Theorem cauappcvgprlemlim 7745

Description: Lemma for cauappcvgpr 7746. 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 276	. . . . 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 276	. . . . 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 276	. . . . 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 529	. . . . 5 |- ( ( ph /\ ( x e. Q. /\ y e. Q. ) ) -> x e. Q. )
9		simprr 531	. . . . 5 |- ( ( ph /\ ( x e. Q. /\ y e. Q. ) ) -> y e. Q. )
10	2, 4, 6, 7, 8, 9	cauappcvgprlem1 7743	. . . 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 7744	. . . 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 306	. . 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 2579	. 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 5561	. . . . . . . 8 |- ( x = q -> ( F ` x ) = ( F ` q ) )
15	14	breq2d 4046	. . . . . . 7 |- ( x = q -> ( l <Q ( F ` x ) <-> l <Q ( F ` q ) ) )
16	15	abbidv 2314	. . . . . 6 |- ( x = q -> { l | l <Q ( F ` x ) } = { l | l <Q ( F ` q ) } )
17	14	breq1d 4044	. . . . . . 7 |- ( x = q -> ( ( F ` x ) <Q u <-> ( F ` q ) <Q u ) )
18	17	abbidv 2314	. . . . . 6 |- ( x = q -> { u | ( F ` x ) <Q u } = { u | ( F ` q ) <Q u } )
19	16, 18	opeq12d 3817	. . . . 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 5932	. . . . . . . . 9 |- ( x = q -> ( x +Q y ) = ( q +Q y ) )
21	20	breq2d 4046	. . . . . . . 8 |- ( x = q -> ( l <Q ( x +Q y ) <-> l <Q ( q +Q y ) ) )
22	21	abbidv 2314	. . . . . . 7 |- ( x = q -> { l | l <Q ( x +Q y ) } = { l | l <Q ( q +Q y ) } )
23	20	breq1d 4044	. . . . . . . 8 |- ( x = q -> ( ( x +Q y ) <Q u <-> ( q +Q y ) <Q u ) )
24	23	abbidv 2314	. . . . . . 7 |- ( x = q -> { u | ( x +Q y ) <Q u } = { u | ( q +Q y ) <Q u } )
25	22, 24	opeq12d 3817	. . . . . 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 5941	. . . . 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 4047	. . . 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 5943	. . . . . . . 8 |- ( x = q -> ( ( F ` x ) +Q ( x +Q y ) ) = ( ( F ` q ) +Q ( q +Q y ) ) )
29	28	breq2d 4046	. . . . . . 7 |- ( x = q -> ( l <Q ( ( F ` x ) +Q ( x +Q y ) ) <-> l <Q ( ( F ` q ) +Q ( q +Q y ) ) ) )
30	29	abbidv 2314	. . . . . 6 |- ( x = q -> { l | l <Q ( ( F ` x ) +Q ( x +Q y ) ) } = { l | l <Q ( ( F ` q ) +Q ( q +Q y ) ) } )
31	28	breq1d 4044	. . . . . . 7 |- ( x = q -> ( ( ( F ` x ) +Q ( x +Q y ) ) <Q u <-> ( ( F ` q ) +Q ( q +Q y ) ) <Q u ) )
32	31	abbidv 2314	. . . . . 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 3817	. . . . 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 4046	. . . 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 473	. . 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 5933	. . . . . . . . 9 |- ( y = r -> ( q +Q y ) = ( q +Q r ) )
37	36	breq2d 4046	. . . . . . . 8 |- ( y = r -> ( l <Q ( q +Q y ) <-> l <Q ( q +Q r ) ) )
38	37	abbidv 2314	. . . . . . 7 |- ( y = r -> { l | l <Q ( q +Q y ) } = { l | l <Q ( q +Q r ) } )
39	36	breq1d 4044	. . . . . . . 8 |- ( y = r -> ( ( q +Q y ) <Q u <-> ( q +Q r ) <Q u ) )
40	39	abbidv 2314	. . . . . . 7 |- ( y = r -> { u | ( q +Q y ) <Q u } = { u | ( q +Q r ) <Q u } )
41	38, 40	opeq12d 3817	. . . . . 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 5941	. . . . 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 4046	. . . 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 5941	. . . . . . . 8 |- ( y = r -> ( ( F ` q ) +Q ( q +Q y ) ) = ( ( F ` q ) +Q ( q +Q r ) ) )
45	44	breq2d 4046	. . . . . . 7 |- ( y = r -> ( l <Q ( ( F ` q ) +Q ( q +Q y ) ) <-> l <Q ( ( F ` q ) +Q ( q +Q r ) ) ) )
46	45	abbidv 2314	. . . . . 6 |- ( y = r -> { l | l <Q ( ( F ` q ) +Q ( q +Q y ) ) } = { l | l <Q ( ( F ` q ) +Q ( q +Q r ) ) } )
47	44	breq1d 4044	. . . . . . 7 |- ( y = r -> ( ( ( F ` q ) +Q ( q +Q y ) ) <Q u <-> ( ( F ` q ) +Q ( q +Q r ) ) <Q u ) )
48	47	abbidv 2314	. . . . . 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 3817	. . . . 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 4046	. . . 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 473	. . 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 2742	. 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 122	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 104   = wceq 1364   e. wcel 2167   {cab 2182   A.wral 2475   E.wrex 2476   {crab 2479   <.cop 3626   class class class wbr 4034   -->wf 5255   ` cfv 5259  (class class class)co 5925   Q.cnq 7364   +Q cplq 7366   <Q cltq 7369   +P. cpp 7377   <P cltp 7379

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-eprel 4325  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-irdg 6437  df-1o 6483  df-2o 6484  df-oadd 6487  df-omul 6488  df-er 6601  df-ec 6603  df-qs 6607  df-ni 7388  df-pli 7389  df-mi 7390  df-lti 7391  df-plpq 7428  df-mpq 7429  df-enq 7431  df-nqqs 7432  df-plqqs 7433  df-mqqs 7434  df-1nqqs 7435  df-rq 7436  df-ltnqqs 7437  df-enq0 7508  df-nq0 7509  df-0nq0 7510  df-plq0 7511  df-mq0 7512  df-inp 7550  df-iplp 7552  df-iltp 7554

This theorem is referenced by:  cauappcvgpr  7746