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Theorem ltpopr 7351
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 7352. (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 7231 . . . . . . . 8  |-  ( s  e.  P.  ->  <. ( 1st `  s ) ,  ( 2nd `  s
) >.  e.  P. )
2 prdisj 7248 . . . . . . . 8  |-  ( (
<. ( 1st `  s
) ,  ( 2nd `  s ) >.  e.  P.  /\  q  e.  Q. )  ->  -.  ( q  e.  ( 1st `  s
)  /\  q  e.  ( 2nd `  s ) ) )
31, 2sylan 279 . . . . . . 7  |-  ( ( s  e.  P.  /\  q  e.  Q. )  ->  -.  ( q  e.  ( 1st `  s
)  /\  q  e.  ( 2nd `  s ) ) )
4 ancom 264 . . . . . . 7  |-  ( ( q  e.  ( 1st `  s )  /\  q  e.  ( 2nd `  s
) )  <->  ( q  e.  ( 2nd `  s
)  /\  q  e.  ( 1st `  s ) ) )
53, 4sylnib 648 . . . . . 6  |-  ( ( s  e.  P.  /\  q  e.  Q. )  ->  -.  ( q  e.  ( 2nd `  s
)  /\  q  e.  ( 1st `  s ) ) )
65nrexdv 2499 . . . . 5  |-  ( s  e.  P.  ->  -.  E. q  e.  Q.  (
q  e.  ( 2nd `  s )  /\  q  e.  ( 1st `  s
) ) )
7 ltdfpr 7262 . . . . . 6  |-  ( ( s  e.  P.  /\  s  e.  P. )  ->  ( s  <P  s  <->  E. q  e.  Q.  (
q  e.  ( 2nd `  s )  /\  q  e.  ( 1st `  s
) ) ) )
87anidms 392 . . . . 5  |-  ( s  e.  P.  ->  (
s  <P  s  <->  E. q  e.  Q.  ( q  e.  ( 2nd `  s
)  /\  q  e.  ( 1st `  s ) ) ) )
96, 8mtbird 645 . . . 4  |-  ( s  e.  P.  ->  -.  s  <P  s )
109adantl 273 . . 3  |-  ( ( T.  /\  s  e. 
P. )  ->  -.  s  <P  s )
11 ltdfpr 7262 . . . . . . . . . . 11  |-  ( ( s  e.  P.  /\  t  e.  P. )  ->  ( s  <P  t  <->  E. q  e.  Q.  (
q  e.  ( 2nd `  s )  /\  q  e.  ( 1st `  t
) ) ) )
12113adant3 984 . . . . . . . . . 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 7262 . . . . . . . . . . 11  |-  ( ( t  e.  P.  /\  u  e.  P. )  ->  ( t  <P  u  <->  E. r  e.  Q.  (
r  e.  ( 2nd `  t )  /\  r  e.  ( 1st `  u
) ) ) )
14133adant1 982 . . . . . . . . . 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 462 . . . . . . . . 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 2574 . . . . . . . . 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 197 . . . . . . . 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 292 . . . . . . 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 509 . . . . . . . . . . 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 7231 . . . . . . . . . . . . . . . . . 18  |-  ( t  e.  P.  ->  <. ( 1st `  t ) ,  ( 2nd `  t
) >.  e.  P. )
21 prltlu 7243 . . . . . . . . . . . . . . . . . 18  |-  ( (
<. ( 1st `  t
) ,  ( 2nd `  t ) >.  e.  P.  /\  q  e.  ( 1st `  t )  /\  r  e.  ( 2nd `  t
) )  ->  q  <Q  r )
2220, 21syl3an1 1232 . . . . . . . . . . . . . . . . 17  |-  ( ( t  e.  P.  /\  q  e.  ( 1st `  t )  /\  r  e.  ( 2nd `  t
) )  ->  q  <Q  r )
23223adant3r 1196 . . . . . . . . . . . . . . . 16  |-  ( ( t  e.  P.  /\  q  e.  ( 1st `  t )  /\  (
r  e.  ( 2nd `  t )  /\  r  e.  ( 1st `  u
) ) )  -> 
q  <Q  r )
24233adant2l 1193 . . . . . . . . . . . . . . 15  |-  ( ( t  e.  P.  /\  ( q  e.  ( 2nd `  s )  /\  q  e.  ( 1st `  t ) )  /\  ( r  e.  ( 2nd `  t
)  /\  r  e.  ( 1st `  u ) ) )  ->  q  <Q  r )
25243expb 1165 . . . . . . . . . . . . . 14  |-  ( ( t  e.  P.  /\  ( ( q  e.  ( 2nd `  s
)  /\  q  e.  ( 1st `  t ) )  /\  ( r  e.  ( 2nd `  t
)  /\  r  e.  ( 1st `  u ) ) ) )  -> 
q  <Q  r )
26253ad2antl2 1127 . . . . . . . . . . . . 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 466 . . . . . . . . . . . 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 7231 . . . . . . . . . . . . . . . . 17  |-  ( u  e.  P.  ->  <. ( 1st `  u ) ,  ( 2nd `  u
) >.  e.  P. )
29 prcdnql 7240 . . . . . . . . . . . . . . . . 17  |-  ( (
<. ( 1st `  u
) ,  ( 2nd `  u ) >.  e.  P.  /\  r  e.  ( 1st `  u ) )  -> 
( q  <Q  r  ->  q  e.  ( 1st `  u ) ) )
3028, 29sylan 279 . . . . . . . . . . . . . . . 16  |-  ( ( u  e.  P.  /\  r  e.  ( 1st `  u ) )  -> 
( q  <Q  r  ->  q  e.  ( 1st `  u ) ) )
3130adantrl 467 . . . . . . . . . . . . . . 15  |-  ( ( u  e.  P.  /\  ( r  e.  ( 2nd `  t )  /\  r  e.  ( 1st `  u ) ) )  ->  (
q  <Q  r  ->  q  e.  ( 1st `  u
) ) )
3231adantrl 467 . . . . . . . . . . . . . 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 1128 . . . . . . . . . . . . 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 466 . . . . . . . . . . . 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 302 . . . . . . . . . 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 114 . . . . . . . . 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 2527 . . . . . . . 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 2507 . . . . . . 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 7262 . . . . . . . . 9  |-  ( ( s  e.  P.  /\  u  e.  P. )  ->  ( s  <P  u  <->  E. q  e.  Q.  (
q  e.  ( 2nd `  s )  /\  q  e.  ( 1st `  u
) ) ) )
42413adant2 983 . . . . . . . 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 157 . . . . . . 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 272 . . . . . 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 114 . . . 4  |-  ( ( s  e.  P.  /\  t  e.  P.  /\  u  e.  P. )  ->  (
( s  <P  t  /\  t  <P  u )  ->  s  <P  u
) )
4746adantl 273 . . 3  |-  ( ( T.  /\  ( s  e.  P.  /\  t  e.  P.  /\  u  e. 
P. ) )  -> 
( ( s  <P 
t  /\  t  <P  u )  ->  s  <P  u ) )
4810, 47ispod 4186 . 2  |-  ( T. 
->  <P  Po  P. )
4948mptru 1323 1  |-  <P  Po  P.
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 945   T. wtru 1315    e. wcel 1463   E.wrex 2391   <.cop 3496   class class class wbr 3895    Po wpo 4176   ` cfv 5081   1stc1st 5990   2ndc2nd 5991   Q.cnq 7036    <Q cltq 7041   P.cnp 7047    <P cltp 7051
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4003  ax-sep 4006  ax-nul 4014  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-setind 4412  ax-iinf 4462
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-ral 2395  df-rex 2396  df-reu 2397  df-rab 2399  df-v 2659  df-sbc 2879  df-csb 2972  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-nul 3330  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-int 3738  df-iun 3781  df-br 3896  df-opab 3950  df-mpt 3951  df-tr 3987  df-eprel 4171  df-id 4175  df-po 4178  df-iso 4179  df-iord 4248  df-on 4250  df-suc 4253  df-iom 4465  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-f1 5086  df-fo 5087  df-f1o 5088  df-fv 5089  df-ov 5731  df-oprab 5732  df-mpo 5733  df-1st 5992  df-2nd 5993  df-recs 6156  df-irdg 6221  df-oadd 6271  df-omul 6272  df-er 6383  df-ec 6385  df-qs 6389  df-ni 7060  df-mi 7062  df-lti 7063  df-enq 7103  df-nqqs 7104  df-ltnqqs 7109  df-inp 7222  df-iltp 7226
This theorem is referenced by:  ltsopr  7352
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