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Theorem ltsopr 6560
Description: Positive real 'less than' is a weak linear order (in the sense of df-iso 4024). Proposition 11.2.3 of [HoTT], p. (varies). (Contributed by Jim Kingdon, 16-Dec-2019.)
Assertion
Ref Expression
ltsopr  <P  Or  P.

Proof of Theorem ltsopr
Dummy variables  r  q  s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltpopr 6559 . 2  <P  Po  P.
2 ltdfpr 6481 . . . . 5  P.  P.  <P  q  Q.  q  2nd `  q  1st `
323adant3 923 . . . 4  P.  P.  P.  <P  q  Q.  q  2nd `  q  1st `
4 prop 6450 . . . . . . . . . . . 12  P.  <. 1st `  ,  2nd `  >.  P.
5 prnminu 6464 . . . . . . . . . . . 12 
<. 1st `  ,  2nd `  >.  P.  q  2nd `  r  2nd `  r 
<Q  q
64, 5sylan 267 . . . . . . . . . . 11  P.  q  2nd `  r  2nd `  r 
<Q  q
7 prop 6450 . . . . . . . . . . . 12  P.  <. 1st `  ,  2nd `  >.  P.
8 prnmaxl 6463 . . . . . . . . . . . 12 
<. 1st `  ,  2nd `  >.  P.  q  1st `  s  1st `  q 
<Q  s
97, 8sylan 267 . . . . . . . . . . 11  P.  q  1st `  s  1st `  q 
<Q  s
106, 9anim12i 321 . . . . . . . . . 10  P.  q  2nd `  P.  q  1st `  r  2nd `  r  <Q  q  s  1st `  q  <Q 
s
1110an4s 522 . . . . . . . . 9  P.  P.  q  2nd `  q  1st `  r  2nd `  r  <Q 
q  s  1st `  q  <Q  s
12 reeanv 2473 . . . . . . . . 9  r  2nd `  s  1st `  r  <Q  q  q  <Q  s  r  2nd `  r  <Q  q  s  1st `  q  <Q 
s
1311, 12sylibr 137 . . . . . . . 8  P.  P.  q  2nd `  q  1st `  r  2nd `  s  1st `  r  <Q  q  q  <Q  s
14133adantl3 1061 . . . . . . 7  P.  P.  P.  q  2nd `  q  1st `  r  2nd `  s  1st `  r  <Q  q  q  <Q  s
15 ltsonq 6375 . . . . . . . . . . . . 13  <Q  Or  Q.
16 ltrelnq 6342 . . . . . . . . . . . . 13  <Q  C_  Q.  X.  Q.
1715, 16sotri 4662 . . . . . . . . . . . 12  r  <Q  q  q  <Q  s  r  <Q  s
1817adantl 262 . . . . . . . . . . 11  P.  P. 
P. 
q  2nd `  q  1st ` 
r  2nd `  s  1st ` 
r  <Q  q  q  <Q  s  r  <Q  s
19 prop 6450 . . . . . . . . . . . . . . . 16  P.  <. 1st `  ,  2nd `  >.  P.
20 prloc 6466 . . . . . . . . . . . . . . . 16 
<. 1st `  ,  2nd `  >.  P.  r  <Q  s  r  1st `  s  2nd `
2119, 20sylan 267 . . . . . . . . . . . . . . 15  P.  r  <Q  s  r  1st `  s  2nd `
22213ad2antl3 1067 . . . . . . . . . . . . . 14  P.  P.  P.  r  <Q  s  r  1st `  s  2nd `
2322ex 108 . . . . . . . . . . . . 13  P.  P.  P. 
r  <Q  s 
r  1st `  s  2nd `
2423adantr 261 . . . . . . . . . . . 12  P.  P.  P.  q  2nd `  q  1st `  r 
<Q  s 
r  1st `  s  2nd `
2524ad2antrr 457 . . . . . . . . . . 11  P.  P. 
P. 
q  2nd `  q  1st ` 
r  2nd `  s  1st ` 
r  <Q  q  q  <Q  s  r  <Q  s  r  1st `  s  2nd `
2618, 25mpd 13 . . . . . . . . . 10  P.  P. 
P. 
q  2nd `  q  1st ` 
r  2nd `  s  1st ` 
r  <Q  q  q  <Q  s  r  1st `  s  2nd `
27 elprnqu 6457 . . . . . . . . . . . . . . . . . . . . 21 
<. 1st `  ,  2nd `  >.  P.  r  2nd `  r  Q.
284, 27sylan 267 . . . . . . . . . . . . . . . . . . . 20  P.  r  2nd `  r  Q.
29 ax-ia3 101 . . . . . . . . . . . . . . . . . . . . 21  r  2nd `  r  1st `  r  2nd `  r  1st `
3029adantl 262 . . . . . . . . . . . . . . . . . . . 20  P.  r  2nd `  r  1st `  r  2nd `  r  1st `
31 19.8a 1479 . . . . . . . . . . . . . . . . . . . 20  r  Q.  r  2nd `  r  1st `  r r  Q. 
r  2nd `  r  1st `
3228, 30, 31syl6an 1320 . . . . . . . . . . . . . . . . . . 19  P.  r  2nd `  r  1st `  r r 
Q.  r  2nd `  r  1st `
33323ad2antl1 1065 . . . . . . . . . . . . . . . . . 18  P.  P.  P.  r  2nd `  r  1st `  r r  Q. 
r  2nd `  r  1st `
3433imp 115 . . . . . . . . . . . . . . . . 17 
P.  P.  P.  r  2nd `  r  1st `  r r  Q.  r  2nd `  r  1st `
35 df-rex 2306 . . . . . . . . . . . . . . . . 17  r  Q. 
r  2nd `  r  1st `  r r  Q.  r  2nd `  r  1st `
3634, 35sylibr 137 . . . . . . . . . . . . . . . 16 
P.  P.  P.  r  2nd `  r  1st `  r  Q.  r  2nd `  r  1st `
37 ltdfpr 6481 . . . . . . . . . . . . . . . . . . 19  P.  P.  <P  r  Q.  r  2nd `  r  1st `
3837biimprd 147 . . . . . . . . . . . . . . . . . 18  P.  P.  r 
Q.  r  2nd `  r  1st `  <P
39383adant2 922 . . . . . . . . . . . . . . . . 17  P.  P.  P.  r  Q.  r  2nd `  r  1st `  <P
4039ad2antrr 457 . . . . . . . . . . . . . . . 16 
P.  P.  P.  r  2nd `  r  1st `  r  Q.  r  2nd `  r  1st `  <P
4136, 40mpd 13 . . . . . . . . . . . . . . 15 
P.  P.  P.  r  2nd `  r  1st `  <P
4241ex 108 . . . . . . . . . . . . . 14  P.  P.  P.  r  2nd `  r  1st `  <P
4342adantrr 448 . . . . . . . . . . . . 13  P.  P.  P.  r  2nd `  s  1st `  r  1st `  <P
44 elprnql 6456 . . . . . . . . . . . . . . . . . . . . 21 
<. 1st `  ,  2nd `  >.  P.  s  1st `  s  Q.
457, 44sylan 267 . . . . . . . . . . . . . . . . . . . 20  P.  s  1st `  s  Q.
46 pm3.21 251 . . . . . . . . . . . . . . . . . . . . 21  s  1st `  s  2nd `  s  2nd `  s  1st `
4746adantl 262 . . . . . . . . . . . . . . . . . . . 20  P.  s  1st `  s  2nd `  s  2nd `  s  1st `
48 19.8a 1479 . . . . . . . . . . . . . . . . . . . 20  s  Q.  s  2nd `  s  1st `  s s  Q. 
s  2nd `  s  1st `
4945, 47, 48syl6an 1320 . . . . . . . . . . . . . . . . . . 19  P.  s  1st `  s  2nd `  s s 
Q.  s  2nd `  s  1st `
50493ad2antl2 1066 . . . . . . . . . . . . . . . . . 18  P.  P.  P.  s  1st `  s  2nd `  s s  Q. 
s  2nd `  s  1st `
5150imp 115 . . . . . . . . . . . . . . . . 17 
P.  P.  P.  s  1st `  s  2nd `  s s  Q.  s  2nd `  s  1st `
52 df-rex 2306 . . . . . . . . . . . . . . . . 17  s  Q. 
s  2nd `  s  1st `  s s  Q.  s  2nd `  s  1st `
5351, 52sylibr 137 . . . . . . . . . . . . . . . 16 
P.  P.  P.  s  1st `  s  2nd `  s  Q.  s  2nd `  s  1st `
54 ltdfpr 6481 . . . . . . . . . . . . . . . . . . . 20  P.  P.  <P  s  Q.  s  2nd `  s  1st `
5554biimprd 147 . . . . . . . . . . . . . . . . . . 19  P.  P.  s 
Q.  s  2nd `  s  1st `  <P
5655ancoms 255 . . . . . . . . . . . . . . . . . 18  P.  P.  s 
Q.  s  2nd `  s  1st `  <P
57563adant1 921 . . . . . . . . . . . . . . . . 17  P.  P.  P.  s  Q.  s  2nd `  s  1st `  <P
5857ad2antrr 457 . . . . . . . . . . . . . . . 16 
P.  P.  P.  s  1st `  s  2nd `  s  Q.  s  2nd `  s  1st `  <P
5953, 58mpd 13 . . . . . . . . . . . . . . 15 
P.  P.  P.  s  1st `  s  2nd `  <P
6059ex 108 . . . . . . . . . . . . . 14  P.  P.  P.  s  1st `  s  2nd `  <P
6160adantrl 447 . . . . . . . . . . . . 13  P.  P.  P.  r  2nd `  s  1st `  s  2nd `  <P
6243, 61orim12d 699 . . . . . . . . . . . 12  P.  P.  P.  r  2nd `  s  1st `  r  1st `  s  2nd `  <P  <P
6362adantlr 446 . . . . . . . . . . 11 
P.  P.  P.  q  2nd `  q  1st ` 
r  2nd `  s  1st `  r  1st `  s  2nd `  <P  <P
6463adantr 261 . . . . . . . . . 10  P.  P. 
P. 
q  2nd `  q  1st ` 
r  2nd `  s  1st ` 
r  <Q  q  q  <Q  s  r  1st `  s  2nd ` 
<P  <P
6526, 64mpd 13 . . . . . . . . 9  P.  P. 
P. 
q  2nd `  q  1st ` 
r  2nd `  s  1st ` 
r  <Q  q  q  <Q  s  <P  <P
6665ex 108 . . . . . . . 8 
P.  P.  P.  q  2nd `  q  1st ` 
r  2nd `  s  1st `  r  <Q  q  q  <Q  s  <P  <P
6766rexlimdvva 2434 . . . . . . 7  P.  P.  P.  q  2nd `  q  1st `  r  2nd `  s  1st `  r  <Q  q  q  <Q  s  <P  <P
6814, 67mpd 13 . . . . . 6  P.  P.  P.  q  2nd `  q  1st ` 
<P  <P
6968ex 108 . . . . 5  P.  P.  P.  q  2nd `  q  1st `  <P  <P
7069rexlimdvw 2430 . . . 4  P.  P.  P.  q  Q.  q  2nd `  q  1st `  <P  <P
713, 70sylbid 139 . . 3  P.  P.  P.  <P  <P  <P
7271rgen3 2400 . 2  P. 
P.  P.  <P  <P  <P
73 df-iso 4024 . 2  <P  Or 
P.  <P  Po  P.  P.  P.  P.  <P  <P  <P
741, 72, 73mpbir2an 848 1  <P  Or  P.
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   wo 628   w3a 884  wex 1378   wcel 1390  wral 2300  wrex 2301   <.cop 3369   class class class wbr 3754    Po wpo 4021    Or wor 4022   ` cfv 4844   1stc1st 5704   2ndc2nd 5705   Q.cnq 6257    <Q cltq 6262   P.cnp 6268    <P cltp 6272
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 3862  ax-sep 3865  ax-nul 3873  ax-pow 3917  ax-pr 3934  ax-un 4135  ax-setind 4219  ax-iinf 4253
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 3352  df-sn 3372  df-pr 3373  df-op 3375  df-uni 3571  df-int 3606  df-iun 3649  df-br 3755  df-opab 3809  df-mpt 3810  df-tr 3845  df-eprel 4016  df-id 4020  df-po 4023  df-iso 4024  df-iord 4068  df-on 4070  df-suc 4073  df-iom 4256  df-xp 4293  df-rel 4294  df-cnv 4295  df-co 4296  df-dm 4297  df-rn 4298  df-res 4299  df-ima 4300  df-iota 4809  df-fun 4846  df-fn 4847  df-f 4848  df-f1 4849  df-fo 4850  df-f1o 4851  df-fv 4852  df-ov 5455  df-oprab 5456  df-mpt2 5457  df-1st 5706  df-2nd 5707  df-recs 5858  df-irdg 5894  df-oadd 5937  df-omul 5938  df-er 6035  df-ec 6037  df-qs 6041  df-ni 6281  df-mi 6283  df-lti 6284  df-enq 6324  df-nqqs 6325  df-ltnqqs 6330  df-inp 6441  df-iltp 6445
This theorem is referenced by:  addextpr  6583  lttrsr  6642  ltposr  6643  ltsosr  6644  archsr  6660
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