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Theorem 1idprl 6566
Description: Lemma for 1idpr 6568. (Contributed by Jim Kingdon, 13-Dec-2019.)
Assertion
Ref Expression
1idprl  P.  1st `  .P.  1P  1st `

Proof of Theorem 1idprl
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssid 2958 . . . . . 6  1st `  1P  C_  1st `  1P
2 rexss 3001 . . . . . 6  1st `  1P 
C_  1st `  1P  1st `  1P  .Q  1st `  1P  1st `  1P  .Q
31, 2ax-mp 7 . . . . 5  1st `  1P  .Q  1st `  1P  1st `  1P  .Q
4 r19.42v 2461 . . . . . 6  1st `  1P 
<Q  .Q  <Q  1st `  1P  .Q
5 1pr 6535 . . . . . . . . . . 11  1P  P.
6 prop 6458 . . . . . . . . . . . 12  1P  P.  <. 1st `  1P ,  2nd `  1P >.  P.
7 elprnql 6464 . . . . . . . . . . . 12 
<. 1st `  1P ,  2nd `  1P >.  P.  1st `  1P  Q.
86, 7sylan 267 . . . . . . . . . . 11  1P  P.  1st `  1P  Q.
95, 8mpan 400 . . . . . . . . . 10  1st `  1P  Q.
10 prop 6458 . . . . . . . . . . . 12  P.  <. 1st `  ,  2nd `  >.  P.
11 elprnql 6464 . . . . . . . . . . . 12 
<. 1st `  ,  2nd `  >.  P.  1st `  Q.
1210, 11sylan 267 . . . . . . . . . . 11  P.  1st `  Q.
13 breq1 3758 . . . . . . . . . . . . 13  .Q  <Q  .Q  <Q
14133ad2ant3 926 . . . . . . . . . . . 12  Q.  Q.  .Q  <Q  .Q  <Q
15 1prl 6536 . . . . . . . . . . . . . . 15  1st `  1P  {  |  <Q  1Q }
1615abeq2i 2145 . . . . . . . . . . . . . 14  1st `  1P  <Q  1Q
17 1nq 6350 . . . . . . . . . . . . . . . . 17  1Q  Q.
18 ltmnqg 6385 . . . . . . . . . . . . . . . . 17  Q.  1Q  Q.  Q.  <Q  1Q  .Q  <Q  .Q  1Q
1917, 18mp3an2 1219 . . . . . . . . . . . . . . . 16  Q.  Q.  <Q  1Q  .Q  <Q  .Q  1Q
2019ancoms 255 . . . . . . . . . . . . . . 15  Q.  Q.  <Q  1Q  .Q  <Q  .Q  1Q
21 mulidnq 6373 . . . . . . . . . . . . . . . . 17  Q.  .Q  1Q
2221breq2d 3767 . . . . . . . . . . . . . . . 16  Q.  .Q  <Q  .Q  1Q  .Q  <Q
2322adantr 261 . . . . . . . . . . . . . . 15  Q.  Q.  .Q  <Q  .Q  1Q  .Q  <Q
2420, 23bitrd 177 . . . . . . . . . . . . . 14  Q.  Q.  <Q  1Q  .Q  <Q
2516, 24syl5rbb 182 . . . . . . . . . . . . 13  Q.  Q.  .Q  <Q  1st `  1P
26253adant3 923 . . . . . . . . . . . 12  Q.  Q.  .Q  .Q  <Q  1st `  1P
2714, 26bitrd 177 . . . . . . . . . . 11  Q.  Q.  .Q  <Q  1st `  1P
2812, 27syl3an1 1167 . . . . . . . . . 10  P.  1st `  Q.  .Q  <Q  1st `  1P
299, 28syl3an2 1168 . . . . . . . . 9  P.  1st `  1st `  1P  .Q  <Q  1st `  1P
30293expia 1105 . . . . . . . 8  P.  1st `  1st `  1P  .Q  <Q  1st `  1P
3130pm5.32rd 424 . . . . . . 7  P.  1st `  1st `  1P  <Q  .Q  1st `  1P  .Q
3231rexbidva 2317 . . . . . 6  P.  1st `  1st `  1P  <Q  .Q  1st `  1P  1st `  1P  .Q
334, 32syl5rbbr 184 . . . . 5  P.  1st `  1st `  1P  1st `  1P  .Q  <Q  1st `  1P  .Q
343, 33syl5bb 181 . . . 4  P.  1st `  1st `  1P  .Q 
<Q  1st `  1P  .Q
3534rexbidva 2317 . . 3  P.  1st `  1st `  1P  .Q  1st `  <Q  1st `  1P  .Q
36 df-imp 6452 . . . . 5  .P.  P. ,  P.  |->  <. {  Q.  |  Q.  Q.  1st `  1st `  .Q  } ,  {  Q.  |  Q.  Q.  2nd `  2nd `  .Q  } >.
37 mulclnq 6360 . . . . 5  Q.  Q.  .Q  Q.
3836, 37genpelvl 6495 . . . 4  P.  1P  P.  1st `  .P.  1P  1st `  1st `  1P  .Q
395, 38mpan2 401 . . 3  P.  1st `  .P.  1P  1st `  1st `  1P  .Q
40 prnmaxl 6471 . . . . . . 7 
<. 1st `  ,  2nd `  >.  P.  1st `  1st ` 
<Q
4110, 40sylan 267 . . . . . 6  P.  1st `  1st ` 
<Q
42 ltrelnq 6349 . . . . . . . . . . . . 13  <Q  C_  Q.  X.  Q.
4342brel 4335 . . . . . . . . . . . 12 
<Q  Q.  Q.
44 ltmnqg 6385 . . . . . . . . . . . . . . . 16  Q.  Q.  Q.  <Q  .Q  <Q  .Q
4544adantl 262 . . . . . . . . . . . . . . 15  Q.  Q.  Q.  Q.  Q.  <Q  .Q  <Q  .Q
46 simpl 102 . . . . . . . . . . . . . . 15  Q.  Q.  Q.
47 simpr 103 . . . . . . . . . . . . . . 15  Q.  Q.  Q.
48 recclnq 6376 . . . . . . . . . . . . . . . 16  Q.  *Q ` 
Q.
4948adantl 262 . . . . . . . . . . . . . . 15  Q.  Q.  *Q `  Q.
50 mulcomnqg 6367 . . . . . . . . . . . . . . . 16  Q.  Q.  .Q  .Q
5150adantl 262 . . . . . . . . . . . . . . 15  Q.  Q.  Q.  Q.  .Q  .Q
5245, 46, 47, 49, 51caovord2d 5612 . . . . . . . . . . . . . 14  Q.  Q.  <Q  .Q  *Q ` 
<Q  .Q  *Q `
53 recidnq 6377 . . . . . . . . . . . . . . . 16  Q.  .Q  *Q `  1Q
5453breq2d 3767 . . . . . . . . . . . . . . 15  Q.  .Q  *Q ` 
<Q  .Q  *Q `  .Q  *Q ` 
<Q  1Q
5554adantl 262 . . . . . . . . . . . . . 14  Q.  Q.  .Q  *Q `  <Q  .Q  *Q `  .Q  *Q `  <Q  1Q
5652, 55bitrd 177 . . . . . . . . . . . . 13  Q.  Q.  <Q  .Q  *Q ` 
<Q  1Q
5756biimpd 132 . . . . . . . . . . . 12  Q.  Q.  <Q  .Q  *Q ` 
<Q  1Q
5843, 57mpcom 32 . . . . . . . . . . 11 
<Q  .Q  *Q `  <Q  1Q
59 mulclnq 6360 . . . . . . . . . . . . . 14  Q.  *Q `  Q.  .Q  *Q `  Q.
6048, 59sylan2 270 . . . . . . . . . . . . 13  Q.  Q.  .Q  *Q `  Q.
6143, 60syl 14 . . . . . . . . . . . 12 
<Q  .Q  *Q `  Q.
62 breq1 3758 . . . . . . . . . . . . 13  .Q  *Q ` 
<Q  1Q  .Q  *Q `  <Q  1Q
6362, 15elab2g 2683 . . . . . . . . . . . 12  .Q  *Q ` 
Q.  .Q  *Q `  1st `  1P  .Q  *Q ` 
<Q  1Q
6461, 63syl 14 . . . . . . . . . . 11 
<Q  .Q  *Q `  1st `  1P  .Q  *Q ` 
<Q  1Q
6558, 64mpbird 156 . . . . . . . . . 10 
<Q  .Q  *Q `  1st `  1P
66 mulassnqg 6368 . . . . . . . . . . . . . 14  Q.  Q.  Q.  .Q  .Q  .Q  .Q
6766adantl 262 . . . . . . . . . . . . 13  Q.  Q.  Q.  Q.  Q.  .Q  .Q  .Q  .Q
6847, 46, 49, 51, 67caov12d 5624 . . . . . . . . . . . 12  Q.  Q.  .Q  .Q  *Q `  .Q  .Q  *Q `
6953oveq2d 5471 . . . . . . . . . . . . 13  Q.  .Q  .Q  *Q `  .Q  1Q
7069adantl 262 . . . . . . . . . . . 12  Q.  Q.  .Q  .Q  *Q `  .Q  1Q
71 mulidnq 6373 . . . . . . . . . . . . 13  Q.  .Q  1Q
7271adantr 261 . . . . . . . . . . . 12  Q.  Q.  .Q  1Q
7368, 70, 723eqtrrd 2074 . . . . . . . . . . 11  Q.  Q.  .Q  .Q  *Q `
7443, 73syl 14 . . . . . . . . . 10 
<Q  .Q  .Q  *Q `
75 oveq2 5463 . . . . . . . . . . . 12  .Q  *Q `  .Q  .Q  .Q  *Q `
7675eqeq2d 2048 . . . . . . . . . . 11  .Q  *Q `  .Q  .Q  .Q  *Q `
7776rspcev 2650 . . . . . . . . . 10  .Q  *Q `  1st `  1P  .Q  .Q  *Q `  1st `  1P  .Q
7865, 74, 77syl2anc 391 . . . . . . . . 9 
<Q  1st `  1P  .Q
7978a1i 9 . . . . . . . 8  1st `  <Q  1st `  1P  .Q
8079ancld 308 . . . . . . 7  1st `  <Q  <Q  1st `  1P  .Q
8180reximia 2408 . . . . . 6  1st `  <Q  1st `  <Q  1st `  1P  .Q
8241, 81syl 14 . . . . 5  P.  1st `  1st `  <Q  1st `  1P  .Q
8382ex 108 . . . 4  P.  1st `  1st ` 
<Q  1st `  1P  .Q
84 prcdnql 6467 . . . . . . 7 
<. 1st `  ,  2nd `  >.  P.  1st `  <Q  1st `
8510, 84sylan 267 . . . . . 6  P.  1st `  <Q  1st `
8685adantrd 264 . . . . 5  P.  1st `  <Q  1st `  1P  .Q  1st `
8786rexlimdva 2427 . . . 4  P.  1st `  <Q  1st `  1P  .Q  1st `
8883, 87impbid 120 . . 3  P.  1st `  1st `  <Q  1st `  1P  .Q
8935, 39, 883bitr4d 209 . 2  P.  1st `  .P.  1P  1st `
9089eqrdv 2035 1  P.  1st `  .P.  1P  1st `
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   w3a 884   wceq 1242   wcel 1390  wrex 2301    C_ wss 2911   <.cop 3370   class class class wbr 3755   ` cfv 4845  (class class class)co 5455   1stc1st 5707   2ndc2nd 5708   Q.cnq 6264   1Qc1q 6265    .Q cmq 6267   *Qcrq 6268    <Q cltq 6269   P.cnp 6275   1Pc1p 6276    .P. cmp 6278
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 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254
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 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-eprel 4017  df-id 4021  df-po 4024  df-iso 4025  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-1o 5940  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-pli 6289  df-mi 6290  df-lti 6291  df-plpq 6328  df-mpq 6329  df-enq 6331  df-nqqs 6332  df-plqqs 6333  df-mqqs 6334  df-1nqqs 6335  df-rq 6336  df-ltnqqs 6337  df-inp 6449  df-i1p 6450  df-imp 6452
This theorem is referenced by:  1idpr  6568
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