Theorem cauappcvgprlemladd 7871

Description: Lemma for cauappcvgpr 7875. 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 7868	. . 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 7870	. . 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 3242	. 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 7867	. . 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 7869	. . 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 3242	. 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 7866	. . . 4 |- ( ph -> L e. P. )
13		nqprlu 7760	. . . . 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 7750	. . . 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 7684	. . . . . . 7 |- P. C_ ( ~P Q. X. ~P Q. )
18	17	sseli 3221	. . . . . 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 6333	. . . . . 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 3310	. . . . . . . 8 |- { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( ( F ` q ) +Q S ) } C_ Q.
22		nqex 7576	. . . . . . . . 9 |- Q. e. _V
23	22	elpw2 4245	. . . . . . . 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 3310	. . . . . . . 8 |- { u e. Q. | E. q e. Q. ( ( ( F ` q ) +Q q ) +Q S ) <Q u } C_ Q.
26	22	elpw2 4245	. . . . . . . 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 4755	. . . . . . 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 6333	. . . . . 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 2244	. . . 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 6323	. . . . . 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 6324	. . . . . 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 4328	. . . . 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 950	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 1395   e. wcel 2200   {cab 2215   A.wral 2508   E.wrex 2509   {crab 2512   C_ wss 3198   ~Pcpw 3650   <.cop 3670   class class class wbr 4086   X. cxp 4721   -->wf 5320   ` cfv 5324  (class class class)co 6013   1stc1st 6296   2ndc2nd 6297   Q.cnq 7493   +Q cplq 7495   <Q cltq 7498   P.cnp 7504   +P. cpp 7506

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-eprel 4384  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-irdg 6531  df-1o 6577  df-2o 6578  df-oadd 6581  df-omul 6582  df-er 6697  df-ec 6699  df-qs 6703  df-ni 7517  df-pli 7518  df-mi 7519  df-lti 7520  df-plpq 7557  df-mpq 7558  df-enq 7560  df-nqqs 7561  df-plqqs 7562  df-mqqs 7563  df-1nqqs 7564  df-rq 7565  df-ltnqqs 7566  df-enq0 7637  df-nq0 7638  df-0nq0 7639  df-plq0 7640  df-mq0 7641  df-inp 7679  df-iplp 7681  df-iltp 7683

This theorem is referenced by:  cauappcvgprlem1  7872