Theorem ltsrpr 9898

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 9888	. 2 |- ~R e. _V
2		enrer 9886	. . 3 |- ~R Er ( P. X. P. )
3		erdm 7752	. . 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 9878	. 2 |- R. = ( ( P. X. P. ) /. ~R )
6		ltrelsr 9889	. 2 |- <R C_ ( R. X. R. )
7		ltrelpr 9820	. 2 |- <P C_ ( P. X. P. )
8		0npr 9814	. 2 |- -. (/) e. P.
9		dmplp 9834	. 2 |- dom +P. = ( P. X. P. )
10		df-ltr 9881	. . 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 9840	. . . . . . 7 |- ( ( w e. P. /\ v e. P. ) -> ( w +P. v ) e. P. )
12	11	ad2ant2lr 784	. . . . . 6 |- ( ( ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( w +P. v ) e. P. )
13		addclpr 9840	. . . . . . 7 |- ( ( B e. P. /\ C e. P. ) -> ( B +P. C ) e. P. )
14	13	ad2ant2lr 784	. . . . . 6 |- ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) -> ( B +P. C ) e. P. )
15	12, 14	anim12ci 591	. . . . 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 869	. . . 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 9887	. . . . . 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 9887	. . . . . . 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 2629	. . . . . . 7 |- ( ( v +P. D ) = ( u +P. C ) <-> ( u +P. C ) = ( v +P. D ) )
20	18, 19	syl6bb 276	. . . . . 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 917	. . . . 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 6659	. . . . . 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 9843	. . . . . . . . . 10 |- ( u +P. B ) = ( B +P. u )
24	23	oveq1i 6660	. . . . . . . . 9 |- ( ( u +P. B ) +P. C ) = ( ( B +P. u ) +P. C )
25		addasspr 9844	. . . . . . . . 9 |- ( ( u +P. B ) +P. C ) = ( u +P. ( B +P. C ) )
26		addasspr 9844	. . . . . . . . 9 |- ( ( B +P. u ) +P. C ) = ( B +P. ( u +P. C ) )
27	24, 25, 26	3eqtr3i 2652	. . . . . . . 8 |- ( u +P. ( B +P. C ) ) = ( B +P. ( u +P. C ) )
28	27	oveq2i 6661	. . . . . . 7 |- ( z +P. ( u +P. ( B +P. C ) ) ) = ( z +P. ( B +P. ( u +P. C ) ) )
29		addasspr 9844	. . . . . . 7 |- ( ( z +P. u ) +P. ( B +P. C ) ) = ( z +P. ( u +P. ( B +P. C ) ) )
30		addasspr 9844	. . . . . . 7 |- ( ( z +P. B ) +P. ( u +P. C ) ) = ( z +P. ( B +P. ( u +P. C ) ) )
31	28, 29, 30	3eqtr4i 2654	. . . . . 6 |- ( ( z +P. u ) +P. ( B +P. C ) ) = ( ( z +P. B ) +P. ( u +P. C ) )
32		addcompr 9843	. . . . . . . . . 10 |- ( v +P. A ) = ( A +P. v )
33	32	oveq1i 6660	. . . . . . . . 9 |- ( ( v +P. A ) +P. D ) = ( ( A +P. v ) +P. D )
34		addasspr 9844	. . . . . . . . 9 |- ( ( v +P. A ) +P. D ) = ( v +P. ( A +P. D ) )
35		addasspr 9844	. . . . . . . . 9 |- ( ( A +P. v ) +P. D ) = ( A +P. ( v +P. D ) )
36	33, 34, 35	3eqtr3i 2652	. . . . . . . 8 |- ( v +P. ( A +P. D ) ) = ( A +P. ( v +P. D ) )
37	36	oveq2i 6661	. . . . . . 7 |- ( w +P. ( v +P. ( A +P. D ) ) ) = ( w +P. ( A +P. ( v +P. D ) ) )
38		addasspr 9844	. . . . . . 7 |- ( ( w +P. v ) +P. ( A +P. D ) ) = ( w +P. ( v +P. ( A +P. D ) ) )
39		addasspr 9844	. . . . . . 7 |- ( ( w +P. A ) +P. ( v +P. D ) ) = ( w +P. ( A +P. ( v +P. D ) ) )
40	37, 38, 39	3eqtr4i 2654	. . . . . 6 |- ( ( w +P. v ) +P. ( A +P. D ) ) = ( ( w +P. A ) +P. ( v +P. D ) )
41	22, 31, 40	3eqtr4g 2681	. . . . 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 243	. . . 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 6678	. . . . 5 |- ( z +P. u ) e. _V
44		ovex 6678	. . . . 5 |- ( B +P. C ) e. _V
45		ltapr 9867	. . . . 5 |- ( f e. P. -> ( x <P y <-> ( f +P. x ) <P ( f +P. y ) ) )
46		ovex 6678	. . . . 5 |- ( w +P. v ) e. _V
47		addcompr 9843	. . . . 5 |- ( x +P. y ) = ( y +P. x )
48		ovex 6678	. . . . 5 |- ( A +P. D ) e. _V
49	43, 44, 45, 46, 47, 48	caovord3 6847	. . . 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 568	. . 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 7840	. 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 7841	1 |- ( [ <. A , B >. ] ~R <R [ <. C , D >. ] ~R <-> ( A +P. D ) <P ( B +P. C ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   <.cop 4183   class class class wbr 4653   X. cxp 5112   dom cdm 5114  (class class class)co 6650   Er wer 7739   [cec 7740   P.cnp 9681   +P. cpp 9683   <P cltp 9685   ~R cer 9686   R.cnr 9687   <R cltr 9693

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  ax-un 6949  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-omul 7565  df-er 7742  df-ec 7744  df-qs 7748  df-ni 9694  df-pli 9695  df-mi 9696  df-lti 9697  df-plpq 9730  df-mpq 9731  df-ltpq 9732  df-enq 9733  df-nq 9734  df-erq 9735  df-plq 9736  df-mq 9737  df-1nq 9738  df-rq 9739  df-ltnq 9740  df-np 9803  df-plp 9805  df-ltp 9807  df-enr 9877  df-nr 9878  df-ltr 9881

This theorem is referenced by:  gt0srpr  9899  ltsosr  9915  0lt1sr  9916  ltasr  9921  mappsrpr  9929  ltpsrpr  9930  map2psrpr  9931