Theorem cauappcvgprlemlim 7462

Description: Lemma for cauappcvgpr 7463. The putative limit is a limit. (Contributed by Jim Kingdon, 20-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 } >.

Assertion

Ref	Expression
cauappcvgprlemlim	|- ( ph -> A. q e. Q. A. r e. Q. ( <. { l | l <Q ( F ` q ) } , { u | ( F ` q ) <Q u } >. <P ( L +P. <. { l | l <Q ( q +Q r ) } , { u | ( q +Q r ) <Q u } >. ) /\ L <P <. { l | l <Q ( ( F ` q ) +Q ( q +Q r ) ) } , { u | ( ( F ` q ) +Q ( q +Q r ) ) <Q u } >. ) )

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

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

Proof of Theorem cauappcvgprlemlim

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cauappcvgpr.f	. . . . . 6 |- ( ph -> F : Q. --> Q. )
2	1	adantr 274	. . . . 5 |- ( ( ph /\ ( x e. Q. /\ y e. Q. ) ) -> F : Q. --> Q. )
3		cauappcvgpr.app	. . . . . 6 |- ( 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 ) ) ) )
4	3	adantr 274	. . . . 5 |- ( ( ph /\ ( x e. Q. /\ y e. Q. ) ) -> 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 ) ) ) )
5		cauappcvgpr.bnd	. . . . . 6 |- ( ph -> A. p e. Q. A <Q ( F ` p ) )
6	5	adantr 274	. . . . 5 |- ( ( ph /\ ( x e. Q. /\ y e. Q. ) ) -> A. p e. Q. A <Q ( F ` p ) )
7		cauappcvgpr.lim	. . . . 5 |- 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 } >.
8		simprl 520	. . . . 5 |- ( ( ph /\ ( x e. Q. /\ y e. Q. ) ) -> x e. Q. )
9		simprr 521	. . . . 5 |- ( ( ph /\ ( x e. Q. /\ y e. Q. ) ) -> y e. Q. )
10	2, 4, 6, 7, 8, 9	cauappcvgprlem1 7460	. . . 4 |- ( ( ph /\ ( x e. Q. /\ y e. Q. ) ) -> <. { l | l <Q ( F ` x ) } , { u | ( F ` x ) <Q u } >. <P ( L +P. <. { l | l <Q ( x +Q y ) } , { u | ( x +Q y ) <Q u } >. ) )
11	2, 4, 6, 7, 8, 9	cauappcvgprlem2 7461	. . . 4 |- ( ( ph /\ ( x e. Q. /\ y e. Q. ) ) -> L <P <. { l | l <Q ( ( F ` x ) +Q ( x +Q y ) ) } , { u | ( ( F ` x ) +Q ( x +Q y ) ) <Q u } >. )
12	10, 11	jca 304	. . 3 |- ( ( ph /\ ( x e. Q. /\ y e. Q. ) ) -> ( <. { l | l <Q ( F ` x ) } , { u | ( F ` x ) <Q u } >. <P ( L +P. <. { l | l <Q ( x +Q y ) } , { u | ( x +Q y ) <Q u } >. ) /\ L <P <. { l | l <Q ( ( F ` x ) +Q ( x +Q y ) ) } , { u | ( ( F ` x ) +Q ( x +Q y ) ) <Q u } >. ) )
13	12	ralrimivva 2512	. 2 |- ( ph -> A. x e. Q. A. y e. Q. ( <. { l | l <Q ( F ` x ) } , { u | ( F ` x ) <Q u } >. <P ( L +P. <. { l | l <Q ( x +Q y ) } , { u | ( x +Q y ) <Q u } >. ) /\ L <P <. { l | l <Q ( ( F ` x ) +Q ( x +Q y ) ) } , { u | ( ( F ` x ) +Q ( x +Q y ) ) <Q u } >. ) )
14		fveq2 5414	. . . . . . . 8 |- ( x = q -> ( F ` x ) = ( F ` q ) )
15	14	breq2d 3936	. . . . . . 7 |- ( x = q -> ( l <Q ( F ` x ) <-> l <Q ( F ` q ) ) )
16	15	abbidv 2255	. . . . . 6 |- ( x = q -> { l | l <Q ( F ` x ) } = { l | l <Q ( F ` q ) } )
17	14	breq1d 3934	. . . . . . 7 |- ( x = q -> ( ( F ` x ) <Q u <-> ( F ` q ) <Q u ) )
18	17	abbidv 2255	. . . . . 6 |- ( x = q -> { u | ( F ` x ) <Q u } = { u | ( F ` q ) <Q u } )
19	16, 18	opeq12d 3708	. . . . 5 |- ( x = q -> <. { l | l <Q ( F ` x ) } , { u | ( F ` x ) <Q u } >. = <. { l | l <Q ( F ` q ) } , { u | ( F ` q ) <Q u } >. )
20		oveq1 5774	. . . . . . . . 9 |- ( x = q -> ( x +Q y ) = ( q +Q y ) )
21	20	breq2d 3936	. . . . . . . 8 |- ( x = q -> ( l <Q ( x +Q y ) <-> l <Q ( q +Q y ) ) )
22	21	abbidv 2255	. . . . . . 7 |- ( x = q -> { l | l <Q ( x +Q y ) } = { l | l <Q ( q +Q y ) } )
23	20	breq1d 3934	. . . . . . . 8 |- ( x = q -> ( ( x +Q y ) <Q u <-> ( q +Q y ) <Q u ) )
24	23	abbidv 2255	. . . . . . 7 |- ( x = q -> { u | ( x +Q y ) <Q u } = { u | ( q +Q y ) <Q u } )
25	22, 24	opeq12d 3708	. . . . . 6 |- ( x = q -> <. { l | l <Q ( x +Q y ) } , { u | ( x +Q y ) <Q u } >. = <. { l | l <Q ( q +Q y ) } , { u | ( q +Q y ) <Q u } >. )
26	25	oveq2d 5783	. . . . 5 |- ( x = q -> ( L +P. <. { l | l <Q ( x +Q y ) } , { u | ( x +Q y ) <Q u } >. ) = ( L +P. <. { l | l <Q ( q +Q y ) } , { u | ( q +Q y ) <Q u } >. ) )
27	19, 26	breq12d 3937	. . . 4 |- ( x = q -> ( <. { l | l <Q ( F ` x ) } , { u | ( F ` x ) <Q u } >. <P ( L +P. <. { l | l <Q ( x +Q y ) } , { u | ( x +Q y ) <Q u } >. ) <-> <. { l | l <Q ( F ` q ) } , { u | ( F ` q ) <Q u } >. <P ( L +P. <. { l | l <Q ( q +Q y ) } , { u | ( q +Q y ) <Q u } >. ) ) )
28	14, 20	oveq12d 5785	. . . . . . . 8 |- ( x = q -> ( ( F ` x ) +Q ( x +Q y ) ) = ( ( F ` q ) +Q ( q +Q y ) ) )
29	28	breq2d 3936	. . . . . . 7 |- ( x = q -> ( l <Q ( ( F ` x ) +Q ( x +Q y ) ) <-> l <Q ( ( F ` q ) +Q ( q +Q y ) ) ) )
30	29	abbidv 2255	. . . . . 6 |- ( x = q -> { l | l <Q ( ( F ` x ) +Q ( x +Q y ) ) } = { l | l <Q ( ( F ` q ) +Q ( q +Q y ) ) } )
31	28	breq1d 3934	. . . . . . 7 |- ( x = q -> ( ( ( F ` x ) +Q ( x +Q y ) ) <Q u <-> ( ( F ` q ) +Q ( q +Q y ) ) <Q u ) )
32	31	abbidv 2255	. . . . . 6 |- ( x = q -> { u | ( ( F ` x ) +Q ( x +Q y ) ) <Q u } = { u | ( ( F ` q ) +Q ( q +Q y ) ) <Q u } )
33	30, 32	opeq12d 3708	. . . . 5 |- ( x = q -> <. { l | l <Q ( ( F ` x ) +Q ( x +Q y ) ) } , { u | ( ( F ` x ) +Q ( x +Q y ) ) <Q u } >. = <. { l | l <Q ( ( F ` q ) +Q ( q +Q y ) ) } , { u | ( ( F ` q ) +Q ( q +Q y ) ) <Q u } >. )
34	33	breq2d 3936	. . . 4 |- ( x = q -> ( L <P <. { l | l <Q ( ( F ` x ) +Q ( x +Q y ) ) } , { u | ( ( F ` x ) +Q ( x +Q y ) ) <Q u } >. <-> L <P <. { l | l <Q ( ( F ` q ) +Q ( q +Q y ) ) } , { u | ( ( F ` q ) +Q ( q +Q y ) ) <Q u } >. ) )
35	27, 34	anbi12d 464	. . 3 |- ( x = q -> ( ( <. { l | l <Q ( F ` x ) } , { u | ( F ` x ) <Q u } >. <P ( L +P. <. { l | l <Q ( x +Q y ) } , { u | ( x +Q y ) <Q u } >. ) /\ L <P <. { l | l <Q ( ( F ` x ) +Q ( x +Q y ) ) } , { u | ( ( F ` x ) +Q ( x +Q y ) ) <Q u } >. ) <-> ( <. { l | l <Q ( F ` q ) } , { u | ( F ` q ) <Q u } >. <P ( L +P. <. { l | l <Q ( q +Q y ) } , { u | ( q +Q y ) <Q u } >. ) /\ L <P <. { l | l <Q ( ( F ` q ) +Q ( q +Q y ) ) } , { u | ( ( F ` q ) +Q ( q +Q y ) ) <Q u } >. ) ) )
36		oveq2 5775	. . . . . . . . 9 |- ( y = r -> ( q +Q y ) = ( q +Q r ) )
37	36	breq2d 3936	. . . . . . . 8 |- ( y = r -> ( l <Q ( q +Q y ) <-> l <Q ( q +Q r ) ) )
38	37	abbidv 2255	. . . . . . 7 |- ( y = r -> { l | l <Q ( q +Q y ) } = { l | l <Q ( q +Q r ) } )
39	36	breq1d 3934	. . . . . . . 8 |- ( y = r -> ( ( q +Q y ) <Q u <-> ( q +Q r ) <Q u ) )
40	39	abbidv 2255	. . . . . . 7 |- ( y = r -> { u | ( q +Q y ) <Q u } = { u | ( q +Q r ) <Q u } )
41	38, 40	opeq12d 3708	. . . . . 6 |- ( y = r -> <. { l | l <Q ( q +Q y ) } , { u | ( q +Q y ) <Q u } >. = <. { l | l <Q ( q +Q r ) } , { u | ( q +Q r ) <Q u } >. )
42	41	oveq2d 5783	. . . . 5 |- ( y = r -> ( L +P. <. { l | l <Q ( q +Q y ) } , { u | ( q +Q y ) <Q u } >. ) = ( L +P. <. { l | l <Q ( q +Q r ) } , { u | ( q +Q r ) <Q u } >. ) )
43	42	breq2d 3936	. . . 4 |- ( y = r -> ( <. { l | l <Q ( F ` q ) } , { u | ( F ` q ) <Q u } >. <P ( L +P. <. { l | l <Q ( q +Q y ) } , { u | ( q +Q y ) <Q u } >. ) <-> <. { l | l <Q ( F ` q ) } , { u | ( F ` q ) <Q u } >. <P ( L +P. <. { l | l <Q ( q +Q r ) } , { u | ( q +Q r ) <Q u } >. ) ) )
44	36	oveq2d 5783	. . . . . . . 8 |- ( y = r -> ( ( F ` q ) +Q ( q +Q y ) ) = ( ( F ` q ) +Q ( q +Q r ) ) )
45	44	breq2d 3936	. . . . . . 7 |- ( y = r -> ( l <Q ( ( F ` q ) +Q ( q +Q y ) ) <-> l <Q ( ( F ` q ) +Q ( q +Q r ) ) ) )
46	45	abbidv 2255	. . . . . 6 |- ( y = r -> { l | l <Q ( ( F ` q ) +Q ( q +Q y ) ) } = { l | l <Q ( ( F ` q ) +Q ( q +Q r ) ) } )
47	44	breq1d 3934	. . . . . . 7 |- ( y = r -> ( ( ( F ` q ) +Q ( q +Q y ) ) <Q u <-> ( ( F ` q ) +Q ( q +Q r ) ) <Q u ) )
48	47	abbidv 2255	. . . . . 6 |- ( y = r -> { u | ( ( F ` q ) +Q ( q +Q y ) ) <Q u } = { u | ( ( F ` q ) +Q ( q +Q r ) ) <Q u } )
49	46, 48	opeq12d 3708	. . . . 5 |- ( y = r -> <. { l | l <Q ( ( F ` q ) +Q ( q +Q y ) ) } , { u | ( ( F ` q ) +Q ( q +Q y ) ) <Q u } >. = <. { l | l <Q ( ( F ` q ) +Q ( q +Q r ) ) } , { u | ( ( F ` q ) +Q ( q +Q r ) ) <Q u } >. )
50	49	breq2d 3936	. . . 4 |- ( y = r -> ( L <P <. { l | l <Q ( ( F ` q ) +Q ( q +Q y ) ) } , { u | ( ( F ` q ) +Q ( q +Q y ) ) <Q u } >. <-> L <P <. { l | l <Q ( ( F ` q ) +Q ( q +Q r ) ) } , { u | ( ( F ` q ) +Q ( q +Q r ) ) <Q u } >. ) )
51	43, 50	anbi12d 464	. . 3 |- ( y = r -> ( ( <. { l | l <Q ( F ` q ) } , { u | ( F ` q ) <Q u } >. <P ( L +P. <. { l | l <Q ( q +Q y ) } , { u | ( q +Q y ) <Q u } >. ) /\ L <P <. { l | l <Q ( ( F ` q ) +Q ( q +Q y ) ) } , { u | ( ( F ` q ) +Q ( q +Q y ) ) <Q u } >. ) <-> ( <. { l | l <Q ( F ` q ) } , { u | ( F ` q ) <Q u } >. <P ( L +P. <. { l | l <Q ( q +Q r ) } , { u | ( q +Q r ) <Q u } >. ) /\ L <P <. { l | l <Q ( ( F ` q ) +Q ( q +Q r ) ) } , { u | ( ( F ` q ) +Q ( q +Q r ) ) <Q u } >. ) ) )
52	35, 51	cbvral2v 2660	. 2 |- ( A. x e. Q. A. y e. Q. ( <. { l | l <Q ( F ` x ) } , { u | ( F ` x ) <Q u } >. <P ( L +P. <. { l | l <Q ( x +Q y ) } , { u | ( x +Q y ) <Q u } >. ) /\ L <P <. { l | l <Q ( ( F ` x ) +Q ( x +Q y ) ) } , { u | ( ( F ` x ) +Q ( x +Q y ) ) <Q u } >. ) <-> A. q e. Q. A. r e. Q. ( <. { l | l <Q ( F ` q ) } , { u | ( F ` q ) <Q u } >. <P ( L +P. <. { l | l <Q ( q +Q r ) } , { u | ( q +Q r ) <Q u } >. ) /\ L <P <. { l | l <Q ( ( F ` q ) +Q ( q +Q r ) ) } , { u | ( ( F ` q ) +Q ( q +Q r ) ) <Q u } >. ) )
53	13, 52	sylib 121	1 |- ( ph -> A. q e. Q. A. r e. Q. ( <. { l | l <Q ( F ` q ) } , { u | ( F ` q ) <Q u } >. <P ( L +P. <. { l | l <Q ( q +Q r ) } , { u | ( q +Q r ) <Q u } >. ) /\ L <P <. { l | l <Q ( ( F ` q ) +Q ( q +Q r ) ) } , { u | ( ( F ` q ) +Q ( q +Q r ) ) <Q u } >. ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   e. wcel 1480   {cab 2123   A.wral 2414   E.wrex 2415   {crab 2418   <.cop 3525   class class class wbr 3924   -->wf 5114   ` cfv 5118  (class class class)co 5767   Q.cnq 7081   +Q cplq 7083   <Q cltq 7086   +P. cpp 7094   <P cltp 7096

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-eprel 4206  df-id 4210  df-po 4213  df-iso 4214  df-iord 4283  df-on 4285  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-irdg 6260  df-1o 6306  df-2o 6307  df-oadd 6310  df-omul 6311  df-er 6422  df-ec 6424  df-qs 6428  df-ni 7105  df-pli 7106  df-mi 7107  df-lti 7108  df-plpq 7145  df-mpq 7146  df-enq 7148  df-nqqs 7149  df-plqqs 7150  df-mqqs 7151  df-1nqqs 7152  df-rq 7153  df-ltnqqs 7154  df-enq0 7225  df-nq0 7226  df-0nq0 7227  df-plq0 7228  df-mq0 7229  df-inp 7267  df-iplp 7269  df-iltp 7271

This theorem is referenced by:  cauappcvgpr  7463