Theorem ltsrpr 9452

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 9442	. 2 |- ~R e. _V
2		enrer 9440	. . 3 |- ~R Er ( P. X. P. )
3		erdm 7319	. . 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 9432	. 2 |- R. = ( ( P. X. P. ) /. ~R )
6		ltrelsr 9443	. 2 |- <R C_ ( R. X. R. )
7		ltrelpr 9374	. 2 |- <P C_ ( P. X. P. )
8		0npr 9368	. 2 |- -. (/) e. P.
9		dmplp 9388	. 2 |- dom +P. = ( P. X. P. )
10		df-ltr 9435	. . 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 9394	. . . . . . 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 9394	. . . . . . 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 9441	. . . . . 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 9441	. . . . . . 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 2450	. . . . . . 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 6286	. . . . . 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 9397	. . . . . . . . . 10 |- ( u +P. B ) = ( B +P. u )
24	23	oveq1i 6287	. . . . . . . . 9 |- ( ( u +P. B ) +P. C ) = ( ( B +P. u ) +P. C )
25		addasspr 9398	. . . . . . . . 9 |- ( ( u +P. B ) +P. C ) = ( u +P. ( B +P. C ) )
26		addasspr 9398	. . . . . . . . 9 |- ( ( B +P. u ) +P. C ) = ( B +P. ( u +P. C ) )
27	24, 25, 26	3eqtr3i 2478	. . . . . . . 8 |- ( u +P. ( B +P. C ) ) = ( B +P. ( u +P. C ) )
28	27	oveq2i 6288	. . . . . . 7 |- ( z +P. ( u +P. ( B +P. C ) ) ) = ( z +P. ( B +P. ( u +P. C ) ) )
29		addasspr 9398	. . . . . . 7 |- ( ( z +P. u ) +P. ( B +P. C ) ) = ( z +P. ( u +P. ( B +P. C ) ) )
30		addasspr 9398	. . . . . . 7 |- ( ( z +P. B ) +P. ( u +P. C ) ) = ( z +P. ( B +P. ( u +P. C ) ) )
31	28, 29, 30	3eqtr4i 2480	. . . . . 6 |- ( ( z +P. u ) +P. ( B +P. C ) ) = ( ( z +P. B ) +P. ( u +P. C ) )
32		addcompr 9397	. . . . . . . . . 10 |- ( v +P. A ) = ( A +P. v )
33	32	oveq1i 6287	. . . . . . . . 9 |- ( ( v +P. A ) +P. D ) = ( ( A +P. v ) +P. D )
34		addasspr 9398	. . . . . . . . 9 |- ( ( v +P. A ) +P. D ) = ( v +P. ( A +P. D ) )
35		addasspr 9398	. . . . . . . . 9 |- ( ( A +P. v ) +P. D ) = ( A +P. ( v +P. D ) )
36	33, 34, 35	3eqtr3i 2478	. . . . . . . 8 |- ( v +P. ( A +P. D ) ) = ( A +P. ( v +P. D ) )
37	36	oveq2i 6288	. . . . . . 7 |- ( w +P. ( v +P. ( A +P. D ) ) ) = ( w +P. ( A +P. ( v +P. D ) ) )
38		addasspr 9398	. . . . . . 7 |- ( ( w +P. v ) +P. ( A +P. D ) ) = ( w +P. ( v +P. ( A +P. D ) ) )
39		addasspr 9398	. . . . . . 7 |- ( ( w +P. A ) +P. ( v +P. D ) ) = ( w +P. ( A +P. ( v +P. D ) ) )
40	37, 38, 39	3eqtr4i 2480	. . . . . 6 |- ( ( w +P. v ) +P. ( A +P. D ) ) = ( ( w +P. A ) +P. ( v +P. D ) )
41	22, 31, 40	3eqtr4g 2507	. . . . 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 6305	. . . . 5 |- ( z +P. u ) e. _V
44		ovex 6305	. . . . 5 |- ( B +P. C ) e. _V
45		ltapr 9421	. . . . 5 |- ( f e. P. -> ( x <P y <-> ( f +P. x ) <P ( f +P. y ) ) )
46		ovex 6305	. . . . 5 |- ( w +P. v ) e. _V
47		addcompr 9397	. . . . 5 |- ( x +P. y ) = ( y +P. x )
48		ovex 6305	. . . . 5 |- ( A +P. D ) e. _V
49	43, 44, 45, 46, 47, 48	caovord3 6469	. . . 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 7402	. 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 7403	1 |- ( [ <. A , B >. ] ~R <R [ <. C , D >. ] ~R <-> ( A +P. D ) <P ( B +P. C ) )

Syntax hints:   <-> wb 184   /\ wa 369   = wceq 1381   e. wcel 1802   <.cop 4016   class class class wbr 4433   X. cxp 4983   dom cdm 4985  (class class class)co 6277   Er wer 7306   [cec 7307   P.cnp 9235   +P. cpp 9237   <P cltp 9239   ~R cer 9240   R.cnr 9241   <R cltr 9247

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-inf2 8056

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-int 4268  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-1st 6781  df-2nd 6782  df-recs 7040  df-rdg 7074  df-1o 7128  df-oadd 7132  df-omul 7133  df-er 7309  df-ec 7311  df-qs 7315  df-ni 9248  df-pli 9249  df-mi 9250  df-lti 9251  df-plpq 9284  df-mpq 9285  df-ltpq 9286  df-enq 9287  df-nq 9288  df-erq 9289  df-plq 9290  df-mq 9291  df-1nq 9292  df-rq 9293  df-ltnq 9294  df-np 9357  df-plp 9359  df-ltp 9361  df-enr 9431  df-nr 9432  df-ltr 9435

This theorem is referenced by:  gt0srpr  9453  ltsosr  9469  0lt1sr  9470  ltasr  9475  mappsrpr  9483  ltpsrpr  9484  map2psrpr  9485