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Theorem ltsosr 6692
Description: Signed real 'less than' is a strict ordering. (Contributed by NM, 19-Feb-1996.)
Assertion
Ref Expression
ltsosr  <R  Or  R.

Proof of Theorem ltsosr
Dummy variables  a  b  c  d  e  r  s  t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltposr 6691 . 2  <R  Po  R.
2 df-nr 6655 . . . 4  R.  P.  X.  P. /.  ~R
3 breq1 3758 . . . . 5  <. a ,  b >.  ~R  <. a ,  b
>.  ~R  <R  <. c ,  d >.  ~R  <R  <. c ,  d >.  ~R
4 breq1 3758 . . . . . 6  <. a ,  b >.  ~R  <. a ,  b
>.  ~R  <R  <. e ,  >.  ~R  <R  <. e ,  >.  ~R
54orbi1d 704 . . . . 5  <. a ,  b >.  ~R  <. a ,  b >.  ~R  <R  <. e , 
>.  ~R  <. e ,  >. 
~R  <R  <. c ,  d >.  ~R  <R  <. e ,  >.  ~R  <. e , 
>.  ~R  <R  <. c ,  d >.  ~R
63, 5imbi12d 223 . . . 4  <. a ,  b >.  ~R  <. a ,  b >.  ~R  <R  <. c ,  d
>.  ~R  <. a ,  b
>.  ~R  <R  <. e ,  >.  ~R  <. e , 
>.  ~R  <R  <. c ,  d >.  ~R  <R  <. c ,  d
>.  ~R  <R  <. e ,  >.  ~R 
<. e ,  >. 
~R  <R  <. c ,  d >.  ~R
7 breq2 3759 . . . . 5  <. c ,  d >.  ~R  <R  <. c ,  d >.  ~R  <R
8 breq2 3759 . . . . . 6  <. c ,  d >.  ~R  <. e , 
>.  ~R  <R  <. c ,  d >.  ~R  <. e , 
>.  ~R  <R
98orbi2d 703 . . . . 5  <. c ,  d >.  ~R  <R  <. e ,  >.  ~R  <. e , 
>.  ~R  <R  <. c ,  d >.  ~R  <R  <. e ,  >.  ~R  <. e , 
>.  ~R  <R
107, 9imbi12d 223 . . . 4  <. c ,  d >.  ~R  <R  <. c ,  d >.  ~R  <R  <. e ,  >.  ~R  <. e , 
>.  ~R  <R  <. c ,  d >.  ~R  <R 
<R  <. e , 
>.  ~R  <. e ,  >. 
~R  <R
11 breq2 3759 . . . . . 6  <. e ,  >. 
~R  <R  <. e ,  >.  ~R  <R
12 breq1 3758 . . . . . 6  <. e ,  >. 
~R  <. e , 
>.  ~R  <R  <R
1311, 12orbi12d 706 . . . . 5  <. e ,  >. 
~R  <R  <. e ,  >.  ~R  <. e , 
>.  ~R  <R  <R  <R
1413imbi2d 219 . . . 4  <. e ,  >. 
~R  <R  <R  <. e ,  >.  ~R  <. e , 
>.  ~R  <R  <R 
<R  <R
15 simp1l 927 . . . . . . . . 9  a  P.  b  P.  c  P.  d  P.  e  P.  P.  a  P.
16 simp3r 932 . . . . . . . . 9  a  P.  b  P.  c  P.  d  P.  e  P.  P.  P.
17 addclpr 6520 . . . . . . . . 9  a  P.  P.  a  +P.  P.
1815, 16, 17syl2anc 391 . . . . . . . 8  a  P.  b  P.  c  P.  d  P.  e  P.  P.  a  +P.  P.
19 simp2r 930 . . . . . . . 8  a  P.  b  P.  c  P.  d  P.  e  P.  P.  d  P.
20 addclpr 6520 . . . . . . . 8  a  +P.  P.  d  P.  a  +P. 
+P.  d 
P.
2118, 19, 20syl2anc 391 . . . . . . 7  a  P.  b  P.  c  P.  d  P.  e  P.  P.  a  +P.  +P.  d  P.
22 simp2l 929 . . . . . . . . 9  a  P.  b  P.  c  P.  d  P.  e  P.  P.  c  P.
23 addclpr 6520 . . . . . . . . 9  P.  c  P.  +P.  c  P.
2416, 22, 23syl2anc 391 . . . . . . . 8  a  P.  b  P.  c  P.  d  P.  e  P.  P.  +P.  c  P.
25 simp1r 928 . . . . . . . 8  a  P.  b  P.  c  P.  d  P.  e  P.  P.  b  P.
26 addclpr 6520 . . . . . . . 8  +P.  c  P.  b  P.  +P.  c 
+P.  b 
P.
2724, 25, 26syl2anc 391 . . . . . . 7  a  P.  b  P.  c  P.  d  P.  e  P.  P. 
+P.  c  +P.  b  P.
28 simp3l 931 . . . . . . . . 9  a  P.  b  P.  c  P.  d  P.  e  P.  P.  e  P.
29 addclpr 6520 . . . . . . . . 9  b  P.  e  P.  b  +P.  e  P.
3025, 28, 29syl2anc 391 . . . . . . . 8  a  P.  b  P.  c  P.  d  P.  e  P.  P.  b  +P.  e  P.
31 addclpr 6520 . . . . . . . 8  b  +P.  e  P.  d  P.  b  +P.  e 
+P.  d 
P.
3230, 19, 31syl2anc 391 . . . . . . 7  a  P.  b  P.  c  P.  d  P.  e  P.  P.  b  +P.  e  +P.  d  P.
33 ltsopr 6570 . . . . . . . 8  <P  Or  P.
34 sowlin 4048 . . . . . . . 8 
<P  Or  P.  a  +P.  +P.  d 
P.  +P.  c  +P.  b  P.  b  +P.  e  +P.  d  P.  a  +P. 
+P.  d  <P  +P.  c  +P.  b  a  +P.  +P.  d  <P  b  +P.  e  +P.  d  b  +P.  e 
+P.  d  <P  +P.  c  +P.  b
3533, 34mpan 400 . . . . . . 7  a  +P.  +P.  d  P.  +P.  c 
+P.  b 
P. 
b  +P.  e  +P.  d  P.  a  +P. 
+P.  d  <P  +P.  c  +P.  b  a  +P.  +P.  d  <P  b  +P.  e  +P.  d  b  +P.  e 
+P.  d  <P  +P.  c  +P.  b
3621, 27, 32, 35syl3anc 1134 . . . . . 6  a  P.  b  P.  c  P.  d  P.  e  P.  P.  a  +P. 
+P.  d  <P  +P.  c  +P.  b  a  +P.  +P.  d  <P  b  +P.  e  +P.  d  b  +P.  e 
+P.  d  <P  +P.  c  +P.  b
37 addclpr 6520 . . . . . . . . 9  a  P.  d  P.  a  +P.  d  P.
3815, 19, 37syl2anc 391 . . . . . . . 8  a  P.  b  P.  c  P.  d  P.  e  P.  P.  a  +P.  d  P.
39 addclpr 6520 . . . . . . . . 9  b  P.  c  P.  b  +P.  c  P.
4025, 22, 39syl2anc 391 . . . . . . . 8  a  P.  b  P.  c  P.  d  P.  e  P.  P.  b  +P.  c  P.
41 ltaprg 6592 . . . . . . . 8  a  +P.  d  P. 
b  +P.  c  P. 
P.  a  +P.  d 
<P  b  +P.  c  +P. 
a  +P.  d  <P  +P.  b  +P.  c
4238, 40, 16, 41syl3anc 1134 . . . . . . 7  a  P.  b  P.  c  P.  d  P.  e  P.  P.  a  +P.  d  <P  b  +P.  c  +P. 
a  +P.  d  <P  +P.  b  +P.  c
43 addcomprg 6554 . . . . . . . . . . 11  r  P.  s  P.  r  +P.  s  s  +P.  r
4443adantl 262 . . . . . . . . . 10  a 
P.  b  P. 
c  P.  d  P.  e  P.  P.  r 
P.  s  P. 
r  +P.  s  s  +P.  r
45 addassprg 6555 . . . . . . . . . . 11  r  P.  s  P.  t  P.  r  +P.  s 
+P.  t  r  +P. 
s  +P.  t
4645adantl 262 . . . . . . . . . 10  a 
P.  b  P. 
c  P.  d  P.  e  P.  P.  r 
P.  s  P.  t  P.  r  +P.  s 
+P.  t  r  +P. 
s  +P.  t
4716, 15, 19, 44, 46caov12d 5624 . . . . . . . . 9  a  P.  b  P.  c  P.  d  P.  e  P.  P.  +P.  a  +P.  d  a  +P.  +P.  d
4846, 15, 16, 19caovassd 5602 . . . . . . . . 9  a  P.  b  P.  c  P.  d  P.  e  P.  P.  a  +P.  +P.  d 
a  +P.  +P.  d
4947, 48eqtr4d 2072 . . . . . . . 8  a  P.  b  P.  c  P.  d  P.  e  P.  P.  +P.  a  +P.  d  a  +P. 
+P.  d
5046, 16, 25, 22caovassd 5602 . . . . . . . . 9  a  P.  b  P.  c  P.  d  P.  e  P.  P. 
+P.  b  +P.  c  +P.  b  +P.  c
5116, 25, 22, 44, 46caov32d 5623 . . . . . . . . 9  a  P.  b  P.  c  P.  d  P.  e  P.  P. 
+P.  b  +P.  c  +P.  c 
+P.  b
5250, 51eqtr3d 2071 . . . . . . . 8  a  P.  b  P.  c  P.  d  P.  e  P.  P.  +P.  b  +P.  c  +P.  c 
+P.  b
5349, 52breq12d 3768 . . . . . . 7  a  P.  b  P.  c  P.  d  P.  e  P.  P. 
+P.  a  +P.  d  <P  +P.  b  +P.  c  a  +P.  +P.  d  <P 
+P.  c  +P.  b
5442, 53bitrd 177 . . . . . 6  a  P.  b  P.  c  P.  d  P.  e  P.  P.  a  +P.  d  <P  b  +P.  c  a  +P.  +P.  d  <P 
+P.  c  +P.  b
55 ltaprg 6592 . . . . . . . . 9  r  P.  s  P.  t  P. 
r  <P  s  t  +P.  r  <P  t  +P.  s
5655adantl 262 . . . . . . . 8  a 
P.  b  P. 
c  P.  d  P.  e  P.  P.  r 
P.  s  P.  t  P. 
r  <P  s  t  +P.  r  <P  t  +P.  s
5756, 18, 30, 19, 44caovord2d 5612 . . . . . . 7  a  P.  b  P.  c  P.  d  P.  e  P.  P.  a  +P.  <P  b  +P.  e  a  +P.  +P.  d  <P  b  +P.  e  +P.  d
58 addclpr 6520 . . . . . . . . . 10  e  P.  d  P.  e  +P.  d  P.
5928, 19, 58syl2anc 391 . . . . . . . . 9  a  P.  b  P.  c  P.  d  P.  e  P.  P.  e  +P.  d  P.
6056, 59, 24, 25, 44caovord2d 5612 . . . . . . . 8  a  P.  b  P.  c  P.  d  P.  e  P.  P.  e  +P.  d  <P  +P.  c  e  +P.  d  +P.  b  <P 
+P.  c  +P.  b
6146, 25, 28, 19caovassd 5602 . . . . . . . . . 10  a  P.  b  P.  c  P.  d  P.  e  P.  P.  b  +P.  e  +P.  d 
b  +P.  e  +P.  d
6244, 25, 59caovcomd 5599 . . . . . . . . . 10  a  P.  b  P.  c  P.  d  P.  e  P.  P.  b  +P.  e  +P.  d  e  +P.  d 
+P.  b
6361, 62eqtrd 2069 . . . . . . . . 9  a  P.  b  P.  c  P.  d  P.  e  P.  P.  b  +P.  e  +P.  d  e  +P.  d 
+P.  b
6463breq1d 3765 . . . . . . . 8  a  P.  b  P.  c  P.  d  P.  e  P.  P.  b  +P.  e 
+P.  d  <P  +P.  c  +P.  b  e  +P.  d 
+P.  b  <P  +P.  c  +P.  b
6560, 64bitr4d 180 . . . . . . 7  a  P.  b  P.  c  P.  d  P.  e  P.  P.  e  +P.  d  <P  +P.  c  b  +P.  e  +P.  d  <P 
+P.  c  +P.  b
6657, 65orbi12d 706 . . . . . 6  a  P.  b  P.  c  P.  d  P.  e  P.  P.  a  +P. 
<P  b  +P.  e  e  +P.  d  <P  +P.  c  a  +P.  +P.  d  <P  b  +P.  e 
+P.  d  b  +P.  e  +P.  d  <P 
+P.  c  +P.  b
6736, 54, 663imtr4d 192 . . . . 5  a  P.  b  P.  c  P.  d  P.  e  P.  P.  a  +P.  d  <P  b  +P.  c  a  +P.  <P 
b  +P.  e  e  +P.  d  <P  +P.  c
68 ltsrprg 6675 . . . . . 6  a  P.  b  P.  c  P.  d  P.  <. a ,  b >.  ~R  <R  <. c ,  d
>.  ~R  a  +P.  d  <P  b  +P.  c
69683adant3 923 . . . . 5  a  P.  b  P.  c  P.  d  P.  e  P.  P.  <. a ,  b >.  ~R  <R  <. c ,  d
>.  ~R  a  +P.  d  <P  b  +P.  c
70 ltsrprg 6675 . . . . . . 7  a  P.  b  P.  e  P.  P.  <. a ,  b >.  ~R  <R  <. e , 
>.  ~R  a  +P.  <P  b  +P.  e
71703adant2 922 . . . . . 6  a  P.  b  P.  c  P.  d  P.  e  P.  P.  <. a ,  b >.  ~R  <R  <. e , 
>.  ~R  a  +P.  <P  b  +P.  e
72 ltsrprg 6675 . . . . . . . 8  e  P.  P.  c  P.  d  P.  <. e ,  >.  ~R  <R  <. c ,  d
>.  ~R  e  +P.  d  <P  +P.  c
7372ancoms 255 . . . . . . 7  c  P.  d  P.  e  P.  P.  <. e ,  >.  ~R  <R  <. c ,  d
>.  ~R  e  +P.  d  <P  +P.  c
74733adant1 921 . . . . . 6  a  P.  b  P.  c  P.  d  P.  e  P.  P.  <. e ,  >.  ~R  <R  <. c ,  d
>.  ~R  e  +P.  d  <P  +P.  c
7571, 74orbi12d 706 . . . . 5  a  P.  b  P.  c  P.  d  P.  e  P.  P.  <. a ,  b >.  ~R  <R  <. e ,  >.  ~R 
<. e ,  >. 
~R  <R  <. c ,  d >.  ~R  a  +P.  <P 
b  +P.  e  e  +P.  d  <P  +P.  c
7667, 69, 753imtr4d 192 . . . 4  a  P.  b  P.  c  P.  d  P.  e  P.  P.  <. a ,  b >.  ~R  <R  <. c ,  d
>.  ~R  <. a ,  b
>.  ~R  <R  <. e ,  >.  ~R  <. e , 
>.  ~R  <R  <. c ,  d >.  ~R
772, 6, 10, 14, 763ecoptocl 6131 . . 3  R.  R.  R.  <R  <R  <R
7877rgen3 2400 . 2  R. 
R.  R.  <R  <R  <R
79 df-iso 4025 . 2  <R  Or 
R.  <R  Po  R.  R.  R.  R.  <R  <R  <R
801, 78, 79mpbir2an 848 1  <R  Or  R.
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   wo 628   w3a 884   wceq 1242   wcel 1390  wral 2300   <.cop 3370   class class class wbr 3755    Po wpo 4022    Or wor 4023  (class class class)co 5455  cec 6040   P.cnp 6275    +P. cpp 6277    <P cltp 6279    ~R cer 6280   R.cnr 6281    <R cltr 6287
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254
This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-eprel 4017  df-id 4021  df-po 4024  df-iso 4025  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-1o 5940  df-2o 5941  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-pli 6289  df-mi 6290  df-lti 6291  df-plpq 6328  df-mpq 6329  df-enq 6331  df-nqqs 6332  df-plqqs 6333  df-mqqs 6334  df-1nqqs 6335  df-rq 6336  df-ltnqqs 6337  df-enq0 6407  df-nq0 6408  df-0nq0 6409  df-plq0 6410  df-mq0 6411  df-inp 6449  df-iplp 6451  df-iltp 6453  df-enr 6654  df-nr 6655  df-ltr 6658
This theorem is referenced by:  1ne0sr  6694  addgt0sr  6703  axpre-ltirr  6766  axpre-ltwlin  6767  axpre-lttrn  6768
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