Theorem caucvgprlem1 7827

Description: Lemma for caucvgpr 7830. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 3-Oct-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 } >.
caucvgprlemlim.q	|- ( ph -> Q e. Q. )
caucvgprlemlim.jk	|- ( ph -> J <N K )
caucvgprlemlim.jkq	|- ( ph -> ( *Q ` [ <. J , 1o >. ] ~Q ) <Q Q )

Assertion

Ref	Expression
caucvgprlem1	|- ( ph -> <. { l | l <Q ( F ` K ) } , { u | ( F ` K ) <Q u } >. <P ( L +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) )

Distinct variable groups:   A, j   j, F, l, u   j, K, l, u   Q, j, l, u   Q, k   j, L, k   u, j   k, F, n   j, k

Allowed substitution hints:   ph( u, j, k, n, l)   A( u, k, n, l)   Q( n)   J( u, j, k, n, l)   K( k, n)   L( u, n, l)

Proof of Theorem caucvgprlem1

Step	Hyp	Ref	Expression
1		caucvgprlemlim.jk	. . . . . 6 |- ( ph -> J <N K )
2		ltrelpi 7472	. . . . . . 7 |- <N C_ ( N. X. N. )
3	2	brel 4745	. . . . . 6 |- ( J <N K -> ( J e. N. /\ K e. N. ) )
4	1, 3	syl 14	. . . . 5 |- ( ph -> ( J e. N. /\ K e. N. ) )
5	4	simprd 114	. . . 4 |- ( ph -> K e. N. )
6		caucvgprlemlim.jkq	. . . . . 6 |- ( ph -> ( *Q ` [ <. J , 1o >. ] ~Q ) <Q Q )
7	1, 6	caucvgprlemk 7813	. . . . 5 |- ( ph -> ( *Q ` [ <. K , 1o >. ] ~Q ) <Q Q )
8		caucvgpr.f	. . . . . 6 |- ( ph -> F : N. --> Q. )
9	8, 5	ffvelcdmd 5739	. . . . 5 |- ( ph -> ( F ` K ) e. Q. )
10		ltanqi 7550	. . . . 5 |- ( ( ( *Q ` [ <. K , 1o >. ] ~Q ) <Q Q /\ ( F ` K ) e. Q. ) -> ( ( F ` K ) +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q ( ( F ` K ) +Q Q ) )
11	7, 9, 10	syl2anc 411	. . . 4 |- ( ph -> ( ( F ` K ) +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q ( ( F ` K ) +Q Q ) )
12		opeq1 3833	. . . . . . . . 9 |- ( j = K -> <. j , 1o >. = <. K , 1o >. )
13	12	eceq1d 6679	. . . . . . . 8 |- ( j = K -> [ <. j , 1o >. ] ~Q = [ <. K , 1o >. ] ~Q )
14	13	fveq2d 5603	. . . . . . 7 |- ( j = K -> ( *Q ` [ <. j , 1o >. ] ~Q ) = ( *Q ` [ <. K , 1o >. ] ~Q ) )
15	14	oveq2d 5983	. . . . . 6 |- ( j = K -> ( ( F ` K ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) = ( ( F ` K ) +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) )
16		fveq2 5599	. . . . . . 7 |- ( j = K -> ( F ` j ) = ( F ` K ) )
17	16	oveq1d 5982	. . . . . 6 |- ( j = K -> ( ( F ` j ) +Q Q ) = ( ( F ` K ) +Q Q ) )
18	15, 17	breq12d 4072	. . . . 5 |- ( j = K -> ( ( ( F ` K ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( ( F ` j ) +Q Q ) <-> ( ( F ` K ) +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q ( ( F ` K ) +Q Q ) ) )
19	18	rspcev 2884	. . . 4 |- ( ( K e. N. /\ ( ( F ` K ) +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q ( ( F ` K ) +Q Q ) ) -> E. j e. N. ( ( F ` K ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( ( F ` j ) +Q Q ) )
20	5, 11, 19	syl2anc 411	. . 3 |- ( ph -> E. j e. N. ( ( F ` K ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( ( F ` j ) +Q Q ) )
21		oveq1 5974	. . . . . . . 8 |- ( l = ( F ` K ) -> ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) = ( ( F ` K ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) )
22	21	breq1d 4069	. . . . . . 7 |- ( l = ( F ` K ) -> ( ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( ( F ` j ) +Q Q ) <-> ( ( F ` K ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( ( F ` j ) +Q Q ) ) )
23	22	rexbidv 2509	. . . . . 6 |- ( l = ( F ` K ) -> ( E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( ( F ` j ) +Q Q ) <-> E. j e. N. ( ( F ` K ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( ( F ` j ) +Q Q ) ) )
24	23	elrab3 2937	. . . . 5 |- ( ( F ` K ) e. Q. -> ( ( F ` K ) e. { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( ( F ` j ) +Q Q ) } <-> E. j e. N. ( ( F ` K ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( ( F ` j ) +Q Q ) ) )
25	9, 24	syl 14	. . . 4 |- ( ph -> ( ( F ` K ) e. { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( ( F ` j ) +Q Q ) } <-> E. j e. N. ( ( F ` K ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( ( F ` j ) +Q Q ) ) )
26		caucvgpr.cau	. . . . . 6 |- ( 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 ) ) ) ) )
27		caucvgpr.bnd	. . . . . 6 |- ( ph -> A. j e. N. A <Q ( F ` j ) )
28		caucvgpr.lim	. . . . . 6 |- 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 } >.
29		caucvgprlemlim.q	. . . . . 6 |- ( ph -> Q e. Q. )
30	8, 26, 27, 28, 29	caucvgprlemladdrl 7826	. . . . 5 |- ( ph -> { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( ( F ` j ) +Q Q ) } C_ ( 1st ` ( L +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) ) )
31	30	sseld 3200	. . . 4 |- ( ph -> ( ( F ` K ) e. { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( ( F ` j ) +Q Q ) } -> ( F ` K ) e. ( 1st ` ( L +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) ) ) )
32	25, 31	sylbird 170	. . 3 |- ( ph -> ( E. j e. N. ( ( F ` K ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( ( F ` j ) +Q Q ) -> ( F ` K ) e. ( 1st ` ( L +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) ) ) )
33	20, 32	mpd 13	. 2 |- ( ph -> ( F ` K ) e. ( 1st ` ( L +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) ) )
34	8, 26, 27, 28	caucvgprlemcl 7824	. . . 4 |- ( ph -> L e. P. )
35		nqprlu 7695	. . . . 5 |- ( Q e. Q. -> <. { l | l <Q Q } , { u | Q <Q u } >. e. P. )
36	29, 35	syl 14	. . . 4 |- ( ph -> <. { l | l <Q Q } , { u | Q <Q u } >. e. P. )
37		addclpr 7685	. . . 4 |- ( ( L e. P. /\ <. { l | l <Q Q } , { u | Q <Q u } >. e. P. ) -> ( L +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) e. P. )
38	34, 36, 37	syl2anc 411	. . 3 |- ( ph -> ( L +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) e. P. )
39		nqprl 7699	. . 3 |- ( ( ( F ` K ) e. Q. /\ ( L +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) e. P. ) -> ( ( F ` K ) e. ( 1st ` ( L +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) ) <-> <. { l | l <Q ( F ` K ) } , { u | ( F ` K ) <Q u } >. <P ( L +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) ) )
40	9, 38, 39	syl2anc 411	. 2 |- ( ph -> ( ( F ` K ) e. ( 1st ` ( L +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) ) <-> <. { l | l <Q ( F ` K ) } , { u | ( F ` K ) <Q u } >. <P ( L +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) ) )
41	33, 40	mpbid 147	1 |- ( ph -> <. { l | l <Q ( F ` K ) } , { u | ( F ` K ) <Q u } >. <P ( L +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   e. wcel 2178   {cab 2193   A.wral 2486   E.wrex 2487   {crab 2490   <.cop 3646   class class class wbr 4059   -->wf 5286   ` cfv 5290  (class class class)co 5967   1stc1st 6247   1oc1o 6518   [cec 6641   N.cnpi 7420   <N clti 7423   ~Q ceq 7427   Q.cnq 7428   +Q cplq 7430   *Qcrq 7432   <Q cltq 7433   P.cnp 7439   +P. cpp 7441   <P cltp 7443

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654

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 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-eprel 4354  df-id 4358  df-po 4361  df-iso 4362  df-iord 4431  df-on 4433  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-recs 6414  df-irdg 6479  df-1o 6525  df-2o 6526  df-oadd 6529  df-omul 6530  df-er 6643  df-ec 6645  df-qs 6649  df-ni 7452  df-pli 7453  df-mi 7454  df-lti 7455  df-plpq 7492  df-mpq 7493  df-enq 7495  df-nqqs 7496  df-plqqs 7497  df-mqqs 7498  df-1nqqs 7499  df-rq 7500  df-ltnqqs 7501  df-enq0 7572  df-nq0 7573  df-0nq0 7574  df-plq0 7575  df-mq0 7576  df-inp 7614  df-iplp 7616  df-iltp 7618

This theorem is referenced by:  caucvgprlemlim  7829