Theorem cauappcvgprlemlim 7218

Description: Lemma for cauappcvgpr 7219. 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 270	. . . . 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 270	. . . . 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 270	. . . . 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 498	. . . . 5 |- ( ( ph /\ ( x e. Q. /\ y e. Q. ) ) -> x e. Q. )
9		simprr 499	. . . . 5 |- ( ( ph /\ ( x e. Q. /\ y e. Q. ) ) -> y e. Q. )
10	2, 4, 6, 7, 8, 9	cauappcvgprlem1 7216	. . . 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 7217	. . . 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 300	. . 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 2455	. 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 5305	. . . . . . . 8 |- ( x = q -> ( F ` x ) = ( F ` q ) )
15	14	breq2d 3857	. . . . . . 7 |- ( x = q -> ( l <Q ( F ` x ) <-> l <Q ( F ` q ) ) )
16	15	abbidv 2205	. . . . . 6 |- ( x = q -> { l | l <Q ( F ` x ) } = { l | l <Q ( F ` q ) } )
17	14	breq1d 3855	. . . . . . 7 |- ( x = q -> ( ( F ` x ) <Q u <-> ( F ` q ) <Q u ) )
18	17	abbidv 2205	. . . . . 6 |- ( x = q -> { u | ( F ` x ) <Q u } = { u | ( F ` q ) <Q u } )
19	16, 18	opeq12d 3630	. . . . 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 5659	. . . . . . . . 9 |- ( x = q -> ( x +Q y ) = ( q +Q y ) )
21	20	breq2d 3857	. . . . . . . 8 |- ( x = q -> ( l <Q ( x +Q y ) <-> l <Q ( q +Q y ) ) )
22	21	abbidv 2205	. . . . . . 7 |- ( x = q -> { l | l <Q ( x +Q y ) } = { l | l <Q ( q +Q y ) } )
23	20	breq1d 3855	. . . . . . . 8 |- ( x = q -> ( ( x +Q y ) <Q u <-> ( q +Q y ) <Q u ) )
24	23	abbidv 2205	. . . . . . 7 |- ( x = q -> { u | ( x +Q y ) <Q u } = { u | ( q +Q y ) <Q u } )
25	22, 24	opeq12d 3630	. . . . . 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 5668	. . . . 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 3858	. . . 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 5670	. . . . . . . 8 |- ( x = q -> ( ( F ` x ) +Q ( x +Q y ) ) = ( ( F ` q ) +Q ( q +Q y ) ) )
29	28	breq2d 3857	. . . . . . 7 |- ( x = q -> ( l <Q ( ( F ` x ) +Q ( x +Q y ) ) <-> l <Q ( ( F ` q ) +Q ( q +Q y ) ) ) )
30	29	abbidv 2205	. . . . . 6 |- ( x = q -> { l | l <Q ( ( F ` x ) +Q ( x +Q y ) ) } = { l | l <Q ( ( F ` q ) +Q ( q +Q y ) ) } )
31	28	breq1d 3855	. . . . . . 7 |- ( x = q -> ( ( ( F ` x ) +Q ( x +Q y ) ) <Q u <-> ( ( F ` q ) +Q ( q +Q y ) ) <Q u ) )
32	31	abbidv 2205	. . . . . 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 3630	. . . . 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 3857	. . . 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 457	. . 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 5660	. . . . . . . . 9 |- ( y = r -> ( q +Q y ) = ( q +Q r ) )
37	36	breq2d 3857	. . . . . . . 8 |- ( y = r -> ( l <Q ( q +Q y ) <-> l <Q ( q +Q r ) ) )
38	37	abbidv 2205	. . . . . . 7 |- ( y = r -> { l | l <Q ( q +Q y ) } = { l | l <Q ( q +Q r ) } )
39	36	breq1d 3855	. . . . . . . 8 |- ( y = r -> ( ( q +Q y ) <Q u <-> ( q +Q r ) <Q u ) )
40	39	abbidv 2205	. . . . . . 7 |- ( y = r -> { u | ( q +Q y ) <Q u } = { u | ( q +Q r ) <Q u } )
41	38, 40	opeq12d 3630	. . . . . 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 5668	. . . . 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 3857	. . . 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 5668	. . . . . . . 8 |- ( y = r -> ( ( F ` q ) +Q ( q +Q y ) ) = ( ( F ` q ) +Q ( q +Q r ) ) )
45	44	breq2d 3857	. . . . . . 7 |- ( y = r -> ( l <Q ( ( F ` q ) +Q ( q +Q y ) ) <-> l <Q ( ( F ` q ) +Q ( q +Q r ) ) ) )
46	45	abbidv 2205	. . . . . 6 |- ( y = r -> { l | l <Q ( ( F ` q ) +Q ( q +Q y ) ) } = { l | l <Q ( ( F ` q ) +Q ( q +Q r ) ) } )
47	44	breq1d 3855	. . . . . . 7 |- ( y = r -> ( ( ( F ` q ) +Q ( q +Q y ) ) <Q u <-> ( ( F ` q ) +Q ( q +Q r ) ) <Q u ) )
48	47	abbidv 2205	. . . . . 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 3630	. . . . 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 3857	. . . 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 457	. . 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 2598	. 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 120	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 102   = wceq 1289   e. wcel 1438   {cab 2074   A.wral 2359   E.wrex 2360   {crab 2363   <.cop 3449   class class class wbr 3845   -->wf 5011   ` cfv 5015  (class class class)co 5652   Q.cnq 6837   +Q cplq 6839   <Q cltq 6842   +P. cpp 6850   <P cltp 6852

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403

This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-eprel 4116  df-id 4120  df-po 4123  df-iso 4124  df-iord 4193  df-on 4195  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-irdg 6135  df-1o 6181  df-2o 6182  df-oadd 6185  df-omul 6186  df-er 6290  df-ec 6292  df-qs 6296  df-ni 6861  df-pli 6862  df-mi 6863  df-lti 6864  df-plpq 6901  df-mpq 6902  df-enq 6904  df-nqqs 6905  df-plqqs 6906  df-mqqs 6907  df-1nqqs 6908  df-rq 6909  df-ltnqqs 6910  df-enq0 6981  df-nq0 6982  df-0nq0 6983  df-plq0 6984  df-mq0 6985  df-inp 7023  df-iplp 7025  df-iltp 7027

This theorem is referenced by:  cauappcvgpr  7219