Theorem caucvgprprlemloccalc 7903

Description: Lemma for caucvgprpr 7931. Rearranging some expressions for caucvgprprlemloc 7922. (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 7584	. . . . . . 7 |- <Q C_ ( Q. X. Q. )
3	2	brel 4778	. . . . . 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 7641	. . . . 5 |- ( M e. N. -> [ <. M , 1o >. ] ~Q e. Q. )
8		recclnq 7611	. . . . 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 7594	. . . 4 |- ( ( S e. Q. /\ ( *Q ` [ <. M , 1o >. ] ~Q ) e. Q. ) -> ( S +Q ( *Q ` [ <. M , 1o >. ] ~Q ) ) e. Q. )
11	5, 9, 10	syl2anc 411	. . 3 |- ( ph -> ( S +Q ( *Q ` [ <. M , 1o >. ] ~Q ) ) e. Q. )
12		addnqpr 7780	. . 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 411	. 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 7601	. . . . 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 1273	. . . 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 7623	. . . . . . . . 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 1274	. . . . . . . 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 433	. . . . . . 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 7617	. . . . . . . 8 |- <Q Or Q.
23	22, 2	sotri 5132	. . . . . . 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 411	. . . . . 6 |- ( ph -> ( ( *Q ` [ <. M , 1o >. ] ~Q ) +Q ( *Q ` [ <. M , 1o >. ] ~Q ) ) <Q Y )
25		ltanqi 7621	. . . . . 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 411	. . . . 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 4114	. . . 4 |- ( ph -> ( S +Q ( ( *Q ` [ <. M , 1o >. ] ~Q ) +Q ( *Q ` [ <. M , 1o >. ] ~Q ) ) ) <Q T )
29	15, 28	eqbrtrd 4110	. . 3 |- ( ph -> ( ( S +Q ( *Q ` [ <. M , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. M , 1o >. ] ~Q ) ) <Q T )
30		ltnqpri 7813	. . 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 4112	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 104   = wceq 1397   e. wcel 2202   {cab 2217   <.cop 3672   class class class wbr 4088   ` cfv 5326  (class class class)co 6017   1oc1o 6574   [cec 6699   N.cnpi 7491   ~Q ceq 7498   Q.cnq 7499   +Q cplq 7501   *Qcrq 7503   <Q cltq 7504   +P. cpp 7512   <P cltp 7514

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-eprel 4386  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-irdg 6535  df-1o 6581  df-2o 6582  df-oadd 6585  df-omul 6586  df-er 6701  df-ec 6703  df-qs 6707  df-ni 7523  df-pli 7524  df-mi 7525  df-lti 7526  df-plpq 7563  df-mpq 7564  df-enq 7566  df-nqqs 7567  df-plqqs 7568  df-mqqs 7569  df-1nqqs 7570  df-rq 7571  df-ltnqqs 7572  df-enq0 7643  df-nq0 7644  df-0nq0 7645  df-plq0 7646  df-mq0 7647  df-inp 7685  df-iplp 7687  df-iltp 7689

This theorem is referenced by:  caucvgprprlemloc  7922