Theorem cauappcvgprlemladd 6899

Description: Lemma for cauappcvgpr 6903. 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 6896	. . 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 6898	. . 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 3017	. 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 6895	. . 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 6897	. . 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 3017	. 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 6894	. . . 4 |- ( ph -> L e. P. )
13		nqprlu 6788	. . . . 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 6778	. . . 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 403	. . 3 |- ( ph -> ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) e. P. )
17		npsspw 6712	. . . . . . 7 |- P. C_ ( ~P Q. X. ~P Q. )
18	17	sseli 2996	. . . . . 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 5826	. . . . . 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 3080	. . . . . . . 8 |- { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( ( F ` q ) +Q S ) } C_ Q.
22		nqex 6604	. . . . . . . . 9 |- Q. e. _V
23	22	elpw2 3934	. . . . . . . 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 144	. . . . . . 7 |- { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( ( F ` q ) +Q S ) } e. ~P Q.
25		ssrab2 3080	. . . . . . . 8 |- { u e. Q. | E. q e. Q. ( ( ( F ` q ) +Q q ) +Q S ) <Q u } C_ Q.
26	22	elpw2 3934	. . . . . . . 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 144	. . . . . . 7 |- { u e. Q. | E. q e. Q. ( ( ( F ` q ) +Q q ) +Q S ) <Q u } e. ~P Q.
28		opelxpi 4396	. . . . . . 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 417	. . . . . 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 5826	. . . . . 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 2096	. . . 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 5817	. . . . . 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 5818	. . . . . 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 3995	. . . . 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 403	. . . 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 186	. . 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 886	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 102   <-> wb 103   = wceq 1285   e. wcel 1434   {cab 2068   A.wral 2349   E.wrex 2350   {crab 2353   C_ wss 2974   ~Pcpw 3384   <.cop 3403   class class class wbr 3787   X. cxp 4363   -->wf 4922   ` cfv 4926  (class class class)co 5537   1stc1st 5790   2ndc2nd 5791   Q.cnq 6521   +Q cplq 6523   <Q cltq 6526   P.cnp 6532   +P. cpp 6534

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3895  ax-sep 3898  ax-nul 3906  ax-pow 3950  ax-pr 3966  ax-un 4190  ax-setind 4282  ax-iinf 4331

This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3253  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3604  df-int 3639  df-iun 3682  df-br 3788  df-opab 3842  df-mpt 3843  df-tr 3878  df-eprel 4046  df-id 4050  df-po 4053  df-iso 4054  df-iord 4123  df-on 4125  df-suc 4128  df-iom 4334  df-xp 4371  df-rel 4372  df-cnv 4373  df-co 4374  df-dm 4375  df-rn 4376  df-res 4377  df-ima 4378  df-iota 4891  df-fun 4928  df-fn 4929  df-f 4930  df-f1 4931  df-fo 4932  df-f1o 4933  df-fv 4934  df-ov 5540  df-oprab 5541  df-mpt2 5542  df-1st 5792  df-2nd 5793  df-recs 5948  df-irdg 6013  df-1o 6059  df-2o 6060  df-oadd 6063  df-omul 6064  df-er 6165  df-ec 6167  df-qs 6171  df-ni 6545  df-pli 6546  df-mi 6547  df-lti 6548  df-plpq 6585  df-mpq 6586  df-enq 6588  df-nqqs 6589  df-plqqs 6590  df-mqqs 6591  df-1nqqs 6592  df-rq 6593  df-ltnqqs 6594  df-enq0 6665  df-nq0 6666  df-0nq0 6667  df-plq0 6668  df-mq0 6669  df-inp 6707  df-iplp 6709  df-iltp 6711

This theorem is referenced by:  cauappcvgprlem1  6900