Theorem caucvgprlemdisj 7787

Description: Lemma for caucvgpr 7795. 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 5951	. . . . . . . . . . . 12 |- ( l = s -> ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) = ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) )
2	1	breq1d 4054	. . . . . . . . . . 11 |- ( l = s -> ( ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) <-> ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) )
3	2	rexbidv 2507	. . . . . . . . . 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 5579	. . . . . . . . . . 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 7476	. . . . . . . . . . . . 13 |- Q. e. _V
7	6	rabex 4188	. . . . . . . . . . . 12 |- { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } e. _V
8	6	rabex 4188	. . . . . . . . . . . 12 |- { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } e. _V
9	7, 8	op1st 6232	. . . . . . . . . . 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 2226	. . . . . . . . . 10 |- ( 1st ` L ) = { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) }
11	3, 10	elrab2 2932	. . . . . . . . 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 3819	. . . . . . . . . . . . 13 |- ( j = k -> <. j , 1o >. = <. k , 1o >. )
14	13	eceq1d 6656	. . . . . . . . . . . 12 |- ( j = k -> [ <. j , 1o >. ] ~Q = [ <. k , 1o >. ] ~Q )
15	14	fveq2d 5580	. . . . . . . . . . 11 |- ( j = k -> ( *Q ` [ <. j , 1o >. ] ~Q ) = ( *Q ` [ <. k , 1o >. ] ~Q ) )
16	15	oveq2d 5960	. . . . . . . . . 10 |- ( j = k -> ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) = ( s +Q ( *Q ` [ <. k , 1o >. ] ~Q ) ) )
17		fveq2 5576	. . . . . . . . . 10 |- ( j = k -> ( F ` j ) = ( F ` k ) )
18	16, 17	breq12d 4057	. . . . . . . . 9 |- ( j = k -> ( ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) <-> ( s +Q ( *Q ` [ <. k , 1o >. ] ~Q ) ) <Q ( F ` k ) ) )
19	18	cbvrexv 2739	. . . . . . . 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 4048	. . . . . . . . . 10 |- ( u = s -> ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u <-> ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q s ) )
22	21	rexbidv 2507	. . . . . . . . 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 5579	. . . . . . . . . 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 6233	. . . . . . . . . 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 2226	. . . . . . . . 9 |- ( 2nd ` L ) = { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u }
26	22, 25	elrab2 2932	. . . . . . . 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 2676	. . . . . 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 7779	. . . . . 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 2631	. . . 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 1392	. 2 |- ( ph -> -. ( s e. ( 1st ` L ) /\ s e. ( 2nd ` L ) ) )
46	45	ralrimivw 2580	1 |- ( ph -> A. s e. Q. -. ( s e. ( 1st ` L ) /\ s e. ( 2nd ` L ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   = wceq 1373   F. wfal 1378   e. wcel 2176   A.wral 2484   E.wrex 2485   {crab 2488   <.cop 3636   class class class wbr 4044   -->wf 5267   ` cfv 5271  (class class class)co 5944   1stc1st 6224   2ndc2nd 6225   1oc1o 6495   [cec 6618   N.cnpi 7385   <N clti 7388   ~Q ceq 7392   Q.cnq 7393   +Q cplq 7395   *Qcrq 7397   <Q cltq 7398

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-iinf 4636

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-tr 4143  df-eprel 4336  df-id 4340  df-po 4343  df-iso 4344  df-iord 4413  df-on 4415  df-suc 4418  df-iom 4639  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-recs 6391  df-irdg 6456  df-1o 6502  df-oadd 6506  df-omul 6507  df-er 6620  df-ec 6622  df-qs 6626  df-ni 7417  df-pli 7418  df-mi 7419  df-lti 7420  df-plpq 7457  df-mpq 7458  df-enq 7460  df-nqqs 7461  df-plqqs 7462  df-mqqs 7463  df-1nqqs 7464  df-rq 7465  df-ltnqqs 7466

This theorem is referenced by:  caucvgprlemcl  7789