /\ w e. G ) /\ ( v e. G /\ u e. G ) ) -> ( ( ( A e. G /\ B e. G ) /\ ( C e. G /\ D e. G ) ) -> ( ( [ <. z , w >. ] .~ = [ <. A , B >. ] .~ /\ [ <. v , u >. ] .~ = [ <. C , D >. ] .~ ) -> ( ph <-> ps ) ) ) )
44	43	com13 80	. . . . . 6 |- ( ( [ <. z , w >. ] .~ = [ <. A , B >. ] .~ /\ [ <. v , u >. ] .~ = [ <. C , D >. ] .~ ) -> ( ( ( A e. G /\ B e. G ) /\ ( C e. G /\ D e. G ) ) -> ( ( ( z e. G /\ w e. G ) /\ ( v e. G /\ u e. G ) ) -> ( ph <-> ps ) ) ) )
45	40, 44	mpdd 41	. . . . 5 |- ( ( [ <. z , w >. ] .~ = [ <. A , B >. ] .~ /\ [ <. v , u >. ] .~ = [ <. C , D >. ] .~ ) -> ( ( ( A e. G /\ B e. G ) /\ ( C e. G /\ D e. G ) ) -> ( ph <-> ps ) ) )
46	45	pm5.74d 182	. . . 4 |- ( ( [ <. z , w >. ] .~ = [ <. A , B >. ] .~ /\ [ <. v , u >. ] .~ = [ <. C , D >. ] .~ ) -> ( ( ( ( A e. G /\ B e. G ) /\ ( C e. G /\ D e. G ) ) -> ph ) <-> ( ( ( A e. G /\ B e. G ) /\ ( C e. G /\ D e. G ) ) -> ps ) ) )
47	24, 46	cgsex4g 2859	. . 3 |- ( ( ( A e. G /\ B e. G ) /\ ( C e. G /\ D e. G ) ) -> ( E. z E. w E. v E. u ( ( [ <. z , w >. ] .~ = [ <. A , B >. ] .~ /\ [ <. v , u >. ] .~ = [ <. C , D >. ] .~ ) /\ ( ( ( A e. G /\ B e. G ) /\ ( C e. G /\ D e. G ) ) -> ph ) ) <-> ( ( ( A e. G /\ B e. G ) /\ ( C e. G /\ D e. G ) ) -> ps ) ) )
48		eqcom 2240	. . . . . . 7 |- ( [ <. A , B >. ] .~ = [ <. z , w >. ] .~ <-> [ <. z , w >. ] .~ = [ <. A , B >. ] .~ )
49		eqcom 2240	. . . . . . 7 |- ( [ <. C , D >. ] .~ = [ <. v , u >. ] .~ <-> [ <. v , u >. ] .~ = [ <. C , D >. ] .~ )
50	48, 49	anbi12i 464	. . . . . 6 |- ( ( [ <. A , B >. ] .~ = [ <. z , w >. ] .~ /\ [ <. C , D >. ] .~ = [ <. v , u >. ] .~ ) <-> ( [ <. z , w >. ] .~ = [ <. A , B >. ] .~ /\ [ <. v , u >. ] .~ = [ <. C , D >. ] .~ ) )
51	50	a1i 9	. . . . 5 |- ( ( ( A e. G /\ B e. G ) /\ ( C e. G /\ D e. G ) ) -> ( ( [ <. A , B >. ] .~ = [ <. z , w >. ] .~ /\ [ <. C , D >. ] .~ = [ <. v , u >. ] .~ ) <-> ( [ <. z , w >. ] .~ = [ <. A , B >. ] .~ /\ [ <. v , u >. ] .~ = [ <. C , D >. ] .~ ) ) )
52		biimt 241	. . . . 5 |- ( ( ( A e. G /\ B e. G ) /\ ( C e. G /\ D e. G ) ) -> ( ph <-> ( ( ( A e. G /\ B e. G ) /\ ( C e. G /\ D e. G ) ) -> ph ) ) )
53	51, 52	anbi12d 477	. . . 4 |- ( ( ( A e. G /\ B e. G ) /\ ( C e. G /\ D e. G ) ) -> ( ( ( [ <. A , B >. ] .~ = [ <. z , w >. ] .~ /\ [ <. C , D >. ] .~ = [ <. v , u >. ] .~ ) /\ ph ) <-> ( ( [ <. z , w >. ] .~ = [ <. A , B >. ] .~ /\ [ <. v , u >. ] .~ = [ <. C , D >. ] .~ ) /\ ( ( ( A e. G /\ B e. G ) /\ ( C e. G /\ D e. G ) ) -> ph ) ) ) )
54	53	4exbidv 1923	. . 3 |- ( ( ( A e. G /\ B e. G ) /\ ( C e. G /\ D e. G ) ) -> ( E. z E. w E. v E. u ( ( [ <. A , B >. ] .~ = [ <. z , w >. ] .~ /\ [ <. C , D >. ] .~ = [ <. v , u >. ] .~ ) /\ ph ) <-> E. z E. w E. v E. u ( ( [ <. z , w >. ] .~ = [ <. A , B >. ] .~ /\ [ <. v , u >. ] .~ = [ <. C , D >. ] .~ ) /\ ( ( ( A e. G /\ B e. G ) /\ ( C e. G /\ D e. G ) ) -> ph ) ) ) )
55		biimt 241	. . 3 |- ( ( ( A e. G /\ B e. G ) /\ ( C e. G /\ D e. G ) ) -> ( ps <-> ( ( ( A e. G /\ B e. G ) /\ ( C e. G /\ D e. G ) ) -> ps ) ) )
56	47, 54, 55	3bitr4d 220	. 2 |- ( ( ( A e. G /\ B e. G ) /\ ( C e. G /\ D e. G ) ) -> ( E. z E. w E. v E. u ( ( [ <. A , B >. ] .~ = [ <. z , w >. ] .~ /\ [ <. C , D >. ] .~ = [ <. v , u >. ] .~ ) /\ ph ) <-> ps ) )
57	19, 56	bitrd 188	1 |- ( ( ( A e. G /\ B e. G ) /\ ( C e. G /\ D e. G ) ) -> ( [ <. A , B >. ] .~ .<_ [ <. C , D >. ] .~ <-> ps ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   E.wex 1545   e. wcel 2209   _Vcvv 2821   <.cop 3708   class class class wbr 4125   {copab 4186   X. cxp 4767   Er wer 6794   [cec 6795   /.cqs 6796

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-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

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  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-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  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-br 4126  df-opab 4188  df-xp 4775  df-cnv 4777  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-er 6797  df-ec 6799  df-qs 6803

This theorem is referenced by:  ordpipqqs  7731  ltsrprg  8104