Theorem caucvgprlemdisj 7937

Description: Lemma for caucvgpr 7945. 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 6035	. . . . . . . . . . . 12 |- ( l = s -> ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) = ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) )
2	1	breq1d 4103	. . . . . . . . . . 11 |- ( l = s -> ( ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) <-> ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) )
3	2	rexbidv 2534	. . . . . . . . . 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 5651	. . . . . . . . . . 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 7626	. . . . . . . . . . . . 13 |- Q. e. _V
7	6	rabex 4239	. . . . . . . . . . . 12 |- { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } e. _V
8	6	rabex 4239	. . . . . . . . . . . 12 |- { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } e. _V
9	7, 8	op1st 6318	. . . . . . . . . . 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 2252	. . . . . . . . . 10 |- ( 1st ` L ) = { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) }
11	3, 10	elrab2 2966	. . . . . . . . 9 |- ( s e. ( 1st ` L ) <-> ( s e. Q. /\ E. j e. N. ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) )
12	11	simprbi 275	. . . . . . . 8 |- ( s e. ( 1st ` L ) -> E. j e. N. ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) )
13		opeq1 3867	. . . . . . . . . . . . 13 |- ( j = k -> <. j , 1o >. = <. k , 1o >. )
14	13	eceq1d 6781	. . . . . . . . . . . 12 |- ( j = k -> [ <. j , 1o >. ] ~Q = [ <. k , 1o >. ] ~Q )
15	14	fveq2d 5652	. . . . . . . . . . 11 |- ( j = k -> ( *Q ` [ <. j , 1o >. ] ~Q ) = ( *Q ` [ <. k , 1o >. ] ~Q ) )
16	15	oveq2d 6044	. . . . . . . . . 10 |- ( j = k -> ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) = ( s +Q ( *Q ` [ <. k , 1o >. ] ~Q ) ) )
17		fveq2 5648	. . . . . . . . . 10 |- ( j = k -> ( F ` j ) = ( F ` k ) )
18	16, 17	breq12d 4106	. . . . . . . . 9 |- ( j = k -> ( ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) <-> ( s +Q ( *Q ` [ <. k , 1o >. ] ~Q ) ) <Q ( F ` k ) ) )
19	18	cbvrexv 2769	. . . . . . . 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 122	. . . . . . 7 |- ( s e. ( 1st ` L ) -> E. k e. N. ( s +Q ( *Q ` [ <. k , 1o >. ] ~Q ) ) <Q ( F ` k ) )
21		breq2 4097	. . . . . . . . . 10 |- ( u = s -> ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u <-> ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q s ) )
22	21	rexbidv 2534	. . . . . . . . 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 5651	. . . . . . . . . 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 6319	. . . . . . . . . 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 2252	. . . . . . . . 9 |- ( 2nd ` L ) = { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u }
26	22, 25	elrab2 2966	. . . . . . . 8 |- ( s e. ( 2nd ` L ) <-> ( s e. Q. /\ E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q s ) )
27	26	simprbi 275	. . . . . . 7 |- ( s e. ( 2nd ` L ) -> E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q s )
28	20, 27	anim12i 338	. . . . . 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 2704	. . . . . 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 134	. . . . 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 277	. . . 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 488	. . . . . . 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 488	. . . . . . 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 531	. . . . . . 7 |- ( ( ( ph /\ ( s e. ( 1st ` L ) /\ s e. ( 2nd ` L ) ) ) /\ ( k e. N. /\ j e. N. ) ) -> k e. N. )
37		simprr 533	. . . . . . 7 |- ( ( ( ph /\ ( s e. ( 1st ` L ) /\ s e. ( 2nd ` L ) ) ) /\ ( k e. N. /\ j e. N. ) ) -> j e. N. )
38	11	simplbi 274	. . . . . . . . 9 |- ( s e. ( 1st ` L ) -> s e. Q. )
39	38	ad2antrl 490	. . . . . . . 8 |- ( ( ph /\ ( s e. ( 1st ` L ) /\ s e. ( 2nd ` L ) ) ) -> s e. Q. )
40	39	adantr 276	. . . . . . 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 7929	. . . . . 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 624	. . . . 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 2659	. . . 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 1417	. 2 |- ( ph -> -. ( s e. ( 1st ` L ) /\ s e. ( 2nd ` L ) ) )
46	45	ralrimivw 2607	1 |- ( ph -> A. s e. Q. -. ( s e. ( 1st ` L ) /\ s e. ( 2nd ` L ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   = wceq 1398   F. wfal 1403   e. wcel 2202   A.wral 2511   E.wrex 2512   {crab 2515   <.cop 3676   class class class wbr 4093   -->wf 5329   ` cfv 5333  (class class class)co 6028   1stc1st 6310   2ndc2nd 6311   1oc1o 6618   [cec 6743   N.cnpi 7535   <N clti 7538   ~Q ceq 7542   Q.cnq 7543   +Q cplq 7545   *Qcrq 7547   <Q cltq 7548

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-eprel 4392  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-irdg 6579  df-1o 6625  df-oadd 6629  df-omul 6630  df-er 6745  df-ec 6747  df-qs 6751  df-ni 7567  df-pli 7568  df-mi 7569  df-lti 7570  df-plpq 7607  df-mpq 7608  df-enq 7610  df-nqqs 7611  df-plqqs 7612  df-mqqs 7613  df-1nqqs 7614  df-rq 7615  df-ltnqqs 7616

This theorem is referenced by:  caucvgprlemcl  7939