Theorem cauappcvgprlemlim 7493

Description: Lemma for cauappcvgpr 7494. 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 7491	. . . 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 7492	. . . 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 2517	. 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 5429	. . . . . . . 8 |- ( x = q -> ( F ` x ) = ( F ` q ) )
15	14	breq2d 3949	. . . . . . 7 |- ( x = q -> ( l <Q ( F ` x ) <-> l <Q ( F ` q ) ) )
16	15	abbidv 2258	. . . . . 6 |- ( x = q -> { l | l <Q ( F ` x ) } = { l | l <Q ( F ` q ) } )
17	14	breq1d 3947	. . . . . . 7 |- ( x = q -> ( ( F ` x ) <Q u <-> ( F ` q ) <Q u ) )
18	17	abbidv 2258	. . . . . 6 |- ( x = q -> { u | ( F ` x ) <Q u } = { u | ( F ` q ) <Q u } )
19	16, 18	opeq12d 3721	. . . . 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 5789	. . . . . . . . 9 |- ( x = q -> ( x +Q y ) = ( q +Q y ) )
21	20	breq2d 3949	. . . . . . . 8 |- ( x = q -> ( l <Q ( x +Q y ) <-> l <Q ( q +Q y ) ) )
22	21	abbidv 2258	. . . . . . 7 |- ( x = q -> { l | l <Q ( x +Q y ) } = { l | l <Q ( q +Q y ) } )
23	20	breq1d 3947	. . . . . . . 8 |- ( x = q -> ( ( x +Q y ) <Q u <-> ( q +Q y ) <Q u ) )
24	23	abbidv 2258	. . . . . . 7 |- ( x = q -> { u | ( x +Q y ) <Q u } = { u | ( q +Q y ) <Q u } )
25	22, 24	opeq12d 3721	. . . . . 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 5798	. . . . 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 3950	. . . 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 5800	. . . . . . . 8 |- ( x = q -> ( ( F ` x ) +Q ( x +Q y ) ) = ( ( F ` q ) +Q ( q +Q y ) ) )
29	28	breq2d 3949	. . . . . . 7 |- ( x = q -> ( l <Q ( ( F ` x ) +Q ( x +Q y ) ) <-> l <Q ( ( F ` q ) +Q ( q +Q y ) ) ) )
30	29	abbidv 2258	. . . . . 6 |- ( x = q -> { l | l <Q ( ( F ` x ) +Q ( x +Q y ) ) } = { l | l <Q ( ( F ` q ) +Q ( q +Q y ) ) } )
31	28	breq1d 3947	. . . . . . 7 |- ( x = q -> ( ( ( F ` x ) +Q ( x +Q y ) ) <Q u <-> ( ( F ` q ) +Q ( q +Q y ) ) <Q u ) )
32	31	abbidv 2258	. . . . . 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 3721	. . . . 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 3949	. . . 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 5790	. . . . . . . . 9 |- ( y = r -> ( q +Q y ) = ( q +Q r ) )
37	36	breq2d 3949	. . . . . . . 8 |- ( y = r -> ( l <Q ( q +Q y ) <-> l <Q ( q +Q r ) ) )
38	37	abbidv 2258	. . . . . . 7 |- ( y = r -> { l | l <Q ( q +Q y ) } = { l | l <Q ( q +Q r ) } )
39	36	breq1d 3947	. . . . . . . 8 |- ( y = r -> ( ( q +Q y ) <Q u <-> ( q +Q r ) <Q u ) )
40	39	abbidv 2258	. . . . . . 7 |- ( y = r -> { u | ( q +Q y ) <Q u } = { u | ( q +Q r ) <Q u } )
41	38, 40	opeq12d 3721	. . . . . 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 5798	. . . . 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 3949	. . . 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 5798	. . . . . . . 8 |- ( y = r -> ( ( F ` q ) +Q ( q +Q y ) ) = ( ( F ` q ) +Q ( q +Q r ) ) )
45	44	breq2d 3949	. . . . . . 7 |- ( y = r -> ( l <Q ( ( F ` q ) +Q ( q +Q y ) ) <-> l <Q ( ( F ` q ) +Q ( q +Q r ) ) ) )
46	45	abbidv 2258	. . . . . 6 |- ( y = r -> { l | l <Q ( ( F ` q ) +Q ( q +Q y ) ) } = { l | l <Q ( ( F ` q ) +Q ( q +Q r ) ) } )
47	44	breq1d 3947	. . . . . . 7 |- ( y = r -> ( ( ( F ` q ) +Q ( q +Q y ) ) <Q u <-> ( ( F ` q ) +Q ( q +Q r ) ) <Q u ) )
48	47	abbidv 2258	. . . . . 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 3721	. . . . 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 3949	. . . 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 2668	. 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 1332   e. wcel 1481   {cab 2126   A.wral 2417   E.wrex 2418   {crab 2421   <.cop 3535   class class class wbr 3937   -->wf 5127   ` cfv 5131  (class class class)co 5782   Q.cnq 7112   +Q cplq 7114   <Q cltq 7117   +P. cpp 7125   <P cltp 7127

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-eprel 4219  df-id 4223  df-po 4226  df-iso 4227  df-iord 4296  df-on 4298  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-irdg 6275  df-1o 6321  df-2o 6322  df-oadd 6325  df-omul 6326  df-er 6437  df-ec 6439  df-qs 6443  df-ni 7136  df-pli 7137  df-mi 7138  df-lti 7139  df-plpq 7176  df-mpq 7177  df-enq 7179  df-nqqs 7180  df-plqqs 7181  df-mqqs 7182  df-1nqqs 7183  df-rq 7184  df-ltnqqs 7185  df-enq0 7256  df-nq0 7257  df-0nq0 7258  df-plq0 7259  df-mq0 7260  df-inp 7298  df-iplp 7300  df-iltp 7302

This theorem is referenced by:  cauappcvgpr  7494