Theorem ltsosr 9915

Description: Signed real 'less than' is a strict ordering. (Contributed by NM, 19-Feb-1996.) (New usage is discouraged.)

Assertion

Ref	Expression
ltsosr	|- <R Or R.

Proof of Theorem ltsosr

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

Step	Hyp	Ref	Expression
1		df-nr 9878	. . 3 |- R. = ( ( P. X. P. ) /. ~R )
2		breq1 4656	. . . 4 |- ( [ <. x , y >. ] ~R = f -> ( [ <. x , y >. ] ~R <R [ <. z , w >. ] ~R <-> f <R [ <. z , w >. ] ~R ) )
3		eqeq1 2626	. . . . . 6 |- ( [ <. x , y >. ] ~R = f -> ( [ <. x , y >. ] ~R = [ <. z , w >. ] ~R <-> f = [ <. z , w >. ] ~R ) )
4		breq2 4657	. . . . . 6 |- ( [ <. x , y >. ] ~R = f -> ( [ <. z , w >. ] ~R <R [ <. x , y >. ] ~R <-> [ <. z , w >. ] ~R <R f ) )
5	3, 4	orbi12d 746	. . . . 5 |- ( [ <. x , y >. ] ~R = f -> ( ( [ <. x , y >. ] ~R = [ <. z , w >. ] ~R \/ [ <. z , w >. ] ~R <R [ <. x , y >. ] ~R ) <-> ( f = [ <. z , w >. ] ~R \/ [ <. z , w >. ] ~R <R f ) ) )
6	5	notbid 308	. . . 4 |- ( [ <. x , y >. ] ~R = f -> ( -. ( [ <. x , y >. ] ~R = [ <. z , w >. ] ~R \/ [ <. z , w >. ] ~R <R [ <. x , y >. ] ~R ) <-> -. ( f = [ <. z , w >. ] ~R \/ [ <. z , w >. ] ~R <R f ) ) )
7	2, 6	bibi12d 335	. . 3 |- ( [ <. x , y >. ] ~R = f -> ( ( [ <. x , y >. ] ~R <R [ <. z , w >. ] ~R <-> -. ( [ <. x , y >. ] ~R = [ <. z , w >. ] ~R \/ [ <. z , w >. ] ~R <R [ <. x , y >. ] ~R ) ) <-> ( f <R [ <. z , w >. ] ~R <-> -. ( f = [ <. z , w >. ] ~R \/ [ <. z , w >. ] ~R <R f ) ) ) )
8		breq2 4657	. . . 4 |- ( [ <. z , w >. ] ~R = g -> ( f <R [ <. z , w >. ] ~R <-> f <R g ) )
9		eqeq2 2633	. . . . . 6 |- ( [ <. z , w >. ] ~R = g -> ( f = [ <. z , w >. ] ~R <-> f = g ) )
10		breq1 4656	. . . . . 6 |- ( [ <. z , w >. ] ~R = g -> ( [ <. z , w >. ] ~R <R f <-> g <R f ) )
11	9, 10	orbi12d 746	. . . . 5 |- ( [ <. z , w >. ] ~R = g -> ( ( f = [ <. z , w >. ] ~R \/ [ <. z , w >. ] ~R <R f ) <-> ( f = g \/ g <R f ) ) )
12	11	notbid 308	. . . 4 |- ( [ <. z , w >. ] ~R = g -> ( -. ( f = [ <. z , w >. ] ~R \/ [ <. z , w >. ] ~R <R f ) <-> -. ( f = g \/ g <R f ) ) )
13	8, 12	bibi12d 335	. . 3 |- ( [ <. z , w >. ] ~R = g -> ( ( f <R [ <. z , w >. ] ~R <-> -. ( f = [ <. z , w >. ] ~R \/ [ <. z , w >. ] ~R <R f ) ) <-> ( f <R g <-> -. ( f = g \/ g <R f ) ) ) )
14		ltsrpr 9898	. . . 4 |- ( [ <. x , y >. ] ~R <R [ <. z , w >. ] ~R <-> ( x +P. w ) <P ( y +P. z ) )
15		addclpr 9840	. . . . . . 7 |- ( ( x e. P. /\ w e. P. ) -> ( x +P. w ) e. P. )
16		addclpr 9840	. . . . . . 7 |- ( ( y e. P. /\ z e. P. ) -> ( y +P. z ) e. P. )
17		ltsopr 9854	. . . . . . . 8 |- <P Or P.
18		sotric 5061	. . . . . . . 8 |- ( ( <P Or P. /\ ( ( x +P. w ) e. P. /\ ( y +P. z ) e. P. ) ) -> ( ( x +P. w ) <P ( y +P. z ) <-> -. ( ( x +P. w ) = ( y +P. z ) \/ ( y +P. z ) <P ( x +P. w ) ) ) )
19	17, 18	mpan 706	. . . . . . 7 |- ( ( ( x +P. w ) e. P. /\ ( y +P. z ) e. P. ) -> ( ( x +P. w ) <P ( y +P. z ) <-> -. ( ( x +P. w ) = ( y +P. z ) \/ ( y +P. z ) <P ( x +P. w ) ) ) )
20	15, 16, 19	syl2an 494	. . . . . 6 |- ( ( ( x e. P. /\ w e. P. ) /\ ( y e. P. /\ z e. P. ) ) -> ( ( x +P. w ) <P ( y +P. z ) <-> -. ( ( x +P. w ) = ( y +P. z ) \/ ( y +P. z ) <P ( x +P. w ) ) ) )
21	20	an42s 870	. . . . 5 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( ( x +P. w ) <P ( y +P. z ) <-> -. ( ( x +P. w ) = ( y +P. z ) \/ ( y +P. z ) <P ( x +P. w ) ) ) )
22		enreceq 9887	. . . . . . 7 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( [ <. x , y >. ] ~R = [ <. z , w >. ] ~R <-> ( x +P. w ) = ( y +P. z ) ) )
23		ltsrpr 9898	. . . . . . . . 9 |- ( [ <. z , w >. ] ~R <R [ <. x , y >. ] ~R <-> ( z +P. y ) <P ( w +P. x ) )
24		addcompr 9843	. . . . . . . . . 10 |- ( z +P. y ) = ( y +P. z )
25		addcompr 9843	. . . . . . . . . 10 |- ( w +P. x ) = ( x +P. w )
26	24, 25	breq12i 4662	. . . . . . . . 9 |- ( ( z +P. y ) <P ( w +P. x ) <-> ( y +P. z ) <P ( x +P. w ) )
27	23, 26	bitri 264	. . . . . . . 8 |- ( [ <. z , w >. ] ~R <R [ <. x , y >. ] ~R <-> ( y +P. z ) <P ( x +P. w ) )
28	27	a1i 11	. . . . . . 7 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( [ <. z , w >. ] ~R <R [ <. x , y >. ] ~R <-> ( y +P. z ) <P ( x +P. w ) ) )
29	22, 28	orbi12d 746	. . . . . 6 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( ( [ <. x , y >. ] ~R = [ <. z , w >. ] ~R \/ [ <. z , w >. ] ~R <R [ <. x , y >. ] ~R ) <-> ( ( x +P. w ) = ( y +P. z ) \/ ( y +P. z ) <P ( x +P. w ) ) ) )
30	29	notbid 308	. . . . 5 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( -. ( [ <. x , y >. ] ~R = [ <. z , w >. ] ~R \/ [ <. z , w >. ] ~R <R [ <. x , y >. ] ~R ) <-> -. ( ( x +P. w ) = ( y +P. z ) \/ ( y +P. z ) <P ( x +P. w ) ) ) )
31	21, 30	bitr4d 271	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( ( x +P. w ) <P ( y +P. z ) <-> -. ( [ <. x , y >. ] ~R = [ <. z , w >. ] ~R \/ [ <. z , w >. ] ~R <R [ <. x , y >. ] ~R ) ) )
32	14, 31	syl5bb 272	. . 3 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( [ <. x , y >. ] ~R <R [ <. z , w >. ] ~R <-> -. ( [ <. x , y >. ] ~R = [ <. z , w >. ] ~R \/ [ <. z , w >. ] ~R <R [ <. x , y >. ] ~R ) ) )
33	1, 7, 13, 32	2ecoptocl 7838	. 2 |- ( ( f e. R. /\ g e. R. ) -> ( f <R g <-> -. ( f = g \/ g <R f ) ) )
34	2	anbi1d 741	. . . 4 |- ( [ <. x , y >. ] ~R = f -> ( ( [ <. x , y >. ] ~R <R [ <. z , w >. ] ~R /\ [ <. z , w >. ] ~R <R [ <. v , u >. ] ~R ) <-> ( f <R [ <. z , w >. ] ~R /\ [ <. z , w >. ] ~R <R [ <. v , u >. ] ~R ) ) )
35		breq1 4656	. . . 4 |- ( [ <. x , y >. ] ~R = f -> ( [ <. x , y >. ] ~R <R [ <. v , u >. ] ~R <-> f <R [ <. v , u >. ] ~R ) )
36	34, 35	imbi12d 334	. . 3 |- ( [ <. x , y >. ] ~R = f -> ( ( ( [ <. x , y >. ] ~R <R [ <. z , w >. ] ~R /\ [ <. z , w >. ] ~R <R [ <. v , u >. ] ~R ) -> [ <. x , y >. ] ~R <R [ <. v , u >. ] ~R ) <-> ( ( f <R [ <. z , w >. ] ~R /\ [ <. z , w >. ] ~R <R [ <. v , u >. ] ~R ) -> f <R [ <. v , u >. ] ~R ) ) )
37		breq1 4656	. . . . 5 |- ( [ <. z , w >. ] ~R = g -> ( [ <. z , w >. ] ~R <R [ <. v , u >. ] ~R <-> g <R [ <. v , u >. ] ~R ) )
38	8, 37	anbi12d 747	. . . 4 |- ( [ <. z , w >. ] ~R = g -> ( ( f <R [ <. z , w >. ] ~R /\ [ <. z , w >. ] ~R <R [ <. v , u >. ] ~R ) <-> ( f <R g /\ g <R [ <. v , u >. ] ~R ) ) )
39	38	imbi1d 331	. . 3 |- ( [ <. z , w >. ] ~R = g -> ( ( ( f <R [ <. z , w >. ] ~R /\ [ <. z , w >. ] ~R <R [ <. v , u >. ] ~R ) -> f <R [ <. v , u >. ] ~R ) <-> ( ( f <R g /\ g <R [ <. v , u >. ] ~R ) -> f <R [ <. v , u >. ] ~R ) ) )
40		breq2 4657	. . . . 5 |- ( [ <. v , u >. ] ~R = h -> ( g <R [ <. v , u >. ] ~R <-> g <R h ) )
41	40	anbi2d 740	. . . 4 |- ( [ <. v , u >. ] ~R = h -> ( ( f <R g /\ g <R [ <. v , u >. ] ~R ) <-> ( f <R g /\ g <R h ) ) )
42		breq2 4657	. . . 4 |- ( [ <. v , u >. ] ~R = h -> ( f <R [ <. v , u >. ] ~R <-> f <R h ) )
43	41, 42	imbi12d 334	. . 3 |- ( [ <. v , u >. ] ~R = h -> ( ( ( f <R g /\ g <R [ <. v , u >. ] ~R ) -> f <R [ <. v , u >. ] ~R ) <-> ( ( f <R g /\ g <R h ) -> f <R h ) ) )
44		ovex 6678	. . . . . . . . . 10 |- ( x +P. w ) e. _V
45		ovex 6678	. . . . . . . . . 10 |- ( y +P. z ) e. _V
46		ltapr 9867	. . . . . . . . . 10 |- ( h e. P. -> ( f <P g <-> ( h +P. f ) <P ( h +P. g ) ) )
47		vex 3203	. . . . . . . . . 10 |- u e. _V
48		addcompr 9843	. . . . . . . . . 10 |- ( f +P. g ) = ( g +P. f )
49	44, 45, 46, 47, 48	caovord2 6846	. . . . . . . . 9 |- ( u e. P. -> ( ( x +P. w ) <P ( y +P. z ) <-> ( ( x +P. w ) +P. u ) <P ( ( y +P. z ) +P. u ) ) )
50		addasspr 9844	. . . . . . . . . 10 |- ( ( x +P. w ) +P. u ) = ( x +P. ( w +P. u ) )
51		addasspr 9844	. . . . . . . . . 10 |- ( ( y +P. z ) +P. u ) = ( y +P. ( z +P. u ) )
52	50, 51	breq12i 4662	. . . . . . . . 9 |- ( ( ( x +P. w ) +P. u ) <P ( ( y +P. z ) +P. u ) <-> ( x +P. ( w +P. u ) ) <P ( y +P. ( z +P. u ) ) )
53	49, 52	syl6bb 276	. . . . . . . 8 |- ( u e. P. -> ( ( x +P. w ) <P ( y +P. z ) <-> ( x +P. ( w +P. u ) ) <P ( y +P. ( z +P. u ) ) ) )
54	14, 53	syl5bb 272	. . . . . . 7 |- ( u e. P. -> ( [ <. x , y >. ] ~R <R [ <. z , w >. ] ~R <-> ( x +P. ( w +P. u ) ) <P ( y +P. ( z +P. u ) ) ) )
55		ltsrpr 9898	. . . . . . . 8 |- ( [ <. z , w >. ] ~R <R [ <. v , u >. ] ~R <-> ( z +P. u ) <P ( w +P. v ) )
56		ltapr 9867	. . . . . . . 8 |- ( y e. P. -> ( ( z +P. u ) <P ( w +P. v ) <-> ( y +P. ( z +P. u ) ) <P ( y +P. ( w +P. v ) ) ) )
57	55, 56	syl5bb 272	. . . . . . 7 |- ( y e. P. -> ( [ <. z , w >. ] ~R <R [ <. v , u >. ] ~R <-> ( y +P. ( z +P. u ) ) <P ( y +P. ( w +P. v ) ) ) )
58	54, 57	bi2anan9r 918	. . . . . 6 |- ( ( y e. P. /\ u e. P. ) -> ( ( [ <. x , y >. ] ~R <R [ <. z , w >. ] ~R /\ [ <. z , w >. ] ~R <R [ <. v , u >. ] ~R ) <-> ( ( x +P. ( w +P. u ) ) <P ( y +P. ( z +P. u ) ) /\ ( y +P. ( z +P. u ) ) <P ( y +P. ( w +P. v ) ) ) ) )
59		ltrelpr 9820	. . . . . . . 8 |- <P C_ ( P. X. P. )
60	17, 59	sotri 5523	. . . . . . 7 |- ( ( ( x +P. ( w +P. u ) ) <P ( y +P. ( z +P. u ) ) /\ ( y +P. ( z +P. u ) ) <P ( y +P. ( w +P. v ) ) ) -> ( x +P. ( w +P. u ) ) <P ( y +P. ( w +P. v ) ) )
61		dmplp 9834	. . . . . . . . 9 |- dom +P. = ( P. X. P. )
62		0npr 9814	. . . . . . . . 9 |- -. (/) e. P.
63		ltapr 9867	. . . . . . . . 9 |- ( w e. P. -> ( ( x +P. u ) <P ( y +P. v ) <-> ( w +P. ( x +P. u ) ) <P ( w +P. ( y +P. v ) ) ) )
64	61, 59, 62, 63	ndmovordi 6825	. . . . . . . 8 |- ( ( w +P. ( x +P. u ) ) <P ( w +P. ( y +P. v ) ) -> ( x +P. u ) <P ( y +P. v ) )
65		vex 3203	. . . . . . . . . 10 |- x e. _V
66		vex 3203	. . . . . . . . . 10 |- w e. _V
67		addasspr 9844	. . . . . . . . . 10 |- ( ( f +P. g ) +P. h ) = ( f +P. ( g +P. h ) )
68	65, 66, 47, 48, 67	caov12 6862	. . . . . . . . 9 |- ( x +P. ( w +P. u ) ) = ( w +P. ( x +P. u ) )
69		vex 3203	. . . . . . . . . 10 |- y e. _V
70		vex 3203	. . . . . . . . . 10 |- v e. _V
71	69, 66, 70, 48, 67	caov12 6862	. . . . . . . . 9 |- ( y +P. ( w +P. v ) ) = ( w +P. ( y +P. v ) )
72	68, 71	breq12i 4662	. . . . . . . 8 |- ( ( x +P. ( w +P. u ) ) <P ( y +P. ( w +P. v ) ) <-> ( w +P. ( x +P. u ) ) <P ( w +P. ( y +P. v ) ) )
73		ltsrpr 9898	. . . . . . . 8 |- ( [ <. x , y >. ] ~R <R [ <. v , u >. ] ~R <-> ( x +P. u ) <P ( y +P. v ) )
74	64, 72, 73	3imtr4i 281	. . . . . . 7 |- ( ( x +P. ( w +P. u ) ) <P ( y +P. ( w +P. v ) ) -> [ <. x , y >. ] ~R <R [ <. v , u >. ] ~R )
75	60, 74	syl 17	. . . . . 6 |- ( ( ( x +P. ( w +P. u ) ) <P ( y +P. ( z +P. u ) ) /\ ( y +P. ( z +P. u ) ) <P ( y +P. ( w +P. v ) ) ) -> [ <. x , y >. ] ~R <R [ <. v , u >. ] ~R )
76	58, 75	syl6bi 243	. . . . 5 |- ( ( y e. P. /\ u e. P. ) -> ( ( [ <. x , y >. ] ~R <R [ <. z , w >. ] ~R /\ [ <. z , w >. ] ~R <R [ <. v , u >. ] ~R ) -> [ <. x , y >. ] ~R <R [ <. v , u >. ] ~R ) )
77	76	ad2ant2l 782	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( ( [ <. x , y >. ] ~R <R [ <. z , w >. ] ~R /\ [ <. z , w >. ] ~R <R [ <. v , u >. ] ~R ) -> [ <. x , y >. ] ~R <R [ <. v , u >. ] ~R ) )
78	77	3adant2 1080	. . 3 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( ( [ <. x , y >. ] ~R <R [ <. z , w >. ] ~R /\ [ <. z , w >. ] ~R <R [ <. v , u >. ] ~R ) -> [ <. x , y >. ] ~R <R [ <. v , u >. ] ~R ) )
79	1, 36, 39, 43, 78	3ecoptocl 7839	. 2 |- ( ( f e. R. /\ g e. R. /\ h e. R. ) -> ( ( f <R g /\ g <R h ) -> f <R h ) )
80	33, 79	isso2i 5067	1 |- <R Or R.

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   <.cop 4183   class class class wbr 4653   Or wor 5034  (class class class)co 6650   [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:  1ne0sr  9917  addgt0sr  9925  sqgt0sr  9927  supsrlem  9932  axpre-lttri  9986  axpre-lttrn  9987