Theorem caucvgprlemdisj 7294

Description: Lemma for caucvgpr 7302. The putative limit is disjoint. (Contributed by Jim Kingdon, 27-Sep-2020.)

Hypotheses

Ref	Expression
caucvgpr.f	|- ( ph -> F : N. --> Q. )
caucvgpr.cau	|- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <Q ( ( F ` k ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) /\ ( F ` k ) <Q ( ( F ` n ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) ) ) )
caucvgpr.bnd	|- ( ph -> A. j e. N. A <Q ( F ` j ) )
caucvgpr.lim	|- L = <. { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } , { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } >.

Assertion

Ref	Expression
caucvgprlemdisj	|- ( ph -> A. s e. Q. -. ( s e. ( 1st ` L ) /\ s e. ( 2nd ` L ) ) )

Distinct variable groups:   A, j   j, F, k   F, l, j   u, F, j   n, F   j, L, k   ph, j, s, k   s, l   u, s   k, n

Allowed substitution hints:   ph( u, n, l)   A( u, k, n, s, l)   F( s)   L( u, n, s, l)

Proof of Theorem caucvgprlemdisj

Step	Hyp	Ref	Expression
1		oveq1 5673	. . . . . . . . . . . 12 |- ( l = s -> ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) = ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) )
2	1	breq1d 3861	. . . . . . . . . . 11 |- ( l = s -> ( ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) <-> ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) )
3	2	rexbidv 2382	. . . . . . . . . 10 |- ( l = s -> ( E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) <-> E. j e. N. ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) )
4		caucvgpr.lim	. . . . . . . . . . . 12 |- L = <. { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } , { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } >.
5	4	fveq2i 5321	. . . . . . . . . . 11 |- ( 1st ` L ) = ( 1st ` <. { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } , { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } >. )
6		nqex 6983	. . . . . . . . . . . . 13 |- Q. e. _V
7	6	rabex 3989	. . . . . . . . . . . 12 |- { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } e. _V
8	6	rabex 3989	. . . . . . . . . . . 12 |- { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } e. _V
9	7, 8	op1st 5931	. . . . . . . . . . 11 |- ( 1st ` <. { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } , { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } >. ) = { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) }
10	5, 9	eqtri 2109	. . . . . . . . . 10 |- ( 1st ` L ) = { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) }
11	3, 10	elrab2 2775	. . . . . . . . 9 |- ( s e. ( 1st ` L ) <-> ( s e. Q. /\ E. j e. N. ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) )
12	11	simprbi 270	. . . . . . . 8 |- ( s e. ( 1st ` L ) -> E. j e. N. ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) )
13		opeq1 3628	. . . . . . . . . . . . 13 |- ( j = k -> <. j , 1o >. = <. k , 1o >. )
14	13	eceq1d 6342	. . . . . . . . . . . 12 |- ( j = k -> [ <. j , 1o >. ] ~Q = [ <. k , 1o >. ] ~Q )
15	14	fveq2d 5322	. . . . . . . . . . 11 |- ( j = k -> ( *Q ` [ <. j , 1o >. ] ~Q ) = ( *Q ` [ <. k , 1o >. ] ~Q ) )
16	15	oveq2d 5682	. . . . . . . . . 10 |- ( j = k -> ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) = ( s +Q ( *Q ` [ <. k , 1o >. ] ~Q ) ) )
17		fveq2 5318	. . . . . . . . . 10 |- ( j = k -> ( F ` j ) = ( F ` k ) )
18	16, 17	breq12d 3864	. . . . . . . . 9 |- ( j = k -> ( ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) <-> ( s +Q ( *Q ` [ <. k , 1o >. ] ~Q ) ) <Q ( F ` k ) ) )
19	18	cbvrexv 2592	. . . . . . . 8 |- ( E. j e. N. ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) <-> E. k e. N. ( s +Q ( *Q ` [ <. k , 1o >. ] ~Q ) ) <Q ( F ` k ) )
20	12, 19	sylib 121	. . . . . . 7 |- ( s e. ( 1st ` L ) -> E. k e. N. ( s +Q ( *Q ` [ <. k , 1o >. ] ~Q ) ) <Q ( F ` k ) )
21		breq2 3855	. . . . . . . . . 10 |- ( u = s -> ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u <-> ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q s ) )
22	21	rexbidv 2382	. . . . . . . . 9 |- ( u = s -> ( E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u <-> E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q s ) )
23	4	fveq2i 5321	. . . . . . . . . 10 |- ( 2nd ` L ) = ( 2nd ` <. { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } , { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } >. )
24	7, 8	op2nd 5932	. . . . . . . . . 10 |- ( 2nd ` <. { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } , { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } >. ) = { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u }
25	23, 24	eqtri 2109	. . . . . . . . 9 |- ( 2nd ` L ) = { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u }
26	22, 25	elrab2 2775	. . . . . . . 8 |- ( s e. ( 2nd ` L ) <-> ( s e. Q. /\ E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q s ) )
27	26	simprbi 270	. . . . . . 7 |- ( s e. ( 2nd ` L ) -> E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q s )
28	20, 27	anim12i 332	. . . . . 6 |- ( ( s e. ( 1st ` L ) /\ s e. ( 2nd ` L ) ) -> ( E. k e. N. ( s +Q ( *Q ` [ <. k , 1o >. ] ~Q ) ) <Q ( F ` k ) /\ E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q s ) )
29		reeanv 2537	. . . . . 6 |- ( E. k e. N. E. j e. N. ( ( s +Q ( *Q ` [ <. k , 1o >. ] ~Q ) ) <Q ( F ` k ) /\ ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q s ) <-> ( E. k e. N. ( s +Q ( *Q ` [ <. k , 1o >. ] ~Q ) ) <Q ( F ` k ) /\ E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q s ) )
30	28, 29	sylibr 133	. . . . 5 |- ( ( s e. ( 1st ` L ) /\ s e. ( 2nd ` L ) ) -> E. k e. N. E. j e. N. ( ( s +Q ( *Q ` [ <. k , 1o >. ] ~Q ) ) <Q ( F ` k ) /\ ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q s ) )
31	30	adantl 272	. . . 4 |- ( ( ph /\ ( s e. ( 1st ` L ) /\ s e. ( 2nd ` L ) ) ) -> E. k e. N. E. j e. N. ( ( s +Q ( *Q ` [ <. k , 1o >. ] ~Q ) ) <Q ( F ` k ) /\ ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q s ) )
32		caucvgpr.f	. . . . . . . 8 |- ( ph -> F : N. --> Q. )
33	32	ad2antrr 473	. . . . . . 7 |- ( ( ( ph /\ ( s e. ( 1st ` L ) /\ s e. ( 2nd ` L ) ) ) /\ ( k e. N. /\ j e. N. ) ) -> F : N. --> Q. )
34		caucvgpr.cau	. . . . . . . 8 |- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <Q ( ( F ` k ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) /\ ( F ` k ) <Q ( ( F ` n ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) ) ) )
35	34	ad2antrr 473	. . . . . . 7 |- ( ( ( ph /\ ( s e. ( 1st ` L ) /\ s e. ( 2nd ` L ) ) ) /\ ( k e. N. /\ j e. N. ) ) -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <Q ( ( F ` k ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) /\ ( F ` k ) <Q ( ( F ` n ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) ) ) )
36		simprl 499	. . . . . . 7 |- ( ( ( ph /\ ( s e. ( 1st ` L ) /\ s e. ( 2nd ` L ) ) ) /\ ( k e. N. /\ j e. N. ) ) -> k e. N. )
37		simprr 500	. . . . . . 7 |- ( ( ( ph /\ ( s e. ( 1st ` L ) /\ s e. ( 2nd ` L ) ) ) /\ ( k e. N. /\ j e. N. ) ) -> j e. N. )
38	11	simplbi 269	. . . . . . . . 9 |- ( s e. ( 1st ` L ) -> s e. Q. )
39	38	ad2antrl 475	. . . . . . . 8 |- ( ( ph /\ ( s e. ( 1st ` L ) /\ s e. ( 2nd ` L ) ) ) -> s e. Q. )
40	39	adantr 271	. . . . . . 7 |- ( ( ( ph /\ ( s e. ( 1st ` L ) /\ s e. ( 2nd ` L ) ) ) /\ ( k e. N. /\ j e. N. ) ) -> s e. Q. )
41	33, 35, 36, 37, 40	caucvgprlemnkj 7286	. . . . . 6 |- ( ( ( ph /\ ( s e. ( 1st ` L ) /\ s e. ( 2nd ` L ) ) ) /\ ( k e. N. /\ j e. N. ) ) -> -. ( ( s +Q ( *Q ` [ <. k , 1o >. ] ~Q ) ) <Q ( F ` k ) /\ ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q s ) )
42	41	pm2.21d 585	. . . . 5 |- ( ( ( ph /\ ( s e. ( 1st ` L ) /\ s e. ( 2nd ` L ) ) ) /\ ( k e. N. /\ j e. N. ) ) -> ( ( ( s +Q ( *Q ` [ <. k , 1o >. ] ~Q ) ) <Q ( F ` k ) /\ ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q s ) -> F. ) )
43	42	rexlimdvva 2497	. . . 4 |- ( ( ph /\ ( s e. ( 1st ` L ) /\ s e. ( 2nd ` L ) ) ) -> ( E. k e. N. E. j e. N. ( ( s +Q ( *Q ` [ <. k , 1o >. ] ~Q ) ) <Q ( F ` k ) /\ ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q s ) -> F. ) )
44	31, 43	mpd 13	. . 3 |- ( ( ph /\ ( s e. ( 1st ` L ) /\ s e. ( 2nd ` L ) ) ) -> F. )
45	44	inegd 1309	. 2 |- ( ph -> -. ( s e. ( 1st ` L ) /\ s e. ( 2nd ` L ) ) )
46	45	ralrimivw 2448	1 |- ( ph -> A. s e. Q. -. ( s e. ( 1st ` L ) /\ s e. ( 2nd ` L ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   = wceq 1290   F. wfal 1295   e. wcel 1439   A.wral 2360   E.wrex 2361   {crab 2364   <.cop 3453   class class class wbr 3851   -->wf 5024   ` cfv 5028  (class class class)co 5666   1stc1st 5923   2ndc2nd 5924   1oc1o 6188   [cec 6304   N.cnpi 6892   <N clti 6895   ~Q ceq 6899   Q.cnq 6900   +Q cplq 6902   *Qcrq 6904   <Q cltq 6905

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 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3960  ax-sep 3963  ax-nul 3971  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-iinf 4416

This theorem depends on definitions:  df-bi 116  df-dc 782  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-ral 2365  df-rex 2366  df-reu 2367  df-rab 2369  df-v 2622  df-sbc 2842  df-csb 2935  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-iun 3738  df-br 3852  df-opab 3906  df-mpt 3907  df-tr 3943  df-eprel 4125  df-id 4129  df-po 4132  df-iso 4133  df-iord 4202  df-on 4204  df-suc 4207  df-iom 4419  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-1st 5925  df-2nd 5926  df-recs 6084  df-irdg 6149  df-1o 6195  df-oadd 6199  df-omul 6200  df-er 6306  df-ec 6308  df-qs 6312  df-ni 6924  df-pli 6925  df-mi 6926  df-lti 6927  df-plpq 6964  df-mpq 6965  df-enq 6967  df-nqqs 6968  df-plqqs 6969  df-mqqs 6970  df-1nqqs 6971  df-rq 6972  df-ltnqqs 6973

This theorem is referenced by:  caucvgprlemcl  7296