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Theorem ltpopr 6751
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 6752. (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 6631 . . . . . . . 8  |-  ( s  e.  P.  ->  <. ( 1st `  s ) ,  ( 2nd `  s
) >.  e.  P. )
2 prdisj 6648 . . . . . . . 8  |-  ( (
<. ( 1st `  s
) ,  ( 2nd `  s ) >.  e.  P.  /\  q  e.  Q. )  ->  -.  ( q  e.  ( 1st `  s
)  /\  q  e.  ( 2nd `  s ) ) )
31, 2sylan 271 . . . . . . 7  |-  ( ( s  e.  P.  /\  q  e.  Q. )  ->  -.  ( q  e.  ( 1st `  s
)  /\  q  e.  ( 2nd `  s ) ) )
4 ancom 257 . . . . . . 7  |-  ( ( q  e.  ( 1st `  s )  /\  q  e.  ( 2nd `  s
) )  <->  ( q  e.  ( 2nd `  s
)  /\  q  e.  ( 1st `  s ) ) )
53, 4sylnib 611 . . . . . 6  |-  ( ( s  e.  P.  /\  q  e.  Q. )  ->  -.  ( q  e.  ( 2nd `  s
)  /\  q  e.  ( 1st `  s ) ) )
65nrexdv 2429 . . . . 5  |-  ( s  e.  P.  ->  -.  E. q  e.  Q.  (
q  e.  ( 2nd `  s )  /\  q  e.  ( 1st `  s
) ) )
7 ltdfpr 6662 . . . . . 6  |-  ( ( s  e.  P.  /\  s  e.  P. )  ->  ( s  <P  s  <->  E. q  e.  Q.  (
q  e.  ( 2nd `  s )  /\  q  e.  ( 1st `  s
) ) ) )
87anidms 383 . . . . 5  |-  ( s  e.  P.  ->  (
s  <P  s  <->  E. q  e.  Q.  ( q  e.  ( 2nd `  s
)  /\  q  e.  ( 1st `  s ) ) ) )
96, 8mtbird 608 . . . 4  |-  ( s  e.  P.  ->  -.  s  <P  s )
109adantl 266 . . 3  |-  ( ( T.  /\  s  e. 
P. )  ->  -.  s  <P  s )
11 ltdfpr 6662 . . . . . . . . . . 11  |-  ( ( s  e.  P.  /\  t  e.  P. )  ->  ( s  <P  t  <->  E. q  e.  Q.  (
q  e.  ( 2nd `  s )  /\  q  e.  ( 1st `  t
) ) ) )
12113adant3 935 . . . . . . . . . 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 6662 . . . . . . . . . . 11  |-  ( ( t  e.  P.  /\  u  e.  P. )  ->  ( t  <P  u  <->  E. r  e.  Q.  (
r  e.  ( 2nd `  t )  /\  r  e.  ( 1st `  u
) ) ) )
14133adant1 933 . . . . . . . . . 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 450 . . . . . . . . 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 2496 . . . . . . . . 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 191 . . . . . . . 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 284 . . . . . . 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 497 . . . . . . . . . . 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 6631 . . . . . . . . . . . . . . . . . 18  |-  ( t  e.  P.  ->  <. ( 1st `  t ) ,  ( 2nd `  t
) >.  e.  P. )
21 prltlu 6643 . . . . . . . . . . . . . . . . . 18  |-  ( (
<. ( 1st `  t
) ,  ( 2nd `  t ) >.  e.  P.  /\  q  e.  ( 1st `  t )  /\  r  e.  ( 2nd `  t
) )  ->  q  <Q  r )
2220, 21syl3an1 1179 . . . . . . . . . . . . . . . . 17  |-  ( ( t  e.  P.  /\  q  e.  ( 1st `  t )  /\  r  e.  ( 2nd `  t
) )  ->  q  <Q  r )
23223adant3r 1143 . . . . . . . . . . . . . . . 16  |-  ( ( t  e.  P.  /\  q  e.  ( 1st `  t )  /\  (
r  e.  ( 2nd `  t )  /\  r  e.  ( 1st `  u
) ) )  -> 
q  <Q  r )
24233adant2l 1140 . . . . . . . . . . . . . . 15  |-  ( ( t  e.  P.  /\  ( q  e.  ( 2nd `  s )  /\  q  e.  ( 1st `  t ) )  /\  ( r  e.  ( 2nd `  t
)  /\  r  e.  ( 1st `  u ) ) )  ->  q  <Q  r )
25243expb 1116 . . . . . . . . . . . . . 14  |-  ( ( t  e.  P.  /\  ( ( q  e.  ( 2nd `  s
)  /\  q  e.  ( 1st `  t ) )  /\  ( r  e.  ( 2nd `  t
)  /\  r  e.  ( 1st `  u ) ) ) )  -> 
q  <Q  r )
26253ad2antl2 1078 . . . . . . . . . . . . 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 454 . . . . . . . . . . . 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 6631 . . . . . . . . . . . . . . . . 17  |-  ( u  e.  P.  ->  <. ( 1st `  u ) ,  ( 2nd `  u
) >.  e.  P. )
29 prcdnql 6640 . . . . . . . . . . . . . . . . 17  |-  ( (
<. ( 1st `  u
) ,  ( 2nd `  u ) >.  e.  P.  /\  r  e.  ( 1st `  u ) )  -> 
( q  <Q  r  ->  q  e.  ( 1st `  u ) ) )
3028, 29sylan 271 . . . . . . . . . . . . . . . 16  |-  ( ( u  e.  P.  /\  r  e.  ( 1st `  u ) )  -> 
( q  <Q  r  ->  q  e.  ( 1st `  u ) ) )
3130adantrl 455 . . . . . . . . . . . . . . 15  |-  ( ( u  e.  P.  /\  ( r  e.  ( 2nd `  t )  /\  r  e.  ( 1st `  u ) ) )  ->  (
q  <Q  r  ->  q  e.  ( 1st `  u
) ) )
3231adantrl 455 . . . . . . . . . . . . . 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 1079 . . . . . . . . . . . . 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 454 . . . . . . . . . . . 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 294 . . . . . . . . . 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 112 . . . . . . . . 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 2453 . . . . . . . 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 2437 . . . . . . 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 6662 . . . . . . . . 9  |-  ( ( s  e.  P.  /\  u  e.  P. )  ->  ( s  <P  u  <->  E. q  e.  Q.  (
q  e.  ( 2nd `  s )  /\  q  e.  ( 1st `  u
) ) ) )
42413adant2 934 . . . . . . . 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 151 . . . . . . 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 265 . . . . . 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 112 . . . 4  |-  ( ( s  e.  P.  /\  t  e.  P.  /\  u  e.  P. )  ->  (
( s  <P  t  /\  t  <P  u )  ->  s  <P  u
) )
4746adantl 266 . . 3  |-  ( ( T.  /\  ( s  e.  P.  /\  t  e.  P.  /\  u  e. 
P. ) )  -> 
( ( s  <P 
t  /\  t  <P  u )  ->  s  <P  u ) )
4810, 47ispod 4069 . 2  |-  ( T. 
->  <P  Po  P. )
4948trud 1268 1  |-  <P  Po  P.
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 101    <-> wb 102    /\ w3a 896   T. wtru 1260    e. wcel 1409   E.wrex 2324   <.cop 3406   class class class wbr 3792    Po wpo 4059   ` cfv 4930   1stc1st 5793   2ndc2nd 5794   Q.cnq 6436    <Q cltq 6441   P.cnp 6447    <P cltp 6451
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-eprel 4054  df-id 4058  df-po 4061  df-iso 4062  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-oadd 6036  df-omul 6037  df-er 6137  df-ec 6139  df-qs 6143  df-ni 6460  df-mi 6462  df-lti 6463  df-enq 6503  df-nqqs 6504  df-ltnqqs 6509  df-inp 6622  df-iltp 6626
This theorem is referenced by:  ltsopr  6752
  Copyright terms: Public domain W3C validator