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Theorem recexprlem1ssl 6605
Description: The lower cut of one is a subset of the lower cut of  .P. . Lemma for recexpr 6610. (Contributed by Jim Kingdon, 27-Dec-2019.)
Hypothesis
Ref Expression
recexpr.1  <. {  |  <Q  *Q `  2nd `  } ,  {  |  <Q  *Q `  1st `  } >.
Assertion
Ref Expression
recexprlem1ssl  P.  1st `  1P  C_  1st ` 
.P.
Distinct variable groups:   ,,   ,,

Proof of Theorem recexprlem1ssl
Dummy variables  h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1prl 6536 . . . 4  1st `  1P  {  |  <Q  1Q }
21abeq2i 2145 . . 3  1st `  1P  <Q  1Q
3 rec1nq 6379 . . . . . . 7  *Q `  1Q  1Q
4 ltrnqi 6404 . . . . . . 7 
<Q  1Q  *Q `  1Q  <Q  *Q `
53, 4syl5eqbrr 3789 . . . . . 6 
<Q  1Q  1Q  <Q  *Q `
6 prop 6458 . . . . . . 7  P.  <. 1st `  ,  2nd `  >.  P.
7 prmuloc2 6548 . . . . . . 7 
<. 1st `  ,  2nd `  >.  P.  1Q  <Q  *Q `  1st `  .Q  *Q `  2nd `
86, 7sylan 267 . . . . . 6  P.  1Q  <Q  *Q `  1st `  .Q  *Q `  2nd `
95, 8sylan2 270 . . . . 5  P.  <Q  1Q  1st `  .Q  *Q `  2nd `
10 prnmaxl 6471 . . . . . . . 8 
<. 1st `  ,  2nd `  >.  P.  1st `  1st ` 
<Q
116, 10sylan 267 . . . . . . 7  P.  1st `  1st ` 
<Q
1211ad2ant2r 478 . . . . . 6  P.  <Q  1Q  1st `  .Q  *Q `  2nd `  1st `  <Q
13 elprnql 6464 . . . . . . . . . . . . . 14 
<. 1st `  ,  2nd `  >.  P.  1st `  Q.
146, 13sylan 267 . . . . . . . . . . . . 13  P.  1st `  Q.
1514ad2ant2r 478 . . . . . . . . . . . 12  P.  <Q  1Q  1st `  .Q  *Q `  2nd `  Q.
16153adant3 923 . . . . . . . . . . 11  P.  <Q  1Q  1st `  .Q  *Q `  2nd `  <Q  Q.
17 simp1r 928 . . . . . . . . . . . 12  P.  <Q  1Q  1st `  .Q  *Q `  2nd `  <Q  <Q  1Q
18 ltrelnq 6349 . . . . . . . . . . . . . 14  <Q  C_  Q.  X.  Q.
1918brel 4335 . . . . . . . . . . . . 13 
<Q  1Q  Q.  1Q  Q.
2019simpld 105 . . . . . . . . . . . 12 
<Q  1Q 
Q.
2117, 20syl 14 . . . . . . . . . . 11  P.  <Q  1Q  1st `  .Q  *Q `  2nd `  <Q  Q.
22 simp3 905 . . . . . . . . . . 11  P.  <Q  1Q  1st `  .Q  *Q `  2nd `  <Q  <Q
23 simp2r 930 . . . . . . . . . . 11  P.  <Q  1Q  1st `  .Q  *Q `  2nd `  <Q  .Q  *Q `  2nd `
24 simpr 103 . . . . . . . . . . . 12  Q.  Q.  <Q  .Q  *Q `  2nd `  <Q  .Q  *Q `  2nd `
25 ltrnqi 6404 . . . . . . . . . . . . . 14 
<Q  *Q `  <Q  *Q `
26 ltmnqg 6385 . . . . . . . . . . . . . . . 16  Q.  Q.  h  Q.  <Q  h  .Q  <Q  h  .Q
2726adantl 262 . . . . . . . . . . . . . . 15 
Q.  Q.  <Q  .Q  *Q `  2nd `  Q.  Q.  h 
Q.  <Q  h  .Q  <Q  h  .Q
28 simprl 483 . . . . . . . . . . . . . . . 16  Q.  Q.  <Q  .Q  *Q `  2nd `  <Q
2918brel 4335 . . . . . . . . . . . . . . . . 17 
<Q  Q.  Q.
3029simprd 107 . . . . . . . . . . . . . . . 16 
<Q  Q.
31 recclnq 6376 . . . . . . . . . . . . . . . 16  Q.  *Q ` 
Q.
3228, 30, 313syl 17 . . . . . . . . . . . . . . 15  Q.  Q.  <Q  .Q  *Q `  2nd `  *Q ` 
Q.
33 recclnq 6376 . . . . . . . . . . . . . . . 16  Q.  *Q ` 
Q.
3433ad2antrr 457 . . . . . . . . . . . . . . 15  Q.  Q.  <Q  .Q  *Q `  2nd `  *Q ` 
Q.
35 simplr 482 . . . . . . . . . . . . . . 15  Q.  Q.  <Q  .Q  *Q `  2nd `  Q.
36 mulcomnqg 6367 . . . . . . . . . . . . . . . 16  Q.  Q.  .Q  .Q
3736adantl 262 . . . . . . . . . . . . . . 15 
Q.  Q.  <Q  .Q  *Q `  2nd `  Q.  Q.  .Q  .Q
3827, 32, 34, 35, 37caovord2d 5612 . . . . . . . . . . . . . 14  Q.  Q.  <Q  .Q  *Q `  2nd `  *Q ` 
<Q  *Q `  *Q `  .Q  <Q  *Q `  .Q
3925, 38syl5ib 143 . . . . . . . . . . . . 13  Q.  Q.  <Q  .Q  *Q `  2nd `  <Q  *Q `  .Q  <Q  *Q `  .Q
40 1nq 6350 . . . . . . . . . . . . . . . . . 18  1Q  Q.
41 mulidnq 6373 . . . . . . . . . . . . . . . . . 18  1Q  Q.  1Q  .Q  1Q  1Q
4240, 41ax-mp 7 . . . . . . . . . . . . . . . . 17  1Q 
.Q  1Q  1Q
43 mulcomnqg 6367 . . . . . . . . . . . . . . . . . . . . . 22  Q.  *Q `  Q.  .Q  *Q `  *Q `  .Q
4433, 43mpdan 398 . . . . . . . . . . . . . . . . . . . . 21  Q.  .Q  *Q `  *Q `  .Q
45 recidnq 6377 . . . . . . . . . . . . . . . . . . . . 21  Q.  .Q  *Q `  1Q
4644, 45eqtr3d 2071 . . . . . . . . . . . . . . . . . . . 20  Q.  *Q `  .Q  1Q
47 recidnq 6377 . . . . . . . . . . . . . . . . . . . 20  Q.  .Q  *Q `  1Q
4846, 47oveqan12d 5474 . . . . . . . . . . . . . . . . . . 19  Q.  Q.  *Q `  .Q  .Q  .Q  *Q `  1Q  .Q  1Q
4948adantr 261 . . . . . . . . . . . . . . . . . 18  Q.  Q.  <Q  .Q  *Q `  2nd `  *Q `  .Q  .Q  .Q  *Q `  1Q 
.Q  1Q
50 simpll 481 . . . . . . . . . . . . . . . . . . 19  Q.  Q.  <Q  .Q  *Q `  2nd `  Q.
51 mulassnqg 6368 . . . . . . . . . . . . . . . . . . . 20  Q.  Q.  h  Q.  .Q  .Q  h  .Q  .Q  h
5251adantl 262 . . . . . . . . . . . . . . . . . . 19 
Q.  Q.  <Q  .Q  *Q `  2nd `  Q.  Q.  h 
Q.  .Q  .Q  h  .Q  .Q  h
53 recclnq 6376 . . . . . . . . . . . . . . . . . . . 20  Q.  *Q ` 
Q.
5435, 53syl 14 . . . . . . . . . . . . . . . . . . 19  Q.  Q.  <Q  .Q  *Q `  2nd `  *Q ` 
Q.
55 mulclnq 6360 . . . . . . . . . . . . . . . . . . . 20  Q.  Q.  .Q  Q.
5655adantl 262 . . . . . . . . . . . . . . . . . . 19 
Q.  Q.  <Q  .Q  *Q `  2nd `  Q.  Q.  .Q  Q.
5734, 50, 35, 37, 52, 54, 56caov4d 5627 . . . . . . . . . . . . . . . . . 18  Q.  Q.  <Q  .Q  *Q `  2nd `  *Q `  .Q  .Q  .Q  *Q `  *Q `  .Q  .Q  .Q  *Q `
5849, 57eqtr3d 2071 . . . . . . . . . . . . . . . . 17  Q.  Q.  <Q  .Q  *Q `  2nd `  1Q  .Q  1Q  *Q `  .Q  .Q  .Q  *Q `
5942, 58syl5reqr 2084 . . . . . . . . . . . . . . . 16  Q.  Q.  <Q  .Q  *Q `  2nd `  *Q `  .Q  .Q  .Q  *Q `  1Q
60 mulclnq 6360 . . . . . . . . . . . . . . . . . . 19  *Q `  Q.  Q.  *Q `  .Q 
Q.
6133, 60sylan 267 . . . . . . . . . . . . . . . . . 18  Q.  Q.  *Q `  .Q  Q.
62 mulclnq 6360 . . . . . . . . . . . . . . . . . . 19  Q.  *Q `  Q.  .Q  *Q `  Q.
6353, 62sylan2 270 . . . . . . . . . . . . . . . . . 18  Q.  Q.  .Q  *Q `  Q.
64 recmulnqg 6375 . . . . . . . . . . . . . . . . . 18  *Q `  .Q  Q.  .Q  *Q `  Q.  *Q `  *Q `  .Q  .Q  *Q `  *Q `  .Q  .Q  .Q  *Q `  1Q
6561, 63, 64syl2anc 391 . . . . . . . . . . . . . . . . 17  Q.  Q.  *Q `  *Q `  .Q  .Q  *Q `  *Q `  .Q  .Q  .Q  *Q `  1Q
6665adantr 261 . . . . . . . . . . . . . . . 16  Q.  Q.  <Q  .Q  *Q `  2nd `  *Q `  *Q `  .Q  .Q  *Q `  *Q `  .Q  .Q  .Q  *Q `  1Q
6759, 66mpbird 156 . . . . . . . . . . . . . . 15  Q.  Q.  <Q  .Q  *Q `  2nd `  *Q `  *Q `  .Q  .Q  *Q `
6867eleq1d 2103 . . . . . . . . . . . . . 14  Q.  Q.  <Q  .Q  *Q `  2nd `  *Q `  *Q `  .Q  2nd `  .Q  *Q `  2nd `
6968biimprd 147 . . . . . . . . . . . . 13  Q.  Q.  <Q  .Q  *Q `  2nd `  .Q  *Q `  2nd `  *Q `  *Q `  .Q  2nd `
70 breq2 3759 . . . . . . . . . . . . . . . . . 18  *Q `  .Q  *Q `  .Q  <Q  *Q `  .Q  <Q  *Q `  .Q
71 fveq2 5121 . . . . . . . . . . . . . . . . . . 19  *Q `  .Q  *Q `  *Q `  *Q `  .Q
7271eleq1d 2103 . . . . . . . . . . . . . . . . . 18  *Q `  .Q  *Q `  2nd `  *Q `  *Q `  .Q  2nd `
7370, 72anbi12d 442 . . . . . . . . . . . . . . . . 17  *Q `  .Q  *Q `  .Q  <Q  *Q `  2nd `  *Q `  .Q  <Q  *Q `  .Q  *Q `  *Q `  .Q  2nd `
7473spcegv 2635 . . . . . . . . . . . . . . . 16  *Q `  .Q 
Q.  *Q `  .Q  <Q  *Q `  .Q  *Q `  *Q `  .Q  2nd `  *Q `  .Q  <Q  *Q `  2nd `
7561, 74syl 14 . . . . . . . . . . . . . . 15  Q.  Q.  *Q `  .Q  <Q  *Q `  .Q  *Q `  *Q `  .Q  2nd `  *Q `  .Q  <Q  *Q `  2nd `
76 recexpr.1 . . . . . . . . . . . . . . . 16  <. {  |  <Q  *Q `  2nd `  } ,  {  |  <Q  *Q `  1st `  } >.
7776recexprlemell 6594 . . . . . . . . . . . . . . 15  *Q `  .Q  1st `  *Q `  .Q  <Q  *Q `  2nd `
7875, 77syl6ibr 151 . . . . . . . . . . . . . 14  Q.  Q.  *Q `  .Q  <Q  *Q `  .Q  *Q `  *Q `  .Q  2nd `  *Q `  .Q  1st `
7978adantr 261 . . . . . . . . . . . . 13  Q.  Q.  <Q  .Q  *Q `  2nd `  *Q `  .Q  <Q  *Q `  .Q  *Q `  *Q `  .Q  2nd `  *Q `  .Q  1st `
8039, 69, 79syl2and 279 . . . . . . . . . . . 12  Q.  Q.  <Q  .Q  *Q `  2nd `  <Q  .Q  *Q `  2nd `  *Q `  .Q  1st `
8124, 80mpd 13 . . . . . . . . . . 11  Q.  Q.  <Q  .Q  *Q `  2nd `  *Q `  .Q  1st `
8216, 21, 22, 23, 81syl22anc 1135 . . . . . . . . . 10  P.  <Q  1Q  1st `  .Q  *Q `  2nd `  <Q  *Q `  .Q  1st `
83303ad2ant3 926 . . . . . . . . . . 11  P.  <Q  1Q  1st `  .Q  *Q `  2nd `  <Q  Q.
84 mulidnq 6373 . . . . . . . . . . . . . 14  Q.  .Q  1Q
85 mulcomnqg 6367 . . . . . . . . . . . . . . 15  Q.  1Q  Q.  .Q  1Q  1Q  .Q
8640, 85mpan2 401 . . . . . . . . . . . . . 14  Q.  .Q  1Q  1Q  .Q
8784, 86eqtr3d 2071 . . . . . . . . . . . . 13  Q.  1Q  .Q
8887adantl 262 . . . . . . . . . . . 12  Q.  Q.  1Q 
.Q
89 recidnq 6377 . . . . . . . . . . . . . 14  Q.  .Q  *Q `  1Q
9089oveq1d 5470 . . . . . . . . . . . . 13  Q.  .Q  *Q `  .Q  1Q  .Q
9190adantr 261 . . . . . . . . . . . 12  Q.  Q.  .Q  *Q `  .Q  1Q  .Q
92 mulassnqg 6368 . . . . . . . . . . . . . 14  Q.  *Q `  Q.  Q.  .Q  *Q `  .Q  .Q  *Q `  .Q
9331, 92syl3an2 1168 . . . . . . . . . . . . 13  Q.  Q.  Q.  .Q  *Q `  .Q  .Q  *Q `  .Q
94933anidm12 1191 . . . . . . . . . . . 12  Q.  Q.  .Q  *Q `  .Q  .Q  *Q `  .Q
9588, 91, 943eqtr2d 2075 . . . . . . . . . . 11  Q.  Q.  .Q  *Q `  .Q
9683, 21, 95syl2anc 391 . . . . . . . . . 10  P.  <Q  1Q  1st `  .Q  *Q `  2nd `  <Q  .Q  *Q `  .Q
97 oveq2 5463 . . . . . . . . . . . 12  *Q `  .Q  .Q  .Q  *Q `  .Q
9897eqeq2d 2048 . . . . . . . . . . 11  *Q `  .Q  .Q  .Q  *Q `  .Q
9998rspcev 2650 . . . . . . . . . 10  *Q `  .Q  1st `  .Q  *Q `  .Q  1st `  .Q
10082, 96, 99syl2anc 391 . . . . . . . . 9  P.  <Q  1Q  1st `  .Q  *Q `  2nd `  <Q  1st `  .Q
1011003expia 1105 . . . . . . . 8  P.  <Q  1Q  1st `  .Q  *Q `  2nd `  <Q  1st `  .Q
102101reximdv 2414 . . . . . . 7  P.  <Q  1Q  1st `  .Q  *Q `  2nd `  1st ` 
<Q  1st `  1st `  .Q
10376recexprlempr 6604 . . . . . . . . 9  P.  P.
104 df-imp 6452 . . . . . . . . . 10  .P.  P. ,  P.  |->  <. {  Q.  |  Q.  Q.  1st `  1st `  .Q  } ,  {  Q.  |  Q.  Q.  2nd `  2nd `  .Q  } >.
105104, 55genpelvl 6495 . . . . . . . . 9  P.  P.  1st ` 
.P.  1st `  1st `  .Q
106103, 105mpdan 398 . . . . . . . 8  P.  1st `  .P.  1st `  1st `  .Q
107106ad2antrr 457 . . . . . . 7  P.  <Q  1Q  1st `  .Q  *Q `  2nd `  1st `  .P.  1st `  1st `  .Q
108102, 107sylibrd 158 . . . . . 6  P.  <Q  1Q  1st `  .Q  *Q `  2nd `  1st ` 
<Q  1st `  .P.
10912, 108mpd 13 . . . . 5  P.  <Q  1Q  1st `  .Q  *Q `  2nd `  1st `  .P.
1109, 109rexlimddv 2431 . . . 4  P.  <Q  1Q  1st `  .P.
111110ex 108 . . 3  P.  <Q  1Q  1st `  .P.
1122, 111syl5bi 141 . 2  P.  1st `  1P  1st `  .P.
113112ssrdv 2945 1  P.  1st `  1P  C_  1st ` 
.P.
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   w3a 884   wceq 1242  wex 1378   wcel 1390   {cab 2023  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-bndl 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-2o 5941  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-enq0 6407  df-nq0 6408  df-0nq0 6409  df-plq0 6410  df-mq0 6411  df-inp 6449  df-i1p 6450  df-imp 6452
This theorem is referenced by:  recexprlemex  6609
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