Theorem cauappcvgprlemlim 7581

Description: Lemma for cauappcvgpr 7582. 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 521	. . . . 5 |- ( ( ph /\ ( x e. Q. /\ y e. Q. ) ) -> x e. Q. )
9		simprr 522	. . . . 5 |- ( ( ph /\ ( x e. Q. /\ y e. Q. ) ) -> y e. Q. )
10	2, 4, 6, 7, 8, 9	cauappcvgprlem1 7579	. . . 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 7580	. . . 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 2539	. 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 5468	. . . . . . . 8 |- ( x = q -> ( F ` x ) = ( F ` q ) )
15	14	breq2d 3977	. . . . . . 7 |- ( x = q -> ( l <Q ( F ` x ) <-> l <Q ( F ` q ) ) )
16	15	abbidv 2275	. . . . . 6 |- ( x = q -> { l | l <Q ( F ` x ) } = { l | l <Q ( F ` q ) } )
17	14	breq1d 3975	. . . . . . 7 |- ( x = q -> ( ( F ` x ) <Q u <-> ( F ` q ) <Q u ) )
18	17	abbidv 2275	. . . . . 6 |- ( x = q -> { u | ( F ` x ) <Q u } = { u | ( F ` q ) <Q u } )
19	16, 18	opeq12d 3749	. . . . 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 5831	. . . . . . . . 9 |- ( x = q -> ( x +Q y ) = ( q +Q y ) )
21	20	breq2d 3977	. . . . . . . 8 |- ( x = q -> ( l <Q ( x +Q y ) <-> l <Q ( q +Q y ) ) )
22	21	abbidv 2275	. . . . . . 7 |- ( x = q -> { l | l <Q ( x +Q y ) } = { l | l <Q ( q +Q y ) } )
23	20	breq1d 3975	. . . . . . . 8 |- ( x = q -> ( ( x +Q y ) <Q u <-> ( q +Q y ) <Q u ) )
24	23	abbidv 2275	. . . . . . 7 |- ( x = q -> { u | ( x +Q y ) <Q u } = { u | ( q +Q y ) <Q u } )
25	22, 24	opeq12d 3749	. . . . . 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 5840	. . . . 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 3978	. . . 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 5842	. . . . . . . 8 |- ( x = q -> ( ( F ` x ) +Q ( x +Q y ) ) = ( ( F ` q ) +Q ( q +Q y ) ) )
29	28	breq2d 3977	. . . . . . 7 |- ( x = q -> ( l <Q ( ( F ` x ) +Q ( x +Q y ) ) <-> l <Q ( ( F ` q ) +Q ( q +Q y ) ) ) )
30	29	abbidv 2275	. . . . . 6 |- ( x = q -> { l | l <Q ( ( F ` x ) +Q ( x +Q y ) ) } = { l | l <Q ( ( F ` q ) +Q ( q +Q y ) ) } )
31	28	breq1d 3975	. . . . . . 7 |- ( x = q -> ( ( ( F ` x ) +Q ( x +Q y ) ) <Q u <-> ( ( F ` q ) +Q ( q +Q y ) ) <Q u ) )
32	31	abbidv 2275	. . . . . 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 3749	. . . . 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 3977	. . . 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 465	. . 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 5832	. . . . . . . . 9 |- ( y = r -> ( q +Q y ) = ( q +Q r ) )
37	36	breq2d 3977	. . . . . . . 8 |- ( y = r -> ( l <Q ( q +Q y ) <-> l <Q ( q +Q r ) ) )
38	37	abbidv 2275	. . . . . . 7 |- ( y = r -> { l | l <Q ( q +Q y ) } = { l | l <Q ( q +Q r ) } )
39	36	breq1d 3975	. . . . . . . 8 |- ( y = r -> ( ( q +Q y ) <Q u <-> ( q +Q r ) <Q u ) )
40	39	abbidv 2275	. . . . . . 7 |- ( y = r -> { u | ( q +Q y ) <Q u } = { u | ( q +Q r ) <Q u } )
41	38, 40	opeq12d 3749	. . . . . 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 5840	. . . . 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 3977	. . . 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 5840	. . . . . . . 8 |- ( y = r -> ( ( F ` q ) +Q ( q +Q y ) ) = ( ( F ` q ) +Q ( q +Q r ) ) )
45	44	breq2d 3977	. . . . . . 7 |- ( y = r -> ( l <Q ( ( F ` q ) +Q ( q +Q y ) ) <-> l <Q ( ( F ` q ) +Q ( q +Q r ) ) ) )
46	45	abbidv 2275	. . . . . 6 |- ( y = r -> { l | l <Q ( ( F ` q ) +Q ( q +Q y ) ) } = { l | l <Q ( ( F ` q ) +Q ( q +Q r ) ) } )
47	44	breq1d 3975	. . . . . . 7 |- ( y = r -> ( ( ( F ` q ) +Q ( q +Q y ) ) <Q u <-> ( ( F ` q ) +Q ( q +Q r ) ) <Q u ) )
48	47	abbidv 2275	. . . . . 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 3749	. . . . 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 3977	. . . 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 465	. . 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 2691	. 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 1335   e. wcel 2128   {cab 2143   A.wral 2435   E.wrex 2436   {crab 2439   <.cop 3563   class class class wbr 3965   -->wf 5166   ` cfv 5170  (class class class)co 5824   Q.cnq 7200   +Q cplq 7202   <Q cltq 7205   +P. cpp 7213   <P cltp 7215

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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-iinf 4547

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-eprel 4249  df-id 4253  df-po 4256  df-iso 4257  df-iord 4326  df-on 4328  df-suc 4331  df-iom 4550  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-fo 5176  df-f1o 5177  df-fv 5178  df-ov 5827  df-oprab 5828  df-mpo 5829  df-1st 6088  df-2nd 6089  df-recs 6252  df-irdg 6317  df-1o 6363  df-2o 6364  df-oadd 6367  df-omul 6368  df-er 6480  df-ec 6482  df-qs 6486  df-ni 7224  df-pli 7225  df-mi 7226  df-lti 7227  df-plpq 7264  df-mpq 7265  df-enq 7267  df-nqqs 7268  df-plqqs 7269  df-mqqs 7270  df-1nqqs 7271  df-rq 7272  df-ltnqqs 7273  df-enq0 7344  df-nq0 7345  df-0nq0 7346  df-plq0 7347  df-mq0 7348  df-inp 7386  df-iplp 7388  df-iltp 7390

This theorem is referenced by:  cauappcvgpr  7582