Theorem cauappcvgprlemladd 7697

Description: Lemma for cauappcvgpr 7701. This takes L and offsets it by the positive fraction S. (Contributed by Jim Kingdon, 23-Jun-2020.)

Hypotheses

Ref	Expression
cauappcvgpr.f	|- ( ph -> F : Q. --> Q. )
cauappcvgpr.app	|- ( ph -> A. p e. Q. A. q e. Q. ( ( F ` p ) <Q ( ( F ` q ) +Q ( p +Q q ) ) /\ ( F ` q ) <Q ( ( F ` p ) +Q ( p +Q q ) ) ) )
cauappcvgpr.bnd	|- ( ph -> A. p e. Q. A <Q ( F ` p ) )
cauappcvgpr.lim	|- L = <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) } , { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u } >.
cauappcvgprlemladd.s	|- ( ph -> S e. Q. )

Assertion

Ref	Expression
cauappcvgprlemladd	|- ( ph -> ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) = <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( ( F ` q ) +Q S ) } , { u e. Q. | E. q e. Q. ( ( ( F ` q ) +Q q ) +Q S ) <Q u } >. )

Distinct variable groups:   A, p   L, p, q   ph, p, q   F, l, u, p, q   S, l, q, u, p

Allowed substitution hints:   ph( u, l)   A( u, q, l)   L( u, l)

Proof of Theorem cauappcvgprlemladd

Step	Hyp	Ref	Expression
1		cauappcvgpr.f	. . . 4 |- ( ph -> F : Q. --> Q. )
2		cauappcvgpr.app	. . . 4 |- ( ph -> A. p e. Q. A. q e. Q. ( ( F ` p ) <Q ( ( F ` q ) +Q ( p +Q q ) ) /\ ( F ` q ) <Q ( ( F ` p ) +Q ( p +Q q ) ) ) )
3		cauappcvgpr.bnd	. . . 4 |- ( ph -> A. p e. Q. A <Q ( F ` p ) )
4		cauappcvgpr.lim	. . . 4 |- L = <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) } , { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u } >.
5		cauappcvgprlemladd.s	. . . 4 |- ( ph -> S e. Q. )
6	1, 2, 3, 4, 5	cauappcvgprlemladdfl 7694	. . 3 |- ( ph -> ( 1st ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) C_ ( 1st ` <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( ( F ` q ) +Q S ) } , { u e. Q. | E. q e. Q. ( ( ( F ` q ) +Q q ) +Q S ) <Q u } >. ) )
7	1, 2, 3, 4, 5	cauappcvgprlemladdrl 7696	. . 3 |- ( ph -> ( 1st ` <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( ( F ` q ) +Q S ) } , { u e. Q. | E. q e. Q. ( ( ( F ` q ) +Q q ) +Q S ) <Q u } >. ) C_ ( 1st ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) )
8	6, 7	eqssd 3191	. 2 |- ( ph -> ( 1st ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) = ( 1st ` <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( ( F ` q ) +Q S ) } , { u e. Q. | E. q e. Q. ( ( ( F ` q ) +Q q ) +Q S ) <Q u } >. ) )
9	1, 2, 3, 4, 5	cauappcvgprlemladdfu 7693	. . 3 |- ( ph -> ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) C_ ( 2nd ` <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( ( F ` q ) +Q S ) } , { u e. Q. | E. q e. Q. ( ( ( F ` q ) +Q q ) +Q S ) <Q u } >. ) )
10	1, 2, 3, 4, 5	cauappcvgprlemladdru 7695	. . 3 |- ( ph -> ( 2nd ` <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( ( F ` q ) +Q S ) } , { u e. Q. | E. q e. Q. ( ( ( F ` q ) +Q q ) +Q S ) <Q u } >. ) C_ ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) )
11	9, 10	eqssd 3191	. 2 |- ( ph -> ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) = ( 2nd ` <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( ( F ` q ) +Q S ) } , { u e. Q. | E. q e. Q. ( ( ( F ` q ) +Q q ) +Q S ) <Q u } >. ) )
12	1, 2, 3, 4	cauappcvgprlemcl 7692	. . . 4 |- ( ph -> L e. P. )
13		nqprlu 7586	. . . . 5 |- ( S e. Q. -> <. { l | l <Q S } , { u | S <Q u } >. e. P. )
14	5, 13	syl 14	. . . 4 |- ( ph -> <. { l | l <Q S } , { u | S <Q u } >. e. P. )
15		addclpr 7576	. . . 4 |- ( ( L e. P. /\ <. { l | l <Q S } , { u | S <Q u } >. e. P. ) -> ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) e. P. )
16	12, 14, 15	syl2anc 411	. . 3 |- ( ph -> ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) e. P. )
17		npsspw 7510	. . . . . . 7 |- P. C_ ( ~P Q. X. ~P Q. )
18	17	sseli 3170	. . . . . 6 |- ( ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) e. P. -> ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) e. ( ~P Q. X. ~P Q. ) )
19		1st2nd2 6208	. . . . . 6 |- ( ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) e. ( ~P Q. X. ~P Q. ) -> ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) = <. ( 1st ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) , ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) >. )
20	18, 19	syl 14	. . . . 5 |- ( ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) e. P. -> ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) = <. ( 1st ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) , ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) >. )
21		ssrab2 3259	. . . . . . . 8 |- { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( ( F ` q ) +Q S ) } C_ Q.
22		nqex 7402	. . . . . . . . 9 |- Q. e. _V
23	22	elpw2 4182	. . . . . . . 8 |- ( { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( ( F ` q ) +Q S ) } e. ~P Q. <-> { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( ( F ` q ) +Q S ) } C_ Q. )
24	21, 23	mpbir 146	. . . . . . 7 |- { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( ( F ` q ) +Q S ) } e. ~P Q.
25		ssrab2 3259	. . . . . . . 8 |- { u e. Q. | E. q e. Q. ( ( ( F ` q ) +Q q ) +Q S ) <Q u } C_ Q.
26	22	elpw2 4182	. . . . . . . 8 |- ( { u e. Q. | E. q e. Q. ( ( ( F ` q ) +Q q ) +Q S ) <Q u } e. ~P Q. <-> { u e. Q. | E. q e. Q. ( ( ( F ` q ) +Q q ) +Q S ) <Q u } C_ Q. )
27	25, 26	mpbir 146	. . . . . . 7 |- { u e. Q. | E. q e. Q. ( ( ( F ` q ) +Q q ) +Q S ) <Q u } e. ~P Q.
28		opelxpi 4683	. . . . . . 7 |- ( ( { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( ( F ` q ) +Q S ) } e. ~P Q. /\ { u e. Q. | E. q e. Q. ( ( ( F ` q ) +Q q ) +Q S ) <Q u } e. ~P Q. ) -> <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( ( F ` q ) +Q S ) } , { u e. Q. | E. q e. Q. ( ( ( F ` q ) +Q q ) +Q S ) <Q u } >. e. ( ~P Q. X. ~P Q. ) )
29	24, 27, 28	mp2an 426	. . . . . 6 |- <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( ( F ` q ) +Q S ) } , { u e. Q. | E. q e. Q. ( ( ( F ` q ) +Q q ) +Q S ) <Q u } >. e. ( ~P Q. X. ~P Q. )
30		1st2nd2 6208	. . . . . 6 |- ( <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( ( F ` q ) +Q S ) } , { u e. Q. | E. q e. Q. ( ( ( F ` q ) +Q q ) +Q S ) <Q u } >. e. ( ~P Q. X. ~P Q. ) -> <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( ( F ` q ) +Q S ) } , { u e. Q. | E. q e. Q. ( ( ( F ` q ) +Q q ) +Q S ) <Q u } >. = <. ( 1st ` <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( ( F ` q ) +Q S ) } , { u e. Q. | E. q e. Q. ( ( ( F ` q ) +Q q ) +Q S ) <Q u } >. ) , ( 2nd ` <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( ( F ` q ) +Q S ) } , { u e. Q. | E. q e. Q. ( ( ( F ` q ) +Q q ) +Q S ) <Q u } >. ) >. )
31	29, 30	mp1i 10	. . . . 5 |- ( ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) e. P. -> <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( ( F ` q ) +Q S ) } , { u e. Q. | E. q e. Q. ( ( ( F ` q ) +Q q ) +Q S ) <Q u } >. = <. ( 1st ` <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( ( F ` q ) +Q S ) } , { u e. Q. | E. q e. Q. ( ( ( F ` q ) +Q q ) +Q S ) <Q u } >. ) , ( 2nd ` <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( ( F ` q ) +Q S ) } , { u e. Q. | E. q e. Q. ( ( ( F ` q ) +Q q ) +Q S ) <Q u } >. ) >. )
32	20, 31	eqeq12d 2204	. . . 4 |- ( ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) e. P. -> ( ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) = <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( ( F ` q ) +Q S ) } , { u e. Q. | E. q e. Q. ( ( ( F ` q ) +Q q ) +Q S ) <Q u } >. <-> <. ( 1st ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) , ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) >. = <. ( 1st ` <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( ( F ` q ) +Q S ) } , { u e. Q. | E. q e. Q. ( ( ( F ` q ) +Q q ) +Q S ) <Q u } >. ) , ( 2nd ` <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( ( F ` q ) +Q S ) } , { u e. Q. | E. q e. Q. ( ( ( F ` q ) +Q q ) +Q S ) <Q u } >. ) >. ) )
33		xp1st 6198	. . . . . 6 |- ( ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) e. ( ~P Q. X. ~P Q. ) -> ( 1st ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) e. ~P Q. )
34	18, 33	syl 14	. . . . 5 |- ( ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) e. P. -> ( 1st ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) e. ~P Q. )
35		xp2nd 6199	. . . . . 6 |- ( ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) e. ( ~P Q. X. ~P Q. ) -> ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) e. ~P Q. )
36	18, 35	syl 14	. . . . 5 |- ( ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) e. P. -> ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) e. ~P Q. )
37		opthg 4263	. . . . 5 |- ( ( ( 1st ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) e. ~P Q. /\ ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) e. ~P Q. ) -> ( <. ( 1st ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) , ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) >. = <. ( 1st ` <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( ( F ` q ) +Q S ) } , { u e. Q. | E. q e. Q. ( ( ( F ` q ) +Q q ) +Q S ) <Q u } >. ) , ( 2nd ` <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( ( F ` q ) +Q S ) } , { u e. Q. | E. q e. Q. ( ( ( F ` q ) +Q q ) +Q S ) <Q u } >. ) >. <-> ( ( 1st ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) = ( 1st ` <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( ( F ` q ) +Q S ) } , { u e. Q. | E. q e. Q. ( ( ( F ` q ) +Q q ) +Q S ) <Q u } >. ) /\ ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) = ( 2nd ` <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( ( F ` q ) +Q S ) } , { u e. Q. | E. q e. Q. ( ( ( F ` q ) +Q q ) +Q S ) <Q u } >. ) ) ) )
38	34, 36, 37	syl2anc 411	. . . 4 |- ( ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) e. P. -> ( <. ( 1st ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) , ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) >. = <. ( 1st ` <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( ( F ` q ) +Q S ) } , { u e. Q. | E. q e. Q. ( ( ( F ` q ) +Q q ) +Q S ) <Q u } >. ) , ( 2nd ` <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( ( F ` q ) +Q S ) } , { u e. Q. | E. q e. Q. ( ( ( F ` q ) +Q q ) +Q S ) <Q u } >. ) >. <-> ( ( 1st ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) = ( 1st ` <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( ( F ` q ) +Q S ) } , { u e. Q. | E. q e. Q. ( ( ( F ` q ) +Q q ) +Q S ) <Q u } >. ) /\ ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) = ( 2nd ` <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( ( F ` q ) +Q S ) } , { u e. Q. | E. q e. Q. ( ( ( F ` q ) +Q q ) +Q S ) <Q u } >. ) ) ) )
39	32, 38	bitrd 188	. . 3 |- ( ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) e. P. -> ( ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) = <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( ( F ` q ) +Q S ) } , { u e. Q. | E. q e. Q. ( ( ( F ` q ) +Q q ) +Q S ) <Q u } >. <-> ( ( 1st ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) = ( 1st ` <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( ( F ` q ) +Q S ) } , { u e. Q. | E. q e. Q. ( ( ( F ` q ) +Q q ) +Q S ) <Q u } >. ) /\ ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) = ( 2nd ` <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( ( F ` q ) +Q S ) } , { u e. Q. | E. q e. Q. ( ( ( F ` q ) +Q q ) +Q S ) <Q u } >. ) ) ) )
40	16, 39	syl 14	. 2 |- ( ph -> ( ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) = <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( ( F ` q ) +Q S ) } , { u e. Q. | E. q e. Q. ( ( ( F ` q ) +Q q ) +Q S ) <Q u } >. <-> ( ( 1st ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) = ( 1st ` <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( ( F ` q ) +Q S ) } , { u e. Q. | E. q e. Q. ( ( ( F ` q ) +Q q ) +Q S ) <Q u } >. ) /\ ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) = ( 2nd ` <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( ( F ` q ) +Q S ) } , { u e. Q. | E. q e. Q. ( ( ( F ` q ) +Q q ) +Q S ) <Q u } >. ) ) ) )
41	8, 11, 40	mpbir2and 946	1 |- ( ph -> ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) = <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( ( F ` q ) +Q S ) } , { u e. Q. | E. q e. Q. ( ( ( F ` q ) +Q q ) +Q S ) <Q u } >. )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2160   {cab 2175   A.wral 2468   E.wrex 2469   {crab 2472   C_ wss 3148   ~Pcpw 3597   <.cop 3617   class class class wbr 4025   X. cxp 4649   -->wf 5238   ` cfv 5242  (class class class)co 5904   1stc1st 6171   2ndc2nd 6172   Q.cnq 7319   +Q cplq 7321   <Q cltq 7324   P.cnp 7330   +P. cpp 7332

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 2162  ax-14 2163  ax-ext 2171  ax-coll 4140  ax-sep 4143  ax-nul 4151  ax-pow 4199  ax-pr 4234  ax-un 4458  ax-setind 4561  ax-iinf 4612

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 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2758  df-sbc 2982  df-csb 3077  df-dif 3150  df-un 3152  df-in 3154  df-ss 3161  df-nul 3442  df-pw 3599  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3832  df-int 3867  df-iun 3910  df-br 4026  df-opab 4087  df-mpt 4088  df-tr 4124  df-eprel 4314  df-id 4318  df-po 4321  df-iso 4322  df-iord 4391  df-on 4393  df-suc 4396  df-iom 4615  df-xp 4657  df-rel 4658  df-cnv 4659  df-co 4660  df-dm 4661  df-rn 4662  df-res 4663  df-ima 4664  df-iota 5203  df-fun 5244  df-fn 5245  df-f 5246  df-f1 5247  df-fo 5248  df-f1o 5249  df-fv 5250  df-ov 5907  df-oprab 5908  df-mpo 5909  df-1st 6173  df-2nd 6174  df-recs 6338  df-irdg 6403  df-1o 6449  df-2o 6450  df-oadd 6453  df-omul 6454  df-er 6567  df-ec 6569  df-qs 6573  df-ni 7343  df-pli 7344  df-mi 7345  df-lti 7346  df-plpq 7383  df-mpq 7384  df-enq 7386  df-nqqs 7387  df-plqqs 7388  df-mqqs 7389  df-1nqqs 7390  df-rq 7391  df-ltnqqs 7392  df-enq0 7463  df-nq0 7464  df-0nq0 7465  df-plq0 7466  df-mq0 7467  df-inp 7505  df-iplp 7507  df-iltp 7509

This theorem is referenced by:  cauappcvgprlem1  7698