Theorem caucvgprprlemloccalc 7434

Description: Lemma for caucvgprpr 7462. Rearranging some expressions for caucvgprprlemloc 7453. (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 7115	. . . . . . 7 |- <Q C_ ( Q. X. Q. )
3	2	brel 4549	. . . . . 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 7172	. . . . 5 |- ( M e. N. -> [ <. M , 1o >. ] ~Q e. Q. )
8		recclnq 7142	. . . . 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 7125	. . . 4 |- ( ( S e. Q. /\ ( *Q ` [ <. M , 1o >. ] ~Q ) e. Q. ) -> ( S +Q ( *Q ` [ <. M , 1o >. ] ~Q ) ) e. Q. )
11	5, 9, 10	syl2anc 406	. . 3 |- ( ph -> ( S +Q ( *Q ` [ <. M , 1o >. ] ~Q ) ) e. Q. )
12		addnqpr 7311	. . 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 406	. 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 7132	. . . . 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 1197	. . . 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 7154	. . . . . . . . 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 1198	. . . . . . . 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 427	. . . . . . 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 7148	. . . . . . . 8 |- <Q Or Q.
23	22, 2	sotri 4890	. . . . . . 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 406	. . . . . 6 |- ( ph -> ( ( *Q ` [ <. M , 1o >. ] ~Q ) +Q ( *Q ` [ <. M , 1o >. ] ~Q ) ) <Q Y )
25		ltanqi 7152	. . . . . 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 406	. . . . 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 3917	. . . 4 |- ( ph -> ( S +Q ( ( *Q ` [ <. M , 1o >. ] ~Q ) +Q ( *Q ` [ <. M , 1o >. ] ~Q ) ) ) <Q T )
29	15, 28	eqbrtrd 3913	. . 3 |- ( ph -> ( ( S +Q ( *Q ` [ <. M , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. M , 1o >. ] ~Q ) ) <Q T )
30		ltnqpri 7344	. . 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 3915	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 1312   e. wcel 1461   {cab 2099   <.cop 3494   class class class wbr 3893   ` cfv 5079  (class class class)co 5726   1oc1o 6258   [cec 6379   N.cnpi 7022   ~Q ceq 7029   Q.cnq 7030   +Q cplq 7032   *Qcrq 7034   <Q cltq 7035   +P. cpp 7043   <P cltp 7045

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 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-coll 4001  ax-sep 4004  ax-nul 4012  ax-pow 4056  ax-pr 4089  ax-un 4313  ax-setind 4410  ax-iinf 4460

This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 944  df-3an 945  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ne 2281  df-ral 2393  df-rex 2394  df-reu 2395  df-rab 2397  df-v 2657  df-sbc 2877  df-csb 2970  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-nul 3328  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-int 3736  df-iun 3779  df-br 3894  df-opab 3948  df-mpt 3949  df-tr 3985  df-eprel 4169  df-id 4173  df-po 4176  df-iso 4177  df-iord 4246  df-on 4248  df-suc 4251  df-iom 4463  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-res 4509  df-ima 4510  df-iota 5044  df-fun 5081  df-fn 5082  df-f 5083  df-f1 5084  df-fo 5085  df-f1o 5086  df-fv 5087  df-ov 5729  df-oprab 5730  df-mpo 5731  df-1st 5990  df-2nd 5991  df-recs 6154  df-irdg 6219  df-1o 6265  df-2o 6266  df-oadd 6269  df-omul 6270  df-er 6381  df-ec 6383  df-qs 6387  df-ni 7054  df-pli 7055  df-mi 7056  df-lti 7057  df-plpq 7094  df-mpq 7095  df-enq 7097  df-nqqs 7098  df-plqqs 7099  df-mqqs 7100  df-1nqqs 7101  df-rq 7102  df-ltnqqs 7103  df-enq0 7174  df-nq0 7175  df-0nq0 7176  df-plq0 7177  df-mq0 7178  df-inp 7216  df-iplp 7218  df-iltp 7220

This theorem is referenced by:  caucvgprprlemloc  7453