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Theorem ltpopr 6693
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 6694. (Contributed by Jim Kingdon, 15-Dec-2019.)
Assertion
Ref Expression
ltpopr  |-  <P  Po  P.

Proof of Theorem ltpopr
Dummy variables  r  q  s  t  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prop 6573 . . . . . . . 8  |-  ( s  e.  P.  ->  <. ( 1st `  s ) ,  ( 2nd `  s
) >.  e.  P. )
2 prdisj 6590 . . . . . . . 8  |-  ( (
<. ( 1st `  s
) ,  ( 2nd `  s ) >.  e.  P.  /\  q  e.  Q. )  ->  -.  ( q  e.  ( 1st `  s
)  /\  q  e.  ( 2nd `  s ) ) )
31, 2sylan 267 . . . . . . 7  |-  ( ( s  e.  P.  /\  q  e.  Q. )  ->  -.  ( q  e.  ( 1st `  s
)  /\  q  e.  ( 2nd `  s ) ) )
4 ancom 253 . . . . . . 7  |-  ( ( q  e.  ( 1st `  s )  /\  q  e.  ( 2nd `  s
) )  <->  ( q  e.  ( 2nd `  s
)  /\  q  e.  ( 1st `  s ) ) )
53, 4sylnib 601 . . . . . 6  |-  ( ( s  e.  P.  /\  q  e.  Q. )  ->  -.  ( q  e.  ( 2nd `  s
)  /\  q  e.  ( 1st `  s ) ) )
65nrexdv 2412 . . . . 5  |-  ( s  e.  P.  ->  -.  E. q  e.  Q.  (
q  e.  ( 2nd `  s )  /\  q  e.  ( 1st `  s
) ) )
7 ltdfpr 6604 . . . . . 6  |-  ( ( s  e.  P.  /\  s  e.  P. )  ->  ( s  <P  s  <->  E. q  e.  Q.  (
q  e.  ( 2nd `  s )  /\  q  e.  ( 1st `  s
) ) ) )
87anidms 377 . . . . 5  |-  ( s  e.  P.  ->  (
s  <P  s  <->  E. q  e.  Q.  ( q  e.  ( 2nd `  s
)  /\  q  e.  ( 1st `  s ) ) ) )
96, 8mtbird 598 . . . 4  |-  ( s  e.  P.  ->  -.  s  <P  s )
109adantl 262 . . 3  |-  ( ( T.  /\  s  e. 
P. )  ->  -.  s  <P  s )
11 ltdfpr 6604 . . . . . . . . . . 11  |-  ( ( s  e.  P.  /\  t  e.  P. )  ->  ( s  <P  t  <->  E. q  e.  Q.  (
q  e.  ( 2nd `  s )  /\  q  e.  ( 1st `  t
) ) ) )
12113adant3 924 . . . . . . . . . 10  |-  ( ( s  e.  P.  /\  t  e.  P.  /\  u  e.  P. )  ->  (
s  <P  t  <->  E. q  e.  Q.  ( q  e.  ( 2nd `  s
)  /\  q  e.  ( 1st `  t ) ) ) )
13 ltdfpr 6604 . . . . . . . . . . 11  |-  ( ( t  e.  P.  /\  u  e.  P. )  ->  ( t  <P  u  <->  E. r  e.  Q.  (
r  e.  ( 2nd `  t )  /\  r  e.  ( 1st `  u
) ) ) )
14133adant1 922 . . . . . . . . . 10  |-  ( ( s  e.  P.  /\  t  e.  P.  /\  u  e.  P. )  ->  (
t  <P  u  <->  E. r  e.  Q.  ( r  e.  ( 2nd `  t
)  /\  r  e.  ( 1st `  u ) ) ) )
1512, 14anbi12d 442 . . . . . . . . 9  |-  ( ( s  e.  P.  /\  t  e.  P.  /\  u  e.  P. )  ->  (
( s  <P  t  /\  t  <P  u )  <-> 
( E. q  e. 
Q.  ( q  e.  ( 2nd `  s
)  /\  q  e.  ( 1st `  t ) )  /\  E. r  e.  Q.  ( r  e.  ( 2nd `  t
)  /\  r  e.  ( 1st `  u ) ) ) ) )
16 reeanv 2479 . . . . . . . . 9  |-  ( E. q  e.  Q.  E. r  e.  Q.  (
( q  e.  ( 2nd `  s )  /\  q  e.  ( 1st `  t ) )  /\  ( r  e.  ( 2nd `  t
)  /\  r  e.  ( 1st `  u ) ) )  <->  ( E. q  e.  Q.  (
q  e.  ( 2nd `  s )  /\  q  e.  ( 1st `  t
) )  /\  E. r  e.  Q.  (
r  e.  ( 2nd `  t )  /\  r  e.  ( 1st `  u
) ) ) )
1715, 16syl6bbr 187 . . . . . . . 8  |-  ( ( s  e.  P.  /\  t  e.  P.  /\  u  e.  P. )  ->  (
( s  <P  t  /\  t  <P  u )  <->  E. q  e.  Q.  E. r  e.  Q.  (
( q  e.  ( 2nd `  s )  /\  q  e.  ( 1st `  t ) )  /\  ( r  e.  ( 2nd `  t
)  /\  r  e.  ( 1st `  u ) ) ) ) )
1817biimpa 280 . . . . . . 7  |-  ( ( ( s  e.  P.  /\  t  e.  P.  /\  u  e.  P. )  /\  ( s  <P  t  /\  t  <P  u ) )  ->  E. q  e.  Q.  E. r  e. 
Q.  ( ( q  e.  ( 2nd `  s
)  /\  q  e.  ( 1st `  t ) )  /\  ( r  e.  ( 2nd `  t
)  /\  r  e.  ( 1st `  u ) ) ) )
19 simprll 489 . . . . . . . . . . 11  |-  ( ( ( ( s  e. 
P.  /\  t  e.  P.  /\  u  e.  P. )  /\  ( s  <P 
t  /\  t  <P  u ) )  /\  (
( q  e.  ( 2nd `  s )  /\  q  e.  ( 1st `  t ) )  /\  ( r  e.  ( 2nd `  t
)  /\  r  e.  ( 1st `  u ) ) ) )  -> 
q  e.  ( 2nd `  s ) )
20 prop 6573 . . . . . . . . . . . . . . . . . 18  |-  ( t  e.  P.  ->  <. ( 1st `  t ) ,  ( 2nd `  t
) >.  e.  P. )
21 prltlu 6585 . . . . . . . . . . . . . . . . . 18  |-  ( (
<. ( 1st `  t
) ,  ( 2nd `  t ) >.  e.  P.  /\  q  e.  ( 1st `  t )  /\  r  e.  ( 2nd `  t
) )  ->  q  <Q  r )
2220, 21syl3an1 1168 . . . . . . . . . . . . . . . . 17  |-  ( ( t  e.  P.  /\  q  e.  ( 1st `  t )  /\  r  e.  ( 2nd `  t
) )  ->  q  <Q  r )
23223adant3r 1132 . . . . . . . . . . . . . . . 16  |-  ( ( t  e.  P.  /\  q  e.  ( 1st `  t )  /\  (
r  e.  ( 2nd `  t )  /\  r  e.  ( 1st `  u
) ) )  -> 
q  <Q  r )
24233adant2l 1129 . . . . . . . . . . . . . . 15  |-  ( ( t  e.  P.  /\  ( q  e.  ( 2nd `  s )  /\  q  e.  ( 1st `  t ) )  /\  ( r  e.  ( 2nd `  t
)  /\  r  e.  ( 1st `  u ) ) )  ->  q  <Q  r )
25243expb 1105 . . . . . . . . . . . . . 14  |-  ( ( t  e.  P.  /\  ( ( q  e.  ( 2nd `  s
)  /\  q  e.  ( 1st `  t ) )  /\  ( r  e.  ( 2nd `  t
)  /\  r  e.  ( 1st `  u ) ) ) )  -> 
q  <Q  r )
26253ad2antl2 1067 . . . . . . . . . . . . 13  |-  ( ( ( s  e.  P.  /\  t  e.  P.  /\  u  e.  P. )  /\  ( ( q  e.  ( 2nd `  s
)  /\  q  e.  ( 1st `  t ) )  /\  ( r  e.  ( 2nd `  t
)  /\  r  e.  ( 1st `  u ) ) ) )  -> 
q  <Q  r )
2726adantlr 446 . . . . . . . . . . . 12  |-  ( ( ( ( s  e. 
P.  /\  t  e.  P.  /\  u  e.  P. )  /\  ( s  <P 
t  /\  t  <P  u ) )  /\  (
( q  e.  ( 2nd `  s )  /\  q  e.  ( 1st `  t ) )  /\  ( r  e.  ( 2nd `  t
)  /\  r  e.  ( 1st `  u ) ) ) )  -> 
q  <Q  r )
28 prop 6573 . . . . . . . . . . . . . . . . 17  |-  ( u  e.  P.  ->  <. ( 1st `  u ) ,  ( 2nd `  u
) >.  e.  P. )
29 prcdnql 6582 . . . . . . . . . . . . . . . . 17  |-  ( (
<. ( 1st `  u
) ,  ( 2nd `  u ) >.  e.  P.  /\  r  e.  ( 1st `  u ) )  -> 
( q  <Q  r  ->  q  e.  ( 1st `  u ) ) )
3028, 29sylan 267 . . . . . . . . . . . . . . . 16  |-  ( ( u  e.  P.  /\  r  e.  ( 1st `  u ) )  -> 
( q  <Q  r  ->  q  e.  ( 1st `  u ) ) )
3130adantrl 447 . . . . . . . . . . . . . . 15  |-  ( ( u  e.  P.  /\  ( r  e.  ( 2nd `  t )  /\  r  e.  ( 1st `  u ) ) )  ->  (
q  <Q  r  ->  q  e.  ( 1st `  u
) ) )
3231adantrl 447 . . . . . . . . . . . . . 14  |-  ( ( u  e.  P.  /\  ( ( q  e.  ( 2nd `  s
)  /\  q  e.  ( 1st `  t ) )  /\  ( r  e.  ( 2nd `  t
)  /\  r  e.  ( 1st `  u ) ) ) )  -> 
( q  <Q  r  ->  q  e.  ( 1st `  u ) ) )
33323ad2antl3 1068 . . . . . . . . . . . . 13  |-  ( ( ( s  e.  P.  /\  t  e.  P.  /\  u  e.  P. )  /\  ( ( q  e.  ( 2nd `  s
)  /\  q  e.  ( 1st `  t ) )  /\  ( r  e.  ( 2nd `  t
)  /\  r  e.  ( 1st `  u ) ) ) )  -> 
( q  <Q  r  ->  q  e.  ( 1st `  u ) ) )
3433adantlr 446 . . . . . . . . . . . 12  |-  ( ( ( ( s  e. 
P.  /\  t  e.  P.  /\  u  e.  P. )  /\  ( s  <P 
t  /\  t  <P  u ) )  /\  (
( q  e.  ( 2nd `  s )  /\  q  e.  ( 1st `  t ) )  /\  ( r  e.  ( 2nd `  t
)  /\  r  e.  ( 1st `  u ) ) ) )  -> 
( q  <Q  r  ->  q  e.  ( 1st `  u ) ) )
3527, 34mpd 13 . . . . . . . . . . 11  |-  ( ( ( ( s  e. 
P.  /\  t  e.  P.  /\  u  e.  P. )  /\  ( s  <P 
t  /\  t  <P  u ) )  /\  (
( q  e.  ( 2nd `  s )  /\  q  e.  ( 1st `  t ) )  /\  ( r  e.  ( 2nd `  t
)  /\  r  e.  ( 1st `  u ) ) ) )  -> 
q  e.  ( 1st `  u ) )
3619, 35jca 290 . . . . . . . . . 10  |-  ( ( ( ( s  e. 
P.  /\  t  e.  P.  /\  u  e.  P. )  /\  ( s  <P 
t  /\  t  <P  u ) )  /\  (
( q  e.  ( 2nd `  s )  /\  q  e.  ( 1st `  t ) )  /\  ( r  e.  ( 2nd `  t
)  /\  r  e.  ( 1st `  u ) ) ) )  -> 
( q  e.  ( 2nd `  s )  /\  q  e.  ( 1st `  u ) ) )
3736ex 108 . . . . . . . . 9  |-  ( ( ( s  e.  P.  /\  t  e.  P.  /\  u  e.  P. )  /\  ( s  <P  t  /\  t  <P  u ) )  ->  ( (
( q  e.  ( 2nd `  s )  /\  q  e.  ( 1st `  t ) )  /\  ( r  e.  ( 2nd `  t
)  /\  r  e.  ( 1st `  u ) ) )  ->  (
q  e.  ( 2nd `  s )  /\  q  e.  ( 1st `  u
) ) ) )
3837rexlimdvw 2436 . . . . . . . 8  |-  ( ( ( s  e.  P.  /\  t  e.  P.  /\  u  e.  P. )  /\  ( s  <P  t  /\  t  <P  u ) )  ->  ( E. r  e.  Q.  (
( q  e.  ( 2nd `  s )  /\  q  e.  ( 1st `  t ) )  /\  ( r  e.  ( 2nd `  t
)  /\  r  e.  ( 1st `  u ) ) )  ->  (
q  e.  ( 2nd `  s )  /\  q  e.  ( 1st `  u
) ) ) )
3938reximdv 2420 . . . . . . 7  |-  ( ( ( s  e.  P.  /\  t  e.  P.  /\  u  e.  P. )  /\  ( s  <P  t  /\  t  <P  u ) )  ->  ( E. q  e.  Q.  E. r  e.  Q.  ( ( q  e.  ( 2nd `  s
)  /\  q  e.  ( 1st `  t ) )  /\  ( r  e.  ( 2nd `  t
)  /\  r  e.  ( 1st `  u ) ) )  ->  E. q  e.  Q.  ( q  e.  ( 2nd `  s
)  /\  q  e.  ( 1st `  u ) ) ) )
4018, 39mpd 13 . . . . . 6  |-  ( ( ( s  e.  P.  /\  t  e.  P.  /\  u  e.  P. )  /\  ( s  <P  t  /\  t  <P  u ) )  ->  E. q  e.  Q.  ( q  e.  ( 2nd `  s
)  /\  q  e.  ( 1st `  u ) ) )
41 ltdfpr 6604 . . . . . . . . 9  |-  ( ( s  e.  P.  /\  u  e.  P. )  ->  ( s  <P  u  <->  E. q  e.  Q.  (
q  e.  ( 2nd `  s )  /\  q  e.  ( 1st `  u
) ) ) )
42413adant2 923 . . . . . . . 8  |-  ( ( s  e.  P.  /\  t  e.  P.  /\  u  e.  P. )  ->  (
s  <P  u  <->  E. q  e.  Q.  ( q  e.  ( 2nd `  s
)  /\  q  e.  ( 1st `  u ) ) ) )
4342biimprd 147 . . . . . . 7  |-  ( ( s  e.  P.  /\  t  e.  P.  /\  u  e.  P. )  ->  ( E. q  e.  Q.  ( q  e.  ( 2nd `  s )  /\  q  e.  ( 1st `  u ) )  ->  s  <P  u ) )
4443adantr 261 . . . . . 6  |-  ( ( ( s  e.  P.  /\  t  e.  P.  /\  u  e.  P. )  /\  ( s  <P  t  /\  t  <P  u ) )  ->  ( E. q  e.  Q.  (
q  e.  ( 2nd `  s )  /\  q  e.  ( 1st `  u
) )  ->  s  <P  u ) )
4540, 44mpd 13 . . . . 5  |-  ( ( ( s  e.  P.  /\  t  e.  P.  /\  u  e.  P. )  /\  ( s  <P  t  /\  t  <P  u ) )  ->  s  <P  u )
4645ex 108 . . . 4  |-  ( ( s  e.  P.  /\  t  e.  P.  /\  u  e.  P. )  ->  (
( s  <P  t  /\  t  <P  u )  ->  s  <P  u
) )
4746adantl 262 . . 3  |-  ( ( T.  /\  ( s  e.  P.  /\  t  e.  P.  /\  u  e. 
P. ) )  -> 
( ( s  <P 
t  /\  t  <P  u )  ->  s  <P  u ) )
4810, 47ispod 4041 . 2  |-  ( T. 
->  <P  Po  P. )
4948trud 1252 1  |-  <P  Po  P.
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 97    <-> wb 98    /\ w3a 885   T. wtru 1244    e. wcel 1393   E.wrex 2307   <.cop 3378   class class class wbr 3764    Po wpo 4031   ` cfv 4902   1stc1st 5765   2ndc2nd 5766   Q.cnq 6378    <Q cltq 6383   P.cnp 6389    <P cltp 6393
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-mi 6404  df-lti 6405  df-enq 6445  df-nqqs 6446  df-ltnqqs 6451  df-inp 6564  df-iltp 6568
This theorem is referenced by:  ltsopr  6694
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