Theorem brecop2 7841

Description: Binary relation on a quotient set. Lemma for real number construction. Eliminates antecedent from last hypothesis. (Contributed by NM, 13-Feb-1996.)

Hypotheses

Ref	Expression
brecop2.1	|- .~ e. _V
brecop2.5	|- dom .~ = ( G X. G )
brecop2.6	|- H = ( ( G X. G ) /. .~ )
brecop2.7	|- R C_ ( H X. H )
brecop2.8	|- .<_ C_ ( G X. G )
brecop2.9	|- -. (/) e. G
brecop2.10	|- dom .+ = ( G X. G )
brecop2.11	|- ( ( ( A e. G /\ B e. G ) /\ ( C e. G /\ D e. G ) ) -> ( [ <. A , B >. ] .~ R [ <. C , D >. ] .~ <-> ( A .+ D ) .<_ ( B .+ C ) ) )

Assertion

Ref	Expression
brecop2	|- ( [ <. A , B >. ] .~ R [ <. C , D >. ] .~ <-> ( A .+ D ) .<_ ( B .+ C ) )

Proof of Theorem brecop2

Step	Hyp	Ref	Expression
1		brecop2.7	. . . 4 |- R C_ ( H X. H )
2	1	brel 5168	. . 3 |- ( [ <. A , B >. ] .~ R [ <. C , D >. ] .~ -> ( [ <. A , B >. ] .~ e. H /\ [ <. C , D >. ] .~ e. H ) )
3		brecop2.5	. . . . . . 7 |- dom .~ = ( G X. G )
4		ecelqsdm 7817	. . . . . . 7 |- ( ( dom .~ = ( G X. G ) /\ [ <. A , B >. ] .~ e. ( ( G X. G ) /. .~ ) ) -> <. A , B >. e. ( G X. G ) )
5	3, 4	mpan 706	. . . . . 6 |- ( [ <. A , B >. ] .~ e. ( ( G X. G ) /. .~ ) -> <. A , B >. e. ( G X. G ) )
6		brecop2.6	. . . . . 6 |- H = ( ( G X. G ) /. .~ )
7	5, 6	eleq2s 2719	. . . . 5 |- ( [ <. A , B >. ] .~ e. H -> <. A , B >. e. ( G X. G ) )
8		opelxp 5146	. . . . 5 |- ( <. A , B >. e. ( G X. G ) <-> ( A e. G /\ B e. G ) )
9	7, 8	sylib 208	. . . 4 |- ( [ <. A , B >. ] .~ e. H -> ( A e. G /\ B e. G ) )
10		ecelqsdm 7817	. . . . . . 7 |- ( ( dom .~ = ( G X. G ) /\ [ <. C , D >. ] .~ e. ( ( G X. G ) /. .~ ) ) -> <. C , D >. e. ( G X. G ) )
11	3, 10	mpan 706	. . . . . 6 |- ( [ <. C , D >. ] .~ e. ( ( G X. G ) /. .~ ) -> <. C , D >. e. ( G X. G ) )
12	11, 6	eleq2s 2719	. . . . 5 |- ( [ <. C , D >. ] .~ e. H -> <. C , D >. e. ( G X. G ) )
13		opelxp 5146	. . . . 5 |- ( <. C , D >. e. ( G X. G ) <-> ( C e. G /\ D e. G ) )
14	12, 13	sylib 208	. . . 4 |- ( [ <. C , D >. ] .~ e. H -> ( C e. G /\ D e. G ) )
15	9, 14	anim12i 590	. . 3 |- ( ( [ <. A , B >. ] .~ e. H /\ [ <. C , D >. ] .~ e. H ) -> ( ( A e. G /\ B e. G ) /\ ( C e. G /\ D e. G ) ) )
16	2, 15	syl 17	. 2 |- ( [ <. A , B >. ] .~ R [ <. C , D >. ] .~ -> ( ( A e. G /\ B e. G ) /\ ( C e. G /\ D e. G ) ) )
17		brecop2.8	. . . . 5 |- .<_ C_ ( G X. G )
18	17	brel 5168	. . . 4 |- ( ( A .+ D ) .<_ ( B .+ C ) -> ( ( A .+ D ) e. G /\ ( B .+ C ) e. G ) )
19		brecop2.10	. . . . . 6 |- dom .+ = ( G X. G )
20		brecop2.9	. . . . . 6 |- -. (/) e. G
21	19, 20	ndmovrcl 6820	. . . . 5 |- ( ( A .+ D ) e. G -> ( A e. G /\ D e. G ) )
22	19, 20	ndmovrcl 6820	. . . . 5 |- ( ( B .+ C ) e. G -> ( B e. G /\ C e. G ) )
23	21, 22	anim12i 590	. . . 4 |- ( ( ( A .+ D ) e. G /\ ( B .+ C ) e. G ) -> ( ( A e. G /\ D e. G ) /\ ( B e. G /\ C e. G ) ) )
24	18, 23	syl 17	. . 3 |- ( ( A .+ D ) .<_ ( B .+ C ) -> ( ( A e. G /\ D e. G ) /\ ( B e. G /\ C e. G ) ) )
25		an42 866	. . 3 |- ( ( ( A e. G /\ D e. G ) /\ ( B e. G /\ C e. G ) ) <-> ( ( A e. G /\ B e. G ) /\ ( C e. G /\ D e. G ) ) )
26	24, 25	sylib 208	. 2 |- ( ( A .+ D ) .<_ ( B .+ C ) -> ( ( A e. G /\ B e. G ) /\ ( C e. G /\ D e. G ) ) )
27		brecop2.11	. 2 |- ( ( ( A e. G /\ B e. G ) /\ ( C e. G /\ D e. G ) ) -> ( [ <. A , B >. ] .~ R [ <. C , D >. ] .~ <-> ( A .+ D ) .<_ ( B .+ C ) ) )
28	16, 26, 27	pm5.21nii 368	1 |- ( [ <. A , B >. ] .~ R [ <. C , D >. ] .~ <-> ( A .+ D ) .<_ ( B .+ C ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   (/)c0 3915   <.cop 4183   class class class wbr 4653   X. cxp 5112   dom cdm 5114  (class class class)co 6650   [cec 7740   /.cqs 7741

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fv 5896  df-ov 6653  df-ec 7744  df-qs 7748

This theorem is referenced by:  ltsrpr  9898