Theorem cauappcvgprlemopu 7776

Description: Lemma for cauappcvgpr 7790. The upper cut of the putative limit is open. (Contributed by Jim Kingdon, 4-Aug-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
cauappcvgprlemopu	|- ( ( ph /\ r e. ( 2nd ` L ) ) -> E. s e. Q. ( s <Q r /\ s e. ( 2nd ` L ) ) )

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

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

Proof of Theorem cauappcvgprlemopu

Step	Hyp	Ref	Expression
1		breq2 4054	. . . . . 6 |- ( u = r -> ( ( ( F ` q ) +Q q ) <Q u <-> ( ( F ` q ) +Q q ) <Q r ) )
2	1	rexbidv 2508	. . . . 5 |- ( u = r -> ( E. q e. Q. ( ( F ` q ) +Q q ) <Q u <-> E. q e. Q. ( ( F ` q ) +Q q ) <Q r ) )
3		cauappcvgpr.lim	. . . . . . 7 |- 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 } >.
4	3	fveq2i 5591	. . . . . 6 |- ( 2nd ` L ) = ( 2nd ` <. { 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 } >. )
5		nqex 7491	. . . . . . . 8 |- Q. e. _V
6	5	rabex 4195	. . . . . . 7 |- { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) } e. _V
7	5	rabex 4195	. . . . . . 7 |- { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u } e. _V
8	6, 7	op2nd 6245	. . . . . 6 |- ( 2nd ` <. { 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 } >. ) = { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u }
9	4, 8	eqtri 2227	. . . . 5 |- ( 2nd ` L ) = { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u }
10	2, 9	elrab2 2936	. . . 4 |- ( r e. ( 2nd ` L ) <-> ( r e. Q. /\ E. q e. Q. ( ( F ` q ) +Q q ) <Q r ) )
11	10	simprbi 275	. . 3 |- ( r e. ( 2nd ` L ) -> E. q e. Q. ( ( F ` q ) +Q q ) <Q r )
12	11	adantl 277	. 2 |- ( ( ph /\ r e. ( 2nd ` L ) ) -> E. q e. Q. ( ( F ` q ) +Q q ) <Q r )
13		simprr 531	. . . 4 |- ( ( ( ph /\ r e. ( 2nd ` L ) ) /\ ( q e. Q. /\ ( ( F ` q ) +Q q ) <Q r ) ) -> ( ( F ` q ) +Q q ) <Q r )
14		ltbtwnnqq 7543	. . . 4 |- ( ( ( F ` q ) +Q q ) <Q r <-> E. s e. Q. ( ( ( F ` q ) +Q q ) <Q s /\ s <Q r ) )
15	13, 14	sylib 122	. . 3 |- ( ( ( ph /\ r e. ( 2nd ` L ) ) /\ ( q e. Q. /\ ( ( F ` q ) +Q q ) <Q r ) ) -> E. s e. Q. ( ( ( F ` q ) +Q q ) <Q s /\ s <Q r ) )
16		simprr 531	. . . . . 6 |- ( ( ( ( ( ph /\ r e. ( 2nd ` L ) ) /\ ( q e. Q. /\ ( ( F ` q ) +Q q ) <Q r ) ) /\ s e. Q. ) /\ ( ( ( F ` q ) +Q q ) <Q s /\ s <Q r ) ) -> s <Q r )
17		simplr 528	. . . . . . 7 |- ( ( ( ( ( ph /\ r e. ( 2nd ` L ) ) /\ ( q e. Q. /\ ( ( F ` q ) +Q q ) <Q r ) ) /\ s e. Q. ) /\ ( ( ( F ` q ) +Q q ) <Q s /\ s <Q r ) ) -> s e. Q. )
18		simplrl 535	. . . . . . . . 9 |- ( ( ( ( ph /\ r e. ( 2nd ` L ) ) /\ ( q e. Q. /\ ( ( F ` q ) +Q q ) <Q r ) ) /\ s e. Q. ) -> q e. Q. )
19	18	adantr 276	. . . . . . . 8 |- ( ( ( ( ( ph /\ r e. ( 2nd ` L ) ) /\ ( q e. Q. /\ ( ( F ` q ) +Q q ) <Q r ) ) /\ s e. Q. ) /\ ( ( ( F ` q ) +Q q ) <Q s /\ s <Q r ) ) -> q e. Q. )
20		simprl 529	. . . . . . . 8 |- ( ( ( ( ( ph /\ r e. ( 2nd ` L ) ) /\ ( q e. Q. /\ ( ( F ` q ) +Q q ) <Q r ) ) /\ s e. Q. ) /\ ( ( ( F ` q ) +Q q ) <Q s /\ s <Q r ) ) -> ( ( F ` q ) +Q q ) <Q s )
21		rspe 2556	. . . . . . . 8 |- ( ( q e. Q. /\ ( ( F ` q ) +Q q ) <Q s ) -> E. q e. Q. ( ( F ` q ) +Q q ) <Q s )
22	19, 20, 21	syl2anc 411	. . . . . . 7 |- ( ( ( ( ( ph /\ r e. ( 2nd ` L ) ) /\ ( q e. Q. /\ ( ( F ` q ) +Q q ) <Q r ) ) /\ s e. Q. ) /\ ( ( ( F ` q ) +Q q ) <Q s /\ s <Q r ) ) -> E. q e. Q. ( ( F ` q ) +Q q ) <Q s )
23		breq2 4054	. . . . . . . . 9 |- ( u = s -> ( ( ( F ` q ) +Q q ) <Q u <-> ( ( F ` q ) +Q q ) <Q s ) )
24	23	rexbidv 2508	. . . . . . . 8 |- ( u = s -> ( E. q e. Q. ( ( F ` q ) +Q q ) <Q u <-> E. q e. Q. ( ( F ` q ) +Q q ) <Q s ) )
25	24, 9	elrab2 2936	. . . . . . 7 |- ( s e. ( 2nd ` L ) <-> ( s e. Q. /\ E. q e. Q. ( ( F ` q ) +Q q ) <Q s ) )
26	17, 22, 25	sylanbrc 417	. . . . . 6 |- ( ( ( ( ( ph /\ r e. ( 2nd ` L ) ) /\ ( q e. Q. /\ ( ( F ` q ) +Q q ) <Q r ) ) /\ s e. Q. ) /\ ( ( ( F ` q ) +Q q ) <Q s /\ s <Q r ) ) -> s e. ( 2nd ` L ) )
27	16, 26	jca 306	. . . . 5 |- ( ( ( ( ( ph /\ r e. ( 2nd ` L ) ) /\ ( q e. Q. /\ ( ( F ` q ) +Q q ) <Q r ) ) /\ s e. Q. ) /\ ( ( ( F ` q ) +Q q ) <Q s /\ s <Q r ) ) -> ( s <Q r /\ s e. ( 2nd ` L ) ) )
28	27	ex 115	. . . 4 |- ( ( ( ( ph /\ r e. ( 2nd ` L ) ) /\ ( q e. Q. /\ ( ( F ` q ) +Q q ) <Q r ) ) /\ s e. Q. ) -> ( ( ( ( F ` q ) +Q q ) <Q s /\ s <Q r ) -> ( s <Q r /\ s e. ( 2nd ` L ) ) ) )
29	28	reximdva 2609	. . 3 |- ( ( ( ph /\ r e. ( 2nd ` L ) ) /\ ( q e. Q. /\ ( ( F ` q ) +Q q ) <Q r ) ) -> ( E. s e. Q. ( ( ( F ` q ) +Q q ) <Q s /\ s <Q r ) -> E. s e. Q. ( s <Q r /\ s e. ( 2nd ` L ) ) ) )
30	15, 29	mpd 13	. 2 |- ( ( ( ph /\ r e. ( 2nd ` L ) ) /\ ( q e. Q. /\ ( ( F ` q ) +Q q ) <Q r ) ) -> E. s e. Q. ( s <Q r /\ s e. ( 2nd ` L ) ) )
31	12, 30	rexlimddv 2629	1 |- ( ( ph /\ r e. ( 2nd ` L ) ) -> E. s e. Q. ( s <Q r /\ s e. ( 2nd ` L ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2177   A.wral 2485   E.wrex 2486   {crab 2489   <.cop 3640   class class class wbr 4050   -->wf 5275   ` cfv 5279  (class class class)co 5956   2ndc2nd 6237   Q.cnq 7408   +Q cplq 7410   <Q cltq 7413

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4166  ax-sep 4169  ax-nul 4177  ax-pow 4225  ax-pr 4260  ax-un 4487  ax-setind 4592  ax-iinf 4643

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3622  df-sn 3643  df-pr 3644  df-op 3646  df-uni 3856  df-int 3891  df-iun 3934  df-br 4051  df-opab 4113  df-mpt 4114  df-tr 4150  df-eprel 4343  df-id 4347  df-po 4350  df-iso 4351  df-iord 4420  df-on 4422  df-suc 4425  df-iom 4646  df-xp 4688  df-rel 4689  df-cnv 4690  df-co 4691  df-dm 4692  df-rn 4693  df-res 4694  df-ima 4695  df-iota 5240  df-fun 5281  df-fn 5282  df-f 5283  df-f1 5284  df-fo 5285  df-f1o 5286  df-fv 5287  df-ov 5959  df-oprab 5960  df-mpo 5961  df-1st 6238  df-2nd 6239  df-recs 6403  df-irdg 6468  df-1o 6514  df-oadd 6518  df-omul 6519  df-er 6632  df-ec 6634  df-qs 6638  df-ni 7432  df-pli 7433  df-mi 7434  df-lti 7435  df-plpq 7472  df-mpq 7473  df-enq 7475  df-nqqs 7476  df-plqqs 7477  df-mqqs 7478  df-1nqqs 7479  df-rq 7480  df-ltnqqs 7481

This theorem is referenced by:  cauappcvgprlemrnd  7778