Theorem caucvgprprlemloccalc 7492

Description: Lemma for caucvgprpr 7520. Rearranging some expressions for caucvgprprlemloc 7511. (Contributed by Jim Kingdon, 8-Feb-2021.)

Hypotheses

Ref	Expression
caucvgprprlemloccalc.st	|- ( ph -> S <Q T )
caucvgprprlemloccalc.y	|- ( ph -> Y e. Q. )
caucvgprprlemloccalc.syt	|- ( ph -> ( S +Q Y ) = T )
caucvgprprlemloccalc.x	|- ( ph -> X e. Q. )
caucvgprprlemloccalc.xxy	|- ( ph -> ( X +Q X ) <Q Y )
caucvgprprlemloccalc.m	|- ( ph -> M e. N. )
caucvgprprlemloccalc.mx	|- ( ph -> ( *Q ` [ <. M , 1o >. ] ~Q ) <Q X )

Assertion

Ref	Expression
caucvgprprlemloccalc	|- ( ph -> ( <. { l | l <Q ( S +Q ( *Q ` [ <. M , 1o >. ] ~Q ) ) } , { u | ( S +Q ( *Q ` [ <. M , 1o >. ] ~Q ) ) <Q u } >. +P. <. { l | l <Q ( *Q ` [ <. M , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. M , 1o >. ] ~Q ) <Q u } >. ) <P <. { l | l <Q T } , { u | T <Q u } >. )

Distinct variable groups:   M, l, u   S, l, u   T, l, u

Allowed substitution hints:   ph( u, l)   X( u, l)   Y( u, l)

Proof of Theorem caucvgprprlemloccalc

Step	Hyp	Ref	Expression
1		caucvgprprlemloccalc.st	. . . . . 6 |- ( ph -> S <Q T )
2		ltrelnq 7173	. . . . . . 7 |- <Q C_ ( Q. X. Q. )
3	2	brel 4591	. . . . . 6 |- ( S <Q T -> ( S e. Q. /\ T e. Q. ) )
4	1, 3	syl 14	. . . . 5 |- ( ph -> ( S e. Q. /\ T e. Q. ) )
5	4	simpld 111	. . . 4 |- ( ph -> S e. Q. )
6		caucvgprprlemloccalc.m	. . . . 5 |- ( ph -> M e. N. )
7		nnnq 7230	. . . . 5 |- ( M e. N. -> [ <. M , 1o >. ] ~Q e. Q. )
8		recclnq 7200	. . . . 5 |- ( [ <. M , 1o >. ] ~Q e. Q. -> ( *Q ` [ <. M , 1o >. ] ~Q ) e. Q. )
9	6, 7, 8	3syl 17	. . . 4 |- ( ph -> ( *Q ` [ <. M , 1o >. ] ~Q ) e. Q. )
10		addclnq 7183	. . . 4 |- ( ( S e. Q. /\ ( *Q ` [ <. M , 1o >. ] ~Q ) e. Q. ) -> ( S +Q ( *Q ` [ <. M , 1o >. ] ~Q ) ) e. Q. )
11	5, 9, 10	syl2anc 408	. . 3 |- ( ph -> ( S +Q ( *Q ` [ <. M , 1o >. ] ~Q ) ) e. Q. )
12		addnqpr 7369	. . 3 |- ( ( ( S +Q ( *Q ` [ <. M , 1o >. ] ~Q ) ) e. Q. /\ ( *Q ` [ <. M , 1o >. ] ~Q ) e. Q. ) -> <. { l | l <Q ( ( S +Q ( *Q ` [ <. M , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. M , 1o >. ] ~Q ) ) } , { u | ( ( S +Q ( *Q ` [ <. M , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. M , 1o >. ] ~Q ) ) <Q u } >. = ( <. { l | l <Q ( S +Q ( *Q ` [ <. M , 1o >. ] ~Q ) ) } , { u | ( S +Q ( *Q ` [ <. M , 1o >. ] ~Q ) ) <Q u } >. +P. <. { l | l <Q ( *Q ` [ <. M , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. M , 1o >. ] ~Q ) <Q u } >. ) )
13	11, 9, 12	syl2anc 408	. 2 |- ( ph -> <. { l | l <Q ( ( S +Q ( *Q ` [ <. M , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. M , 1o >. ] ~Q ) ) } , { u | ( ( S +Q ( *Q ` [ <. M , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. M , 1o >. ] ~Q ) ) <Q u } >. = ( <. { l | l <Q ( S +Q ( *Q ` [ <. M , 1o >. ] ~Q ) ) } , { u | ( S +Q ( *Q ` [ <. M , 1o >. ] ~Q ) ) <Q u } >. +P. <. { l | l <Q ( *Q ` [ <. M , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. M , 1o >. ] ~Q ) <Q u } >. ) )
14		addassnqg 7190	. . . . 5 |- ( ( S e. Q. /\ ( *Q ` [ <. M , 1o >. ] ~Q ) e. Q. /\ ( *Q ` [ <. M , 1o >. ] ~Q ) e. Q. ) -> ( ( S +Q ( *Q ` [ <. M , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. M , 1o >. ] ~Q ) ) = ( S +Q ( ( *Q ` [ <. M , 1o >. ] ~Q ) +Q ( *Q ` [ <. M , 1o >. ] ~Q ) ) ) )
15	5, 9, 9, 14	syl3anc 1216	. . . 4 |- ( ph -> ( ( S +Q ( *Q ` [ <. M , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. M , 1o >. ] ~Q ) ) = ( S +Q ( ( *Q ` [ <. M , 1o >. ] ~Q ) +Q ( *Q ` [ <. M , 1o >. ] ~Q ) ) ) )
16		caucvgprprlemloccalc.mx	. . . . . . . 8 |- ( ph -> ( *Q ` [ <. M , 1o >. ] ~Q ) <Q X )
17		caucvgprprlemloccalc.x	. . . . . . . . 9 |- ( ph -> X e. Q. )
18		lt2addnq 7212	. . . . . . . . 9 |- ( ( ( ( *Q ` [ <. M , 1o >. ] ~Q ) e. Q. /\ X e. Q. ) /\ ( ( *Q ` [ <. M , 1o >. ] ~Q ) e. Q. /\ X e. Q. ) ) -> ( ( ( *Q ` [ <. M , 1o >. ] ~Q ) <Q X /\ ( *Q ` [ <. M , 1o >. ] ~Q ) <Q X ) -> ( ( *Q ` [ <. M , 1o >. ] ~Q ) +Q ( *Q ` [ <. M , 1o >. ] ~Q ) ) <Q ( X +Q X ) ) )
19	9, 17, 9, 17, 18	syl22anc 1217	. . . . . . . 8 |- ( ph -> ( ( ( *Q ` [ <. M , 1o >. ] ~Q ) <Q X /\ ( *Q ` [ <. M , 1o >. ] ~Q ) <Q X ) -> ( ( *Q ` [ <. M , 1o >. ] ~Q ) +Q ( *Q ` [ <. M , 1o >. ] ~Q ) ) <Q ( X +Q X ) ) )
20	16, 16, 19	mp2and 429	. . . . . . 7 |- ( ph -> ( ( *Q ` [ <. M , 1o >. ] ~Q ) +Q ( *Q ` [ <. M , 1o >. ] ~Q ) ) <Q ( X +Q X ) )
21		caucvgprprlemloccalc.xxy	. . . . . . 7 |- ( ph -> ( X +Q X ) <Q Y )
22		ltsonq 7206	. . . . . . . 8 |- <Q Or Q.
23	22, 2	sotri 4934	. . . . . . 7 |- ( ( ( ( *Q ` [ <. M , 1o >. ] ~Q ) +Q ( *Q ` [ <. M , 1o >. ] ~Q ) ) <Q ( X +Q X ) /\ ( X +Q X ) <Q Y ) -> ( ( *Q ` [ <. M , 1o >. ] ~Q ) +Q ( *Q ` [ <. M , 1o >. ] ~Q ) ) <Q Y )
24	20, 21, 23	syl2anc 408	. . . . . 6 |- ( ph -> ( ( *Q ` [ <. M , 1o >. ] ~Q ) +Q ( *Q ` [ <. M , 1o >. ] ~Q ) ) <Q Y )
25		ltanqi 7210	. . . . . 6 |- ( ( ( ( *Q ` [ <. M , 1o >. ] ~Q ) +Q ( *Q ` [ <. M , 1o >. ] ~Q ) ) <Q Y /\ S e. Q. ) -> ( S +Q ( ( *Q ` [ <. M , 1o >. ] ~Q ) +Q ( *Q ` [ <. M , 1o >. ] ~Q ) ) ) <Q ( S +Q Y ) )
26	24, 5, 25	syl2anc 408	. . . . 5 |- ( ph -> ( S +Q ( ( *Q ` [ <. M , 1o >. ] ~Q ) +Q ( *Q ` [ <. M , 1o >. ] ~Q ) ) ) <Q ( S +Q Y ) )
27		caucvgprprlemloccalc.syt	. . . . 5 |- ( ph -> ( S +Q Y ) = T )
28	26, 27	breqtrd 3954	. . . 4 |- ( ph -> ( S +Q ( ( *Q ` [ <. M , 1o >. ] ~Q ) +Q ( *Q ` [ <. M , 1o >. ] ~Q ) ) ) <Q T )
29	15, 28	eqbrtrd 3950	. . 3 |- ( ph -> ( ( S +Q ( *Q ` [ <. M , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. M , 1o >. ] ~Q ) ) <Q T )
30		ltnqpri 7402	. . 3 |- ( ( ( S +Q ( *Q ` [ <. M , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. M , 1o >. ] ~Q ) ) <Q T -> <. { l | l <Q ( ( S +Q ( *Q ` [ <. M , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. M , 1o >. ] ~Q ) ) } , { u | ( ( S +Q ( *Q ` [ <. M , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. M , 1o >. ] ~Q ) ) <Q u } >. <P <. { l | l <Q T } , { u | T <Q u } >. )
31	29, 30	syl 14	. 2 |- ( ph -> <. { l | l <Q ( ( S +Q ( *Q ` [ <. M , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. M , 1o >. ] ~Q ) ) } , { u | ( ( S +Q ( *Q ` [ <. M , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. M , 1o >. ] ~Q ) ) <Q u } >. <P <. { l | l <Q T } , { u | T <Q u } >. )
32	13, 31	eqbrtrrd 3952	1 |- ( ph -> ( <. { l | l <Q ( S +Q ( *Q ` [ <. M , 1o >. ] ~Q ) ) } , { u | ( S +Q ( *Q ` [ <. M , 1o >. ] ~Q ) ) <Q u } >. +P. <. { l | l <Q ( *Q ` [ <. M , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. M , 1o >. ] ~Q ) <Q u } >. ) <P <. { l | l <Q T } , { u | T <Q u } >. )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   e. wcel 1480   {cab 2125   <.cop 3530   class class class wbr 3929   ` cfv 5123  (class class class)co 5774   1oc1o 6306   [cec 6427   N.cnpi 7080   ~Q ceq 7087   Q.cnq 7088   +Q cplq 7090   *Qcrq 7092   <Q cltq 7093   +P. cpp 7101   <P cltp 7103

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-2o 6314  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7112  df-pli 7113  df-mi 7114  df-lti 7115  df-plpq 7152  df-mpq 7153  df-enq 7155  df-nqqs 7156  df-plqqs 7157  df-mqqs 7158  df-1nqqs 7159  df-rq 7160  df-ltnqqs 7161  df-enq0 7232  df-nq0 7233  df-0nq0 7234  df-plq0 7235  df-mq0 7236  df-inp 7274  df-iplp 7276  df-iltp 7278

This theorem is referenced by:  caucvgprprlemloc  7511