Theorem caucvgprlemdisj 7734

Description: Lemma for caucvgpr 7742. 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 5925	. . . . . . . . . . . 12 |- ( l = s -> ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) = ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) )
2	1	breq1d 4039	. . . . . . . . . . 11 |- ( l = s -> ( ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) <-> ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) )
3	2	rexbidv 2495	. . . . . . . . . 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 5557	. . . . . . . . . . 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 7423	. . . . . . . . . . . . 13 |- Q. e. _V
7	6	rabex 4173	. . . . . . . . . . . 12 |- { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } e. _V
8	6	rabex 4173	. . . . . . . . . . . 12 |- { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } e. _V
9	7, 8	op1st 6199	. . . . . . . . . . 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 2214	. . . . . . . . . 10 |- ( 1st ` L ) = { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) }
11	3, 10	elrab2 2919	. . . . . . . . 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 3804	. . . . . . . . . . . . 13 |- ( j = k -> <. j , 1o >. = <. k , 1o >. )
14	13	eceq1d 6623	. . . . . . . . . . . 12 |- ( j = k -> [ <. j , 1o >. ] ~Q = [ <. k , 1o >. ] ~Q )
15	14	fveq2d 5558	. . . . . . . . . . 11 |- ( j = k -> ( *Q ` [ <. j , 1o >. ] ~Q ) = ( *Q ` [ <. k , 1o >. ] ~Q ) )
16	15	oveq2d 5934	. . . . . . . . . 10 |- ( j = k -> ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) = ( s +Q ( *Q ` [ <. k , 1o >. ] ~Q ) ) )
17		fveq2 5554	. . . . . . . . . 10 |- ( j = k -> ( F ` j ) = ( F ` k ) )
18	16, 17	breq12d 4042	. . . . . . . . 9 |- ( j = k -> ( ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) <-> ( s +Q ( *Q ` [ <. k , 1o >. ] ~Q ) ) <Q ( F ` k ) ) )
19	18	cbvrexv 2727	. . . . . . . 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 4033	. . . . . . . . . 10 |- ( u = s -> ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u <-> ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q s ) )
22	21	rexbidv 2495	. . . . . . . . 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 5557	. . . . . . . . . 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 6200	. . . . . . . . . 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 2214	. . . . . . . . 9 |- ( 2nd ` L ) = { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u }
26	22, 25	elrab2 2919	. . . . . . . 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 2664	. . . . . 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 529	. . . . . . 7 |- ( ( ( ph /\ ( s e. ( 1st ` L ) /\ s e. ( 2nd ` L ) ) ) /\ ( k e. N. /\ j e. N. ) ) -> k e. N. )
37		simprr 531	. . . . . . 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 7726	. . . . . 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 620	. . . . 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 2619	. . . 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 1383	. 2 |- ( ph -> -. ( s e. ( 1st ` L ) /\ s e. ( 2nd ` L ) ) )
46	45	ralrimivw 2568	1 |- ( ph -> A. s e. Q. -. ( s e. ( 1st ` L ) /\ s e. ( 2nd ` L ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   = wceq 1364   F. wfal 1369   e. wcel 2164   A.wral 2472   E.wrex 2473   {crab 2476   <.cop 3621   class class class wbr 4029   -->wf 5250   ` cfv 5254  (class class class)co 5918   1stc1st 6191   2ndc2nd 6192   1oc1o 6462   [cec 6585   N.cnpi 7332   <N clti 7335   ~Q ceq 7339   Q.cnq 7340   +Q cplq 7342   *Qcrq 7344   <Q cltq 7345

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-eprel 4320  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-irdg 6423  df-1o 6469  df-oadd 6473  df-omul 6474  df-er 6587  df-ec 6589  df-qs 6593  df-ni 7364  df-pli 7365  df-mi 7366  df-lti 7367  df-plpq 7404  df-mpq 7405  df-enq 7407  df-nqqs 7408  df-plqqs 7409  df-mqqs 7410  df-1nqqs 7411  df-rq 7412  df-ltnqqs 7413

This theorem is referenced by:  caucvgprlemcl  7736