Theorem ltsrpr 9455

Description: Ordering of signed reals in terms of positive reals. (Contributed by NM, 20-Feb-1996.) (Revised by Mario Carneiro, 12-Aug-2015.) (New usage is discouraged.)

Assertion

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

Proof of Theorem ltsrpr

Dummy variables x y z w v u f are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		enrex 9445	. 2 |- ~R e. _V
2		enrer 9443	. . 3 |- ~R Er ( P. X. P. )
3		erdm 7322	. . 3 |- ( ~R Er ( P. X. P. ) -> dom ~R = ( P. X. P. ) )
4	2, 3	ax-mp 5	. 2 |- dom ~R = ( P. X. P. )
5		df-nr 9435	. 2 |- R. = ( ( P. X. P. ) /. ~R )
6		ltrelsr 9446	. 2 |- <R C_ ( R. X. R. )
7		ltrelpr 9377	. 2 |- <P C_ ( P. X. P. )
8		0npr 9371	. 2 |- -. (/) e. P.
9		dmplp 9391	. 2 |- dom +P. = ( P. X. P. )
10		df-ltr 9438	. . 3 |- <R = { <. x , y >. | ( ( x e. R. /\ y e. R. ) /\ E. z E. w E. v E. u ( ( x = [ <. z , w >. ] ~R /\ y = [ <. v , u >. ] ~R ) /\ ( z +P. u ) <P ( w +P. v ) ) ) }
11		addclpr 9397	. . . . . . 7 |- ( ( w e. P. /\ v e. P. ) -> ( w +P. v ) e. P. )
12	11	ad2ant2lr 747	. . . . . 6 |- ( ( ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( w +P. v ) e. P. )
13		addclpr 9397	. . . . . . 7 |- ( ( B e. P. /\ C e. P. ) -> ( B +P. C ) e. P. )
14	13	ad2ant2lr 747	. . . . . 6 |- ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) -> ( B +P. C ) e. P. )
15	12, 14	anim12ci 567	. . . . 5 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) /\ ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> ( ( B +P. C ) e. P. /\ ( w +P. v ) e. P. ) )
16	15	an4s 824	. . . 4 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> ( ( B +P. C ) e. P. /\ ( w +P. v ) e. P. ) )
17		enreceq 9444	. . . . . 6 |- ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) -> ( [ <. z , w >. ] ~R = [ <. A , B >. ] ~R <-> ( z +P. B ) = ( w +P. A ) ) )
18		enreceq 9444	. . . . . . 7 |- ( ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) -> ( [ <. v , u >. ] ~R = [ <. C , D >. ] ~R <-> ( v +P. D ) = ( u +P. C ) ) )
19		eqcom 2476	. . . . . . 7 |- ( ( v +P. D ) = ( u +P. C ) <-> ( u +P. C ) = ( v +P. D ) )
20	18, 19	syl6bb 261	. . . . . 6 |- ( ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) -> ( [ <. v , u >. ] ~R = [ <. C , D >. ] ~R <-> ( u +P. C ) = ( v +P. D ) ) )
21	17, 20	bi2anan9 871	. . . . 5 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> ( ( [ <. z , w >. ] ~R = [ <. A , B >. ] ~R /\ [ <. v , u >. ] ~R = [ <. C , D >. ] ~R ) <-> ( ( z +P. B ) = ( w +P. A ) /\ ( u +P. C ) = ( v +P. D ) ) ) )
22		oveq12 6294	. . . . . 6 |- ( ( ( z +P. B ) = ( w +P. A ) /\ ( u +P. C ) = ( v +P. D ) ) -> ( ( z +P. B ) +P. ( u +P. C ) ) = ( ( w +P. A ) +P. ( v +P. D ) ) )
23		addcompr 9400	. . . . . . . . . 10 |- ( u +P. B ) = ( B +P. u )
24	23	oveq1i 6295	. . . . . . . . 9 |- ( ( u +P. B ) +P. C ) = ( ( B +P. u ) +P. C )
25		addasspr 9401	. . . . . . . . 9 |- ( ( u +P. B ) +P. C ) = ( u +P. ( B +P. C ) )
26		addasspr 9401	. . . . . . . . 9 |- ( ( B +P. u ) +P. C ) = ( B +P. ( u +P. C ) )
27	24, 25, 26	3eqtr3i 2504	. . . . . . . 8 |- ( u +P. ( B +P. C ) ) = ( B +P. ( u +P. C ) )
28	27	oveq2i 6296	. . . . . . 7 |- ( z +P. ( u +P. ( B +P. C ) ) ) = ( z +P. ( B +P. ( u +P. C ) ) )
29		addasspr 9401	. . . . . . 7 |- ( ( z +P. u ) +P. ( B +P. C ) ) = ( z +P. ( u +P. ( B +P. C ) ) )
30		addasspr 9401	. . . . . . 7 |- ( ( z +P. B ) +P. ( u +P. C ) ) = ( z +P. ( B +P. ( u +P. C ) ) )
31	28, 29, 30	3eqtr4i 2506	. . . . . 6 |- ( ( z +P. u ) +P. ( B +P. C ) ) = ( ( z +P. B ) +P. ( u +P. C ) )
32		addcompr 9400	. . . . . . . . . 10 |- ( v +P. A ) = ( A +P. v )
33	32	oveq1i 6295	. . . . . . . . 9 |- ( ( v +P. A ) +P. D ) = ( ( A +P. v ) +P. D )
34		addasspr 9401	. . . . . . . . 9 |- ( ( v +P. A ) +P. D ) = ( v +P. ( A +P. D ) )
35		addasspr 9401	. . . . . . . . 9 |- ( ( A +P. v ) +P. D ) = ( A +P. ( v +P. D ) )
36	33, 34, 35	3eqtr3i 2504	. . . . . . . 8 |- ( v +P. ( A +P. D ) ) = ( A +P. ( v +P. D ) )
37	36	oveq2i 6296	. . . . . . 7 |- ( w +P. ( v +P. ( A +P. D ) ) ) = ( w +P. ( A +P. ( v +P. D ) ) )
38		addasspr 9401	. . . . . . 7 |- ( ( w +P. v ) +P. ( A +P. D ) ) = ( w +P. ( v +P. ( A +P. D ) ) )
39		addasspr 9401	. . . . . . 7 |- ( ( w +P. A ) +P. ( v +P. D ) ) = ( w +P. ( A +P. ( v +P. D ) ) )
40	37, 38, 39	3eqtr4i 2506	. . . . . 6 |- ( ( w +P. v ) +P. ( A +P. D ) ) = ( ( w +P. A ) +P. ( v +P. D ) )
41	22, 31, 40	3eqtr4g 2533	. . . . 5 |- ( ( ( z +P. B ) = ( w +P. A ) /\ ( u +P. C ) = ( v +P. D ) ) -> ( ( z +P. u ) +P. ( B +P. C ) ) = ( ( w +P. v ) +P. ( A +P. D ) ) )
42	21, 41	syl6bi 228	. . . 4 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> ( ( [ <. z , w >. ] ~R = [ <. A , B >. ] ~R /\ [ <. v , u >. ] ~R = [ <. C , D >. ] ~R ) -> ( ( z +P. u ) +P. ( B +P. C ) ) = ( ( w +P. v ) +P. ( A +P. D ) ) ) )
43		ovex 6310	. . . . 5 |- ( z +P. u ) e. _V
44		ovex 6310	. . . . 5 |- ( B +P. C ) e. _V
45		ltapr 9424	. . . . 5 |- ( f e. P. -> ( x <P y <-> ( f +P. x ) <P ( f +P. y ) ) )
46		ovex 6310	. . . . 5 |- ( w +P. v ) e. _V
47		addcompr 9400	. . . . 5 |- ( x +P. y ) = ( y +P. x )
48		ovex 6310	. . . . 5 |- ( A +P. D ) e. _V
49	43, 44, 45, 46, 47, 48	caovord3 6473	. . . 4 |- ( ( ( ( B +P. C ) e. P. /\ ( w +P. v ) e. P. ) /\ ( ( z +P. u ) +P. ( B +P. C ) ) = ( ( w +P. v ) +P. ( A +P. D ) ) ) -> ( ( z +P. u ) <P ( w +P. v ) <-> ( A +P. D ) <P ( B +P. C ) ) )
50	16, 42, 49	syl6an 545	. . 3 |- ( ( ( ( z e. P. /\ w e. P. ) /\ ( A e. P. /\ B e. P. ) ) /\ ( ( v e. P. /\ u e. P. ) /\ ( C e. P. /\ D e. P. ) ) ) -> ( ( [ <. z , w >. ] ~R = [ <. A , B >. ] ~R /\ [ <. v , u >. ] ~R = [ <. C , D >. ] ~R ) -> ( ( z +P. u ) <P ( w +P. v ) <-> ( A +P. D ) <P ( B +P. C ) ) ) )
51	1, 2, 5, 10, 50	brecop 7405	. 2 |- ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) -> ( [ <. A , B >. ] ~R <R [ <. C , D >. ] ~R <-> ( A +P. D ) <P ( B +P. C ) ) )
52	1, 4, 5, 6, 7, 8, 9, 51	brecop2 7406	1 |- ( [ <. A , B >. ] ~R <R [ <. C , D >. ] ~R <-> ( A +P. D ) <P ( B +P. C ) )

Syntax hints:   <-> wb 184   /\ wa 369   = wceq 1379   e. wcel 1767   <.cop 4033   class class class wbr 4447   X. cxp 4997   dom cdm 4999  (class class class)co 6285   Er wer 7309   [cec 7310   P.cnp 9238   +P. cpp 9240   <P cltp 9242   ~R cer 9243   R.cnr 9244   <R cltr 9250

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-inf2 8059

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7043  df-rdg 7077  df-1o 7131  df-oadd 7135  df-omul 7136  df-er 7312  df-ec 7314  df-qs 7318  df-ni 9251  df-pli 9252  df-mi 9253  df-lti 9254  df-plpq 9287  df-mpq 9288  df-ltpq 9289  df-enq 9290  df-nq 9291  df-erq 9292  df-plq 9293  df-mq 9294  df-1nq 9295  df-rq 9296  df-ltnq 9297  df-np 9360  df-plp 9362  df-ltp 9364  df-enr 9434  df-nr 9435  df-ltr 9438

This theorem is referenced by:  gt0srpr  9456  ltsosr  9472  0lt1sr  9473  ltasr  9478  mappsrpr  9486  ltpsrpr  9487  map2psrpr  9488