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Theorem ltpopr 7624
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 7625. (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 7504 . . . . . . . 8  |-  ( s  e.  P.  ->  <. ( 1st `  s ) ,  ( 2nd `  s
) >.  e.  P. )
2 prdisj 7521 . . . . . . . 8  |-  ( (
<. ( 1st `  s
) ,  ( 2nd `  s ) >.  e.  P.  /\  q  e.  Q. )  ->  -.  ( q  e.  ( 1st `  s
)  /\  q  e.  ( 2nd `  s ) ) )
31, 2sylan 283 . . . . . . 7  |-  ( ( s  e.  P.  /\  q  e.  Q. )  ->  -.  ( q  e.  ( 1st `  s
)  /\  q  e.  ( 2nd `  s ) ) )
4 ancom 266 . . . . . . 7  |-  ( ( q  e.  ( 1st `  s )  /\  q  e.  ( 2nd `  s
) )  <->  ( q  e.  ( 2nd `  s
)  /\  q  e.  ( 1st `  s ) ) )
53, 4sylnib 677 . . . . . 6  |-  ( ( s  e.  P.  /\  q  e.  Q. )  ->  -.  ( q  e.  ( 2nd `  s
)  /\  q  e.  ( 1st `  s ) ) )
65nrexdv 2583 . . . . 5  |-  ( s  e.  P.  ->  -.  E. q  e.  Q.  (
q  e.  ( 2nd `  s )  /\  q  e.  ( 1st `  s
) ) )
7 ltdfpr 7535 . . . . . 6  |-  ( ( s  e.  P.  /\  s  e.  P. )  ->  ( s  <P  s  <->  E. q  e.  Q.  (
q  e.  ( 2nd `  s )  /\  q  e.  ( 1st `  s
) ) ) )
87anidms 397 . . . . 5  |-  ( s  e.  P.  ->  (
s  <P  s  <->  E. q  e.  Q.  ( q  e.  ( 2nd `  s
)  /\  q  e.  ( 1st `  s ) ) ) )
96, 8mtbird 674 . . . 4  |-  ( s  e.  P.  ->  -.  s  <P  s )
109adantl 277 . . 3  |-  ( ( T.  /\  s  e. 
P. )  ->  -.  s  <P  s )
11 ltdfpr 7535 . . . . . . . . . . 11  |-  ( ( s  e.  P.  /\  t  e.  P. )  ->  ( s  <P  t  <->  E. q  e.  Q.  (
q  e.  ( 2nd `  s )  /\  q  e.  ( 1st `  t
) ) ) )
12113adant3 1019 . . . . . . . . . 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 7535 . . . . . . . . . . 11  |-  ( ( t  e.  P.  /\  u  e.  P. )  ->  ( t  <P  u  <->  E. r  e.  Q.  (
r  e.  ( 2nd `  t )  /\  r  e.  ( 1st `  u
) ) ) )
14133adant1 1017 . . . . . . . . . 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 473 . . . . . . . . 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 2660 . . . . . . . . 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, 16bitr4di 198 . . . . . . . 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 296 . . . . . . 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 537 . . . . . . . . . . 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 7504 . . . . . . . . . . . . . . . . . 18  |-  ( t  e.  P.  ->  <. ( 1st `  t ) ,  ( 2nd `  t
) >.  e.  P. )
21 prltlu 7516 . . . . . . . . . . . . . . . . . 18  |-  ( (
<. ( 1st `  t
) ,  ( 2nd `  t ) >.  e.  P.  /\  q  e.  ( 1st `  t )  /\  r  e.  ( 2nd `  t
) )  ->  q  <Q  r )
2220, 21syl3an1 1282 . . . . . . . . . . . . . . . . 17  |-  ( ( t  e.  P.  /\  q  e.  ( 1st `  t )  /\  r  e.  ( 2nd `  t
) )  ->  q  <Q  r )
23223adant3r 1237 . . . . . . . . . . . . . . . 16  |-  ( ( t  e.  P.  /\  q  e.  ( 1st `  t )  /\  (
r  e.  ( 2nd `  t )  /\  r  e.  ( 1st `  u
) ) )  -> 
q  <Q  r )
24233adant2l 1234 . . . . . . . . . . . . . . 15  |-  ( ( t  e.  P.  /\  ( q  e.  ( 2nd `  s )  /\  q  e.  ( 1st `  t ) )  /\  ( r  e.  ( 2nd `  t
)  /\  r  e.  ( 1st `  u ) ) )  ->  q  <Q  r )
25243expb 1206 . . . . . . . . . . . . . 14  |-  ( ( t  e.  P.  /\  ( ( q  e.  ( 2nd `  s
)  /\  q  e.  ( 1st `  t ) )  /\  ( r  e.  ( 2nd `  t
)  /\  r  e.  ( 1st `  u ) ) ) )  -> 
q  <Q  r )
26253ad2antl2 1162 . . . . . . . . . . . . 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 477 . . . . . . . . . . . 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 7504 . . . . . . . . . . . . . . . . 17  |-  ( u  e.  P.  ->  <. ( 1st `  u ) ,  ( 2nd `  u
) >.  e.  P. )
29 prcdnql 7513 . . . . . . . . . . . . . . . . 17  |-  ( (
<. ( 1st `  u
) ,  ( 2nd `  u ) >.  e.  P.  /\  r  e.  ( 1st `  u ) )  -> 
( q  <Q  r  ->  q  e.  ( 1st `  u ) ) )
3028, 29sylan 283 . . . . . . . . . . . . . . . 16  |-  ( ( u  e.  P.  /\  r  e.  ( 1st `  u ) )  -> 
( q  <Q  r  ->  q  e.  ( 1st `  u ) ) )
3130adantrl 478 . . . . . . . . . . . . . . 15  |-  ( ( u  e.  P.  /\  ( r  e.  ( 2nd `  t )  /\  r  e.  ( 1st `  u ) ) )  ->  (
q  <Q  r  ->  q  e.  ( 1st `  u
) ) )
3231adantrl 478 . . . . . . . . . . . . . 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 1163 . . . . . . . . . . . . 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 477 . . . . . . . . . . . 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 306 . . . . . . . . . 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 115 . . . . . . . . 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 2611 . . . . . . . 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 2591 . . . . . . 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 7535 . . . . . . . . 9  |-  ( ( s  e.  P.  /\  u  e.  P. )  ->  ( s  <P  u  <->  E. q  e.  Q.  (
q  e.  ( 2nd `  s )  /\  q  e.  ( 1st `  u
) ) ) )
42413adant2 1018 . . . . . . . 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 158 . . . . . . 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 276 . . . . . 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 115 . . . 4  |-  ( ( s  e.  P.  /\  t  e.  P.  /\  u  e.  P. )  ->  (
( s  <P  t  /\  t  <P  u )  ->  s  <P  u
) )
4746adantl 277 . . 3  |-  ( ( T.  /\  ( s  e.  P.  /\  t  e.  P.  /\  u  e. 
P. ) )  -> 
( ( s  <P 
t  /\  t  <P  u )  ->  s  <P  u ) )
4810, 47ispod 4322 . 2  |-  ( T. 
->  <P  Po  P. )
4948mptru 1373 1  |-  <P  Po  P.
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    <-> wb 105    /\ w3a 980   T. wtru 1365    e. wcel 2160   E.wrex 2469   <.cop 3610   class class class wbr 4018    Po wpo 4312   ` cfv 5235   1stc1st 6163   2ndc2nd 6164   Q.cnq 7309    <Q cltq 7314   P.cnp 7320    <P cltp 7324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-eprel 4307  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-ov 5899  df-oprab 5900  df-mpo 5901  df-1st 6165  df-2nd 6166  df-recs 6330  df-irdg 6395  df-oadd 6445  df-omul 6446  df-er 6559  df-ec 6561  df-qs 6565  df-ni 7333  df-mi 7335  df-lti 7336  df-enq 7376  df-nqqs 7377  df-ltnqqs 7382  df-inp 7495  df-iltp 7499
This theorem is referenced by:  ltsopr  7625
  Copyright terms: Public domain W3C validator