Theorem caucvgprprlemnjltk 7916

Description: Lemma for caucvgprpr 7937. Part of disjointness. (Contributed by Jim Kingdon, 12-Feb-2021.)

Hypotheses

Ref	Expression
caucvgprpr.f	|- ( ph -> F : N. --> P. )
caucvgprpr.cau	|- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <P ( ( F ` k ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) /\ ( F ` k ) <P ( ( F ` n ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) ) ) )
caucvgprprlemnkj.k	|- ( ph -> K e. N. )
caucvgprprlemnkj.j	|- ( ph -> J e. N. )
caucvgprprlemnkj.s	|- ( ph -> S e. Q. )

Assertion

Ref	Expression
caucvgprprlemnjltk	|- ( ( ph /\ J <N K ) -> -. ( <. { p | p <Q ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) } , { q | ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` K ) /\ ( ( F ` J ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q S } , { q | S <Q q } >. ) )

Distinct variable groups:   k, F, n   J, l, p   J, q, u   K, p   K, q   S, p   S, q   u, k   k, l, n   u, n

Allowed substitution hints:   ph( u, k, n, q, p, l)   S( u, k, n, l)   F( u, q, p, l)   J( k, n)   K( u, k, n, l)

Proof of Theorem caucvgprprlemnjltk

Step	Hyp	Ref	Expression
1		ltsopr 7821	. . 3 |- <P Or P.
2		ltrelpr 7730	. . 3 |- <P C_ ( P. X. P. )
3	1, 2	son2lpi 5135	. 2 |- -. ( <. { p | p <Q S } , { q | S <Q q } >. <P ( ( F ` J ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) /\ ( ( F ` J ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q S } , { q | S <Q q } >. )
4		caucvgprprlemnkj.s	. . . . . . . 8 |- ( ph -> S e. Q. )
5	4	ad2antrr 488	. . . . . . 7 |- ( ( ( ph /\ J <N K ) /\ ( <. { p | p <Q ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) } , { q | ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` K ) /\ ( ( F ` J ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q S } , { q | S <Q q } >. ) ) -> S e. Q. )
6		caucvgprprlemnkj.k	. . . . . . . . 9 |- ( ph -> K e. N. )
7	6	ad2antrr 488	. . . . . . . 8 |- ( ( ( ph /\ J <N K ) /\ ( <. { p | p <Q ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) } , { q | ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` K ) /\ ( ( F ` J ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q S } , { q | S <Q q } >. ) ) -> K e. N. )
8		nnnq 7647	. . . . . . . 8 |- ( K e. N. -> [ <. K , 1o >. ] ~Q e. Q. )
9		recclnq 7617	. . . . . . . 8 |- ( [ <. K , 1o >. ] ~Q e. Q. -> ( *Q ` [ <. K , 1o >. ] ~Q ) e. Q. )
10	7, 8, 9	3syl 17	. . . . . . 7 |- ( ( ( ph /\ J <N K ) /\ ( <. { p | p <Q ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) } , { q | ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` K ) /\ ( ( F ` J ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q S } , { q | S <Q q } >. ) ) -> ( *Q ` [ <. K , 1o >. ] ~Q ) e. Q. )
11		ltaddnq 7632	. . . . . . 7 |- ( ( S e. Q. /\ ( *Q ` [ <. K , 1o >. ] ~Q ) e. Q. ) -> S <Q ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) )
12	5, 10, 11	syl2anc 411	. . . . . 6 |- ( ( ( ph /\ J <N K ) /\ ( <. { p | p <Q ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) } , { q | ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` K ) /\ ( ( F ` J ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q S } , { q | S <Q q } >. ) ) -> S <Q ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) )
13		ltnqpri 7819	. . . . . 6 |- ( S <Q ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) -> <. { p | p <Q S } , { q | S <Q q } >. <P <. { p | p <Q ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) } , { q | ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q q } >. )
14	12, 13	syl 14	. . . . 5 |- ( ( ( ph /\ J <N K ) /\ ( <. { p | p <Q ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) } , { q | ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` K ) /\ ( ( F ` J ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q S } , { q | S <Q q } >. ) ) -> <. { p | p <Q S } , { q | S <Q q } >. <P <. { p | p <Q ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) } , { q | ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q q } >. )
15		simprl 531	. . . . . 6 |- ( ( ( ph /\ J <N K ) /\ ( <. { p | p <Q ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) } , { q | ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` K ) /\ ( ( F ` J ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q S } , { q | S <Q q } >. ) ) -> <. { p | p <Q ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) } , { q | ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` K ) )
16		caucvgprpr.f	. . . . . . . . 9 |- ( ph -> F : N. --> P. )
17		caucvgprpr.cau	. . . . . . . . 9 |- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <P ( ( F ` k ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) /\ ( F ` k ) <P ( ( F ` n ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) ) ) )
18	16, 17	caucvgprprlemval 7913	. . . . . . . 8 |- ( ( ph /\ J <N K ) -> ( ( F ` J ) <P ( ( F ` K ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) /\ ( F ` K ) <P ( ( F ` J ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) ) )
19	18	simprd 114	. . . . . . 7 |- ( ( ph /\ J <N K ) -> ( F ` K ) <P ( ( F ` J ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) )
20	19	adantr 276	. . . . . 6 |- ( ( ( ph /\ J <N K ) /\ ( <. { p | p <Q ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) } , { q | ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` K ) /\ ( ( F ` J ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q S } , { q | S <Q q } >. ) ) -> ( F ` K ) <P ( ( F ` J ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) )
21	1, 2	sotri 5134	. . . . . 6 |- ( ( <. { p | p <Q ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) } , { q | ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` K ) /\ ( F ` K ) <P ( ( F ` J ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) ) -> <. { p | p <Q ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) } , { q | ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q q } >. <P ( ( F ` J ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) )
22	15, 20, 21	syl2anc 411	. . . . 5 |- ( ( ( ph /\ J <N K ) /\ ( <. { p | p <Q ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) } , { q | ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` K ) /\ ( ( F ` J ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q S } , { q | S <Q q } >. ) ) -> <. { p | p <Q ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) } , { q | ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q q } >. <P ( ( F ` J ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) )
23	1, 2	sotri 5134	. . . . 5 |- ( ( <. { p | p <Q S } , { q | S <Q q } >. <P <. { p | p <Q ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) } , { q | ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q q } >. /\ <. { p | p <Q ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) } , { q | ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q q } >. <P ( ( F ` J ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) ) -> <. { p | p <Q S } , { q | S <Q q } >. <P ( ( F ` J ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) )
24	14, 22, 23	syl2anc 411	. . . 4 |- ( ( ( ph /\ J <N K ) /\ ( <. { p | p <Q ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) } , { q | ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` K ) /\ ( ( F ` J ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q S } , { q | S <Q q } >. ) ) -> <. { p | p <Q S } , { q | S <Q q } >. <P ( ( F ` J ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) )
25		simprr 533	. . . 4 |- ( ( ( ph /\ J <N K ) /\ ( <. { p | p <Q ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) } , { q | ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` K ) /\ ( ( F ` J ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q S } , { q | S <Q q } >. ) ) -> ( ( F ` J ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q S } , { q | S <Q q } >. )
26	24, 25	jca 306	. . 3 |- ( ( ( ph /\ J <N K ) /\ ( <. { p | p <Q ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) } , { q | ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` K ) /\ ( ( F ` J ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q S } , { q | S <Q q } >. ) ) -> ( <. { p | p <Q S } , { q | S <Q q } >. <P ( ( F ` J ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) /\ ( ( F ` J ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q S } , { q | S <Q q } >. ) )
27	26	ex 115	. 2 |- ( ( ph /\ J <N K ) -> ( ( <. { p | p <Q ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) } , { q | ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` K ) /\ ( ( F ` J ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q S } , { q | S <Q q } >. ) -> ( <. { p | p <Q S } , { q | S <Q q } >. <P ( ( F ` J ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) /\ ( ( F ` J ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q S } , { q | S <Q q } >. ) ) )
28	3, 27	mtoi 670	1 |- ( ( ph /\ J <N K ) -> -. ( <. { p | p <Q ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) } , { q | ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` K ) /\ ( ( F ` J ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q S } , { q | S <Q q } >. ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   e. wcel 2201   {cab 2216   A.wral 2509   <.cop 3673   class class class wbr 4089   -->wf 5324   ` cfv 5328  (class class class)co 6023   1oc1o 6580   [cec 6705   N.cnpi 7497   <N clti 7500   ~Q ceq 7504   Q.cnq 7505   +Q cplq 7507   *Qcrq 7509   <Q cltq 7510   P.cnp 7516   +P. cpp 7518   <P cltp 7520

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 2203  ax-14 2204  ax-ext 2212  ax-coll 4205  ax-sep 4208  ax-nul 4216  ax-pow 4266  ax-pr 4301  ax-un 4532  ax-setind 4637  ax-iinf 4688

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 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-ral 2514  df-rex 2515  df-reu 2516  df-rab 2518  df-v 2803  df-sbc 3031  df-csb 3127  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-nul 3494  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-int 3930  df-iun 3973  df-br 4090  df-opab 4152  df-mpt 4153  df-tr 4189  df-eprel 4388  df-id 4392  df-po 4395  df-iso 4396  df-iord 4465  df-on 4467  df-suc 4470  df-iom 4691  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-res 4739  df-ima 4740  df-iota 5288  df-fun 5330  df-fn 5331  df-f 5332  df-f1 5333  df-fo 5334  df-f1o 5335  df-fv 5336  df-ov 6026  df-oprab 6027  df-mpo 6028  df-1st 6308  df-2nd 6309  df-recs 6476  df-irdg 6541  df-1o 6587  df-oadd 6591  df-omul 6592  df-er 6707  df-ec 6709  df-qs 6713  df-ni 7529  df-pli 7530  df-mi 7531  df-lti 7532  df-plpq 7569  df-mpq 7570  df-enq 7572  df-nqqs 7573  df-plqqs 7574  df-mqqs 7575  df-1nqqs 7576  df-rq 7577  df-ltnqqs 7578  df-inp 7691  df-iltp 7695

This theorem is referenced by:  caucvgprprlemnkj  7917