Theorem ltsrpr 9383

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 9373	. 2 |- ~R e. _V
2		enrer 9371	. . 3 |- ~R Er ( P. X. P. )
3		erdm 7257	. . 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 9363	. 2 |- R. = ( ( P. X. P. ) /. ~R )
6		ltrelsr 9374	. 2 |- <R C_ ( R. X. R. )
7		ltrelpr 9305	. 2 |- <P C_ ( P. X. P. )
8		0npr 9299	. 2 |- -. (/) e. P.
9		dmplp 9319	. 2 |- dom +P. = ( P. X. P. )
10		df-ltr 9366	. . 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 9325	. . . . . . 7 |- ( ( w e. P. /\ v e. P. ) -> ( w +P. v ) e. P. )
12	11	ad2ant2lr 745	. . . . . 6 |- ( ( ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( w +P. v ) e. P. )
13		addclpr 9325	. . . . . . 7 |- ( ( B e. P. /\ C e. P. ) -> ( B +P. C ) e. P. )
14	13	ad2ant2lr 745	. . . . . 6 |- ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) -> ( B +P. C ) e. P. )
15	12, 14	anim12ci 565	. . . . 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 9372	. . . . . 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 9372	. . . . . . 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 2401	. . . . . . 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 6223	. . . . . 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 9328	. . . . . . . . . 10 |- ( u +P. B ) = ( B +P. u )
24	23	oveq1i 6224	. . . . . . . . 9 |- ( ( u +P. B ) +P. C ) = ( ( B +P. u ) +P. C )
25		addasspr 9329	. . . . . . . . 9 |- ( ( u +P. B ) +P. C ) = ( u +P. ( B +P. C ) )
26		addasspr 9329	. . . . . . . . 9 |- ( ( B +P. u ) +P. C ) = ( B +P. ( u +P. C ) )
27	24, 25, 26	3eqtr3i 2429	. . . . . . . 8 |- ( u +P. ( B +P. C ) ) = ( B +P. ( u +P. C ) )
28	27	oveq2i 6225	. . . . . . 7 |- ( z +P. ( u +P. ( B +P. C ) ) ) = ( z +P. ( B +P. ( u +P. C ) ) )
29		addasspr 9329	. . . . . . 7 |- ( ( z +P. u ) +P. ( B +P. C ) ) = ( z +P. ( u +P. ( B +P. C ) ) )
30		addasspr 9329	. . . . . . 7 |- ( ( z +P. B ) +P. ( u +P. C ) ) = ( z +P. ( B +P. ( u +P. C ) ) )
31	28, 29, 30	3eqtr4i 2431	. . . . . 6 |- ( ( z +P. u ) +P. ( B +P. C ) ) = ( ( z +P. B ) +P. ( u +P. C ) )
32		addcompr 9328	. . . . . . . . . 10 |- ( v +P. A ) = ( A +P. v )
33	32	oveq1i 6224	. . . . . . . . 9 |- ( ( v +P. A ) +P. D ) = ( ( A +P. v ) +P. D )
34		addasspr 9329	. . . . . . . . 9 |- ( ( v +P. A ) +P. D ) = ( v +P. ( A +P. D ) )
35		addasspr 9329	. . . . . . . . 9 |- ( ( A +P. v ) +P. D ) = ( A +P. ( v +P. D ) )
36	33, 34, 35	3eqtr3i 2429	. . . . . . . 8 |- ( v +P. ( A +P. D ) ) = ( A +P. ( v +P. D ) )
37	36	oveq2i 6225	. . . . . . 7 |- ( w +P. ( v +P. ( A +P. D ) ) ) = ( w +P. ( A +P. ( v +P. D ) ) )
38		addasspr 9329	. . . . . . 7 |- ( ( w +P. v ) +P. ( A +P. D ) ) = ( w +P. ( v +P. ( A +P. D ) ) )
39		addasspr 9329	. . . . . . 7 |- ( ( w +P. A ) +P. ( v +P. D ) ) = ( w +P. ( A +P. ( v +P. D ) ) )
40	37, 38, 39	3eqtr4i 2431	. . . . . 6 |- ( ( w +P. v ) +P. ( A +P. D ) ) = ( ( w +P. A ) +P. ( v +P. D ) )
41	22, 31, 40	3eqtr4g 2458	. . . . 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 6242	. . . . 5 |- ( z +P. u ) e. _V
44		ovex 6242	. . . . 5 |- ( B +P. C ) e. _V
45		ltapr 9352	. . . . 5 |- ( f e. P. -> ( x <P y <-> ( f +P. x ) <P ( f +P. y ) ) )
46		ovex 6242	. . . . 5 |- ( w +P. v ) e. _V
47		addcompr 9328	. . . . 5 |- ( x +P. y ) = ( y +P. x )
48		ovex 6242	. . . . 5 |- ( A +P. D ) e. _V
49	43, 44, 45, 46, 47, 48	caovord3 6405	. . . 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 543	. . 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 7340	. 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 7341	1 |- ( [ <. A , B >. ] ~R <R [ <. C , D >. ] ~R <-> ( A +P. D ) <P ( B +P. C ) )

Syntax hints:   <-> wb 184   /\ wa 367   = wceq 1399   e. wcel 1836   <.cop 3963   class class class wbr 4380   X. cxp 4924   dom cdm 4926  (class class class)co 6214   Er wer 7244   [cec 7245   P.cnp 9166   +P. cpp 9168   <P cltp 9170   ~R cer 9171   R.cnr 9172   <R cltr 9178

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2016  ax-ext 2370  ax-sep 4501  ax-nul 4509  ax-pow 4556  ax-pr 4614  ax-un 6509  ax-inf2 7990

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2232  df-mo 2233  df-clab 2378  df-cleq 2384  df-clel 2387  df-nfc 2542  df-ne 2589  df-ral 2747  df-rex 2748  df-reu 2749  df-rmo 2750  df-rab 2751  df-v 3049  df-sbc 3266  df-csb 3362  df-dif 3405  df-un 3407  df-in 3409  df-ss 3416  df-pss 3418  df-nul 3725  df-if 3871  df-pw 3942  df-sn 3958  df-pr 3960  df-tp 3962  df-op 3964  df-uni 4177  df-int 4213  df-iun 4258  df-br 4381  df-opab 4439  df-mpt 4440  df-tr 4474  df-eprel 4718  df-id 4722  df-po 4727  df-so 4728  df-fr 4765  df-we 4767  df-ord 4808  df-on 4809  df-lim 4810  df-suc 4811  df-xp 4932  df-rel 4933  df-cnv 4934  df-co 4935  df-dm 4936  df-rn 4937  df-res 4938  df-ima 4939  df-iota 5473  df-fun 5511  df-fn 5512  df-f 5513  df-f1 5514  df-fo 5515  df-f1o 5516  df-fv 5517  df-ov 6217  df-oprab 6218  df-mpt2 6219  df-om 6618  df-1st 6717  df-2nd 6718  df-recs 6978  df-rdg 7012  df-1o 7066  df-oadd 7070  df-omul 7071  df-er 7247  df-ec 7249  df-qs 7253  df-ni 9179  df-pli 9180  df-mi 9181  df-lti 9182  df-plpq 9215  df-mpq 9216  df-ltpq 9217  df-enq 9218  df-nq 9219  df-erq 9220  df-plq 9221  df-mq 9222  df-1nq 9223  df-rq 9224  df-ltnq 9225  df-np 9288  df-plp 9290  df-ltp 9292  df-enr 9362  df-nr 9363  df-ltr 9366

This theorem is referenced by:  gt0srpr  9384  ltsosr  9400  0lt1sr  9401  ltasr  9406  mappsrpr  9414  ltpsrpr  9415  map2psrpr  9416