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Theorem ltpopr 6567
Description: Positive real 'less than' is a partial ordering. Remark ("< is transitive and irreflexive") preceding Proposition 11.2.3 of [HoTT], p. (varies). Lemma for ltsopr 6568. (Contributed by Jim Kingdon, 15-Dec-2019.)
Assertion
Ref Expression
ltpopr  <P  Po  P.

Proof of Theorem ltpopr
Dummy variables  r  q  s  t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prop 6457 . . . . . . . 8  s  P.  <. 1st `  s ,  2nd `  s >.  P.
2 prdisj 6474 . . . . . . . 8 
<. 1st `  s ,  2nd `  s >.  P.  q  Q.  q  1st `  s  q  2nd `  s
31, 2sylan 267 . . . . . . 7  s  P.  q  Q.  q  1st `  s  q  2nd `  s
4 ancom 253 . . . . . . 7  q  1st `  s  q  2nd `  s  q  2nd `  s  q  1st `  s
53, 4sylnib 600 . . . . . 6  s  P.  q  Q.  q  2nd `  s  q  1st `  s
65nrexdv 2406 . . . . 5  s  P.  q  Q. 
q  2nd `  s  q  1st `  s
7 ltdfpr 6488 . . . . . 6  s  P.  s  P.  s  <P  s  q  Q.  q  2nd `  s  q  1st `  s
87anidms 377 . . . . 5  s  P. 
s  <P  s  q  Q.  q  2nd `  s  q  1st `  s
96, 8mtbird 597 . . . 4  s  P.  s  <P  s
109adantl 262 . . 3  s 
P.  s  <P  s
11 ltdfpr 6488 . . . . . . . . . . 11  s  P.  t  P.  s  <P  t  q  Q.  q  2nd `  s  q  1st `  t
12113adant3 923 . . . . . . . . . 10  s  P.  t  P.  P. 
s  <P  t  q  Q.  q  2nd `  s  q  1st `  t
13 ltdfpr 6488 . . . . . . . . . . 11  t  P.  P.  t  <P  r  Q.  r  2nd `  t  r  1st `
14133adant1 921 . . . . . . . . . 10  s  P.  t  P.  P. 
t  <P  r  Q.  r  2nd `  t  r  1st `
1512, 14anbi12d 442 . . . . . . . . 9  s  P.  t  P.  P.  s  <P  t  t  <P  q  Q. 
q  2nd `  s  q  1st `  t  r  Q.  r  2nd `  t  r  1st `
16 reeanv 2473 . . . . . . . . 9  q  Q.  r  Q.  q  2nd `  s  q  1st `  t  r  2nd `  t  r  1st `  q  Q. 
q  2nd `  s  q  1st `  t  r  Q.  r  2nd `  t  r  1st `
1715, 16syl6bbr 187 . . . . . . . 8  s  P.  t  P.  P.  s  <P  t  t  <P  q  Q.  r 
Q.  q  2nd `  s  q  1st `  t  r  2nd `  t  r  1st `
1817biimpa 280 . . . . . . 7  s  P.  t  P.  P.  s  <P  t  t  <P  q  Q.  r 
Q.  q  2nd `  s  q  1st `  t  r  2nd `  t  r  1st `
19 simprll 489 . . . . . . . . . . 11  s 
P.  t  P.  P.  s  <P 
t  t  <P  q  2nd `  s  q  1st `  t  r  2nd `  t  r  1st `  q  2nd `  s
20 prop 6457 . . . . . . . . . . . . . . . . . 18  t  P.  <. 1st `  t ,  2nd `  t >.  P.
21 prltlu 6469 . . . . . . . . . . . . . . . . . 18 
<. 1st `  t ,  2nd `  t >.  P.  q  1st `  t  r  2nd `  t  q  <Q  r
2220, 21syl3an1 1167 . . . . . . . . . . . . . . . . 17  t  P.  q  1st `  t  r  2nd `  t  q  <Q  r
23223adant3r 1131 . . . . . . . . . . . . . . . 16  t  P.  q  1st `  t 
r  2nd `  t  r  1st `  q  <Q  r
24233adant2l 1128 . . . . . . . . . . . . . . 15  t  P.  q  2nd `  s  q  1st `  t  r  2nd `  t  r  1st `  q  <Q  r
25243expb 1104 . . . . . . . . . . . . . 14  t  P.  q  2nd `  s  q  1st `  t  r  2nd `  t  r  1st `  q  <Q  r
26253ad2antl2 1066 . . . . . . . . . . . . 13  s  P.  t  P.  P.  q  2nd `  s  q  1st `  t  r  2nd `  t  r  1st `  q  <Q  r
2726adantlr 446 . . . . . . . . . . . 12  s 
P.  t  P.  P.  s  <P 
t  t  <P  q  2nd `  s  q  1st `  t  r  2nd `  t  r  1st `  q  <Q 
r
28 prop 6457 . . . . . . . . . . . . . . . . 17  P.  <. 1st `  ,  2nd `  >.  P.
29 prcdnql 6466 . . . . . . . . . . . . . . . . 17 
<. 1st `  ,  2nd `  >.  P.  r  1st `  q  <Q  r  q  1st `
3028, 29sylan 267 . . . . . . . . . . . . . . . 16  P.  r  1st `  q  <Q  r  q  1st `
3130adantrl 447 . . . . . . . . . . . . . . 15  P.  r  2nd `  t  r  1st `  q  <Q  r  q  1st `
3231adantrl 447 . . . . . . . . . . . . . 14  P.  q  2nd `  s  q  1st `  t  r  2nd `  t  r  1st ` 
q  <Q  r  q  1st `
33323ad2antl3 1067 . . . . . . . . . . . . 13  s  P.  t  P.  P.  q  2nd `  s  q  1st `  t  r  2nd `  t  r  1st `  q  <Q  r  q  1st `
3433adantlr 446 . . . . . . . . . . . 12  s 
P.  t  P.  P.  s  <P 
t  t  <P  q  2nd `  s  q  1st `  t  r  2nd `  t  r  1st `  q 
<Q  r  q  1st `
3527, 34mpd 13 . . . . . . . . . . 11  s 
P.  t  P.  P.  s  <P 
t  t  <P  q  2nd `  s  q  1st `  t  r  2nd `  t  r  1st `  q  1st `
3619, 35jca 290 . . . . . . . . . 10  s 
P.  t  P.  P.  s  <P 
t  t  <P  q  2nd `  s  q  1st `  t  r  2nd `  t  r  1st `  q  2nd `  s  q  1st `
3736ex 108 . . . . . . . . 9  s  P.  t  P.  P.  s  <P  t  t  <P  q  2nd `  s  q  1st `  t  r  2nd `  t  r  1st `  q  2nd `  s  q  1st `
3837rexlimdvw 2430 . . . . . . . 8  s  P.  t  P.  P.  s  <P  t  t  <P  r  Q.  q  2nd `  s  q  1st `  t  r  2nd `  t  r  1st `  q  2nd `  s  q  1st `
3938reximdv 2414 . . . . . . 7  s  P.  t  P.  P.  s  <P  t  t  <P  q  Q.  r  Q.  q  2nd `  s  q  1st `  t  r  2nd `  t  r  1st `  q  Q. 
q  2nd `  s  q  1st `
4018, 39mpd 13 . . . . . 6  s  P.  t  P.  P.  s  <P  t  t  <P  q  Q.  q  2nd `  s  q  1st `
41 ltdfpr 6488 . . . . . . . . 9  s  P.  P.  s  <P  q  Q.  q  2nd `  s  q  1st `
42413adant2 922 . . . . . . . 8  s  P.  t  P.  P. 
s  <P  q  Q.  q  2nd `  s  q  1st `
4342biimprd 147 . . . . . . 7  s  P.  t  P.  P.  q  Q.  q  2nd `  s  q  1st `  s  <P
4443adantr 261 . . . . . 6  s  P.  t  P.  P.  s  <P  t  t  <P  q  Q. 
q  2nd `  s  q  1st `  s  <P
4540, 44mpd 13 . . . . 5  s  P.  t  P.  P.  s  <P  t  t  <P  s  <P
4645ex 108 . . . 4  s  P.  t  P.  P.  s  <P  t  t  <P  s  <P
4746adantl 262 . . 3  s  P.  t  P. 
P.  s  <P  t  t  <P  s  <P
4810, 47ispod 4032 . 2  <P  Po  P.
4948trud 1251 1  <P  Po  P.
Colors of variables: wff set class
Syntax hints:   wn 3   wi 4   wa 97   wb 98   w3a 884   wtru 1243   wcel 1390  wrex 2301   <.cop 3370   class class class wbr 3755    Po wpo 4022   ` cfv 4845   1stc1st 5707   2ndc2nd 5708   Q.cnq 6264    <Q cltq 6269   P.cnp 6275    <P cltp 6279
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-bnd 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-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-mi 6290  df-lti 6291  df-enq 6331  df-nqqs 6332  df-ltnqqs 6337  df-inp 6448  df-iltp 6452
This theorem is referenced by:  ltsopr  6568
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