Theorem ltsrpr 9506

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 9496	. 2 |- ~R e. _V
2		enrer 9494	. . 3 |- ~R Er ( P. X. P. )
3		erdm 7378	. . 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 9486	. 2 |- R. = ( ( P. X. P. ) /. ~R )
6		ltrelsr 9497	. 2 |- <R C_ ( R. X. R. )
7		ltrelpr 9428	. 2 |- <P C_ ( P. X. P. )
8		0npr 9422	. 2 |- -. (/) e. P.
9		dmplp 9442	. 2 |- dom +P. = ( P. X. P. )
10		df-ltr 9489	. . 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 9448	. . . . . . 7 |- ( ( w e. P. /\ v e. P. ) -> ( w +P. v ) e. P. )
12	11	ad2ant2lr 755	. . . . . 6 |- ( ( ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( w +P. v ) e. P. )
13		addclpr 9448	. . . . . . 7 |- ( ( B e. P. /\ C e. P. ) -> ( B +P. C ) e. P. )
14	13	ad2ant2lr 755	. . . . . 6 |- ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) -> ( B +P. C ) e. P. )
15	12, 14	anim12ci 571	. . . . 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 836	. . . 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 9495	. . . . . 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 9495	. . . . . . 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 2460	. . . . . . 7 |- ( ( v +P. D ) = ( u +P. C ) <-> ( u +P. C ) = ( v +P. D ) )
20	18, 19	syl6bb 265	. . . . . 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 885	. . . . 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 6304	. . . . . 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 9451	. . . . . . . . . 10 |- ( u +P. B ) = ( B +P. u )
24	23	oveq1i 6305	. . . . . . . . 9 |- ( ( u +P. B ) +P. C ) = ( ( B +P. u ) +P. C )
25		addasspr 9452	. . . . . . . . 9 |- ( ( u +P. B ) +P. C ) = ( u +P. ( B +P. C ) )
26		addasspr 9452	. . . . . . . . 9 |- ( ( B +P. u ) +P. C ) = ( B +P. ( u +P. C ) )
27	24, 25, 26	3eqtr3i 2483	. . . . . . . 8 |- ( u +P. ( B +P. C ) ) = ( B +P. ( u +P. C ) )
28	27	oveq2i 6306	. . . . . . 7 |- ( z +P. ( u +P. ( B +P. C ) ) ) = ( z +P. ( B +P. ( u +P. C ) ) )
29		addasspr 9452	. . . . . . 7 |- ( ( z +P. u ) +P. ( B +P. C ) ) = ( z +P. ( u +P. ( B +P. C ) ) )
30		addasspr 9452	. . . . . . 7 |- ( ( z +P. B ) +P. ( u +P. C ) ) = ( z +P. ( B +P. ( u +P. C ) ) )
31	28, 29, 30	3eqtr4i 2485	. . . . . 6 |- ( ( z +P. u ) +P. ( B +P. C ) ) = ( ( z +P. B ) +P. ( u +P. C ) )
32		addcompr 9451	. . . . . . . . . 10 |- ( v +P. A ) = ( A +P. v )
33	32	oveq1i 6305	. . . . . . . . 9 |- ( ( v +P. A ) +P. D ) = ( ( A +P. v ) +P. D )
34		addasspr 9452	. . . . . . . . 9 |- ( ( v +P. A ) +P. D ) = ( v +P. ( A +P. D ) )
35		addasspr 9452	. . . . . . . . 9 |- ( ( A +P. v ) +P. D ) = ( A +P. ( v +P. D ) )
36	33, 34, 35	3eqtr3i 2483	. . . . . . . 8 |- ( v +P. ( A +P. D ) ) = ( A +P. ( v +P. D ) )
37	36	oveq2i 6306	. . . . . . 7 |- ( w +P. ( v +P. ( A +P. D ) ) ) = ( w +P. ( A +P. ( v +P. D ) ) )
38		addasspr 9452	. . . . . . 7 |- ( ( w +P. v ) +P. ( A +P. D ) ) = ( w +P. ( v +P. ( A +P. D ) ) )
39		addasspr 9452	. . . . . . 7 |- ( ( w +P. A ) +P. ( v +P. D ) ) = ( w +P. ( A +P. ( v +P. D ) ) )
40	37, 38, 39	3eqtr4i 2485	. . . . . 6 |- ( ( w +P. v ) +P. ( A +P. D ) ) = ( ( w +P. A ) +P. ( v +P. D ) )
41	22, 31, 40	3eqtr4g 2512	. . . . 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 232	. . . 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 6323	. . . . 5 |- ( z +P. u ) e. _V
44		ovex 6323	. . . . 5 |- ( B +P. C ) e. _V
45		ltapr 9475	. . . . 5 |- ( f e. P. -> ( x <P y <-> ( f +P. x ) <P ( f +P. y ) ) )
46		ovex 6323	. . . . 5 |- ( w +P. v ) e. _V
47		addcompr 9451	. . . . 5 |- ( x +P. y ) = ( y +P. x )
48		ovex 6323	. . . . 5 |- ( A +P. D ) e. _V
49	43, 44, 45, 46, 47, 48	caovord3 6487	. . . 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 548	. . 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 7461	. 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 7462	1 |- ( [ <. A , B >. ] ~R <R [ <. C , D >. ] ~R <-> ( A +P. D ) <P ( B +P. C ) )

Syntax hints:   <-> wb 188   /\ wa 371   = wceq 1446   e. wcel 1889   <.cop 3976   class class class wbr 4405   X. cxp 4835   dom cdm 4837  (class class class)co 6295   Er wer 7365   [cec 7366   P.cnp 9289   +P. cpp 9291   <P cltp 9293   ~R cer 9294   R.cnr 9295   <R cltr 9301

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-inf2 8151

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-int 4238  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-oadd 7191  df-omul 7192  df-er 7368  df-ec 7370  df-qs 7374  df-ni 9302  df-pli 9303  df-mi 9304  df-lti 9305  df-plpq 9338  df-mpq 9339  df-ltpq 9340  df-enq 9341  df-nq 9342  df-erq 9343  df-plq 9344  df-mq 9345  df-1nq 9346  df-rq 9347  df-ltnq 9348  df-np 9411  df-plp 9413  df-ltp 9415  df-enr 9485  df-nr 9486  df-ltr 9489

This theorem is referenced by:  gt0srpr  9507  ltsosr  9523  0lt1sr  9524  ltasr  9529  mappsrpr  9537  ltpsrpr  9538  map2psrpr  9539