Theorem cauappcvgprlemladd 7660

Description: Lemma for cauappcvgpr 7664. 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 7657	. . 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 7659	. . 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 3174	. 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 7656	. . 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 7658	. . 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 3174	. 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 7655	. . . 4 |- ( ph -> L e. P. )
13		nqprlu 7549	. . . . 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 7539	. . . 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 7473	. . . . . . 7 |- P. C_ ( ~P Q. X. ~P Q. )
18	17	sseli 3153	. . . . . 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 6179	. . . . . 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 3242	. . . . . . . 8 |- { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( ( F ` q ) +Q S ) } C_ Q.
22		nqex 7365	. . . . . . . . 9 |- Q. e. _V
23	22	elpw2 4159	. . . . . . . 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 3242	. . . . . . . 8 |- { u e. Q. | E. q e. Q. ( ( ( F ` q ) +Q q ) +Q S ) <Q u } C_ Q.
26	22	elpw2 4159	. . . . . . . 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 4660	. . . . . . 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 6179	. . . . . 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 2192	. . . 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 6169	. . . . . 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 6170	. . . . . 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 4240	. . . . 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 944	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 1353   e. wcel 2148   {cab 2163   A.wral 2455   E.wrex 2456   {crab 2459   C_ wss 3131   ~Pcpw 3577   <.cop 3597   class class class wbr 4005   X. cxp 4626   -->wf 5214   ` cfv 5218  (class class class)co 5878   1stc1st 6142   2ndc2nd 6143   Q.cnq 7282   +Q cplq 7284   <Q cltq 7287   P.cnp 7293   +P. cpp 7295

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-eprel 4291  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5881  df-oprab 5882  df-mpo 5883  df-1st 6144  df-2nd 6145  df-recs 6309  df-irdg 6374  df-1o 6420  df-2o 6421  df-oadd 6424  df-omul 6425  df-er 6538  df-ec 6540  df-qs 6544  df-ni 7306  df-pli 7307  df-mi 7308  df-lti 7309  df-plpq 7346  df-mpq 7347  df-enq 7349  df-nqqs 7350  df-plqqs 7351  df-mqqs 7352  df-1nqqs 7353  df-rq 7354  df-ltnqqs 7355  df-enq0 7426  df-nq0 7427  df-0nq0 7428  df-plq0 7429  df-mq0 7430  df-inp 7468  df-iplp 7470  df-iltp 7472

This theorem is referenced by:  cauappcvgprlem1  7661