Theorem caucvgprprlemloccalc 8051

Description: Lemma for caucvgprpr 8079. Rearranging some expressions for caucvgprprlemloc 8070. (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 7732	. . . . . . 7 |- <Q C_ ( Q. X. Q. )
3	2	brel 4827	. . . . . 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 112	. . . 4 |- ( ph -> S e. Q. )
6		caucvgprprlemloccalc.m	. . . . 5 |- ( ph -> M e. N. )
7		nnnq 7789	. . . . 5 |- ( M e. N. -> [ <. M , 1o >. ] ~Q e. Q. )
8		recclnq 7759	. . . . 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 7742	. . . 4 |- ( ( S e. Q. /\ ( *Q ` [ <. M , 1o >. ] ~Q ) e. Q. ) -> ( S +Q ( *Q ` [ <. M , 1o >. ] ~Q ) ) e. Q. )
11	5, 9, 10	syl2anc 415	. . 3 |- ( ph -> ( S +Q ( *Q ` [ <. M , 1o >. ] ~Q ) ) e. Q. )
12		addnqpr 7928	. . 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 415	. 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 7749	. . . . 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 1278	. . . 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 7771	. . . . . . . . 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 1279	. . . . . . . 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 437	. . . . . . 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 7765	. . . . . . . 8 |- <Q Or Q.
23	22, 2	sotri 5183	. . . . . . 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 415	. . . . . 6 |- ( ph -> ( ( *Q ` [ <. M , 1o >. ] ~Q ) +Q ( *Q ` [ <. M , 1o >. ] ~Q ) ) <Q Y )
25		ltanqi 7769	. . . . . 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 415	. . . . 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 4156	. . . 4 |- ( ph -> ( S +Q ( ( *Q ` [ <. M , 1o >. ] ~Q ) +Q ( *Q ` [ <. M , 1o >. ] ~Q ) ) ) <Q T )
29	15, 28	eqbrtrd 4152	. . 3 |- ( ph -> ( ( S +Q ( *Q ` [ <. M , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. M , 1o >. ] ~Q ) ) <Q T )
30		ltnqpri 7961	. . 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 4154	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 } >. )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   {cab 2224   <.cop 3712   class class class wbr 4130   ` cfv 5377  (class class class)co 6085   1oc1o 6680   [cec 6805   N.cnpi 7639   ~Q ceq 7646   Q.cnq 7647   +Q cplq 7649   *Qcrq 7651   <Q cltq 7652   +P. cpp 7660   <P cltp 7662

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-eprel 4434  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-1o 6687  df-2o 6688  df-oadd 6691  df-omul 6692  df-er 6807  df-ec 6809  df-qs 6813  df-ni 7671  df-pli 7672  df-mi 7673  df-lti 7674  df-plpq 7711  df-mpq 7712  df-enq 7714  df-nqqs 7715  df-plqqs 7716  df-mqqs 7717  df-1nqqs 7718  df-rq 7719  df-ltnqqs 7720  df-enq0 7791  df-nq0 7792  df-0nq0 7793  df-plq0 7794  df-mq0 7795  df-inp 7833  df-iplp 7835  df-iltp 7837

This theorem is used by:  caucvgprprlemloc  8070