Theorem caucvgprlemopu 7934

Description: Lemma for caucvgpr 7945. The upper cut of the putative limit is open. (Contributed by Jim Kingdon, 20-Oct-2020.)

Hypotheses

Ref	Expression
caucvgpr.f	⊢ (𝜑 → 𝐹:N⟶Q)
caucvgpr.cau	⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <Q ((𝐹‘𝑘) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝑛) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )))))
caucvgpr.bnd	⊢ (𝜑 → ∀𝑗 ∈ N 𝐴 <Q (𝐹‘𝑗))
caucvgpr.lim	⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)}, {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}⟩

Assertion

Ref	Expression
caucvgprlemopu	⊢ ((𝜑 ∧ 𝑟 ∈ (2nd ‘𝐿)) → ∃𝑠 ∈ Q (𝑠 <Q 𝑟 ∧ 𝑠 ∈ (2nd ‘𝐿)))

Distinct variable groups:   𝐴,𝑗   𝐹,𝑙,𝑟,𝑠   𝑢,𝐹   𝑗,𝐿,𝑟,𝑠   𝑗,𝑙,𝑠   𝜑,𝑗,𝑟,𝑠   𝑢,𝑗,𝑟,𝑠

Allowed substitution hints:   𝜑(𝑢,𝑘,𝑛,𝑙)   𝐴(𝑢,𝑘,𝑛,𝑠,𝑟,𝑙)   𝐹(𝑗,𝑘,𝑛)   𝐿(𝑢,𝑘,𝑛,𝑙)

Proof of Theorem caucvgprlemopu

Step	Hyp	Ref	Expression
1		breq2 4097	. . . . . 6 ⊢ (𝑢 = 𝑟 → (((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢 ↔ ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑟))
2	1	rexbidv 2534	. . . . 5 ⊢ (𝑢 = 𝑟 → (∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢 ↔ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑟))
3		caucvgpr.lim	. . . . . . 7 ⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)}, {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}⟩
4	3	fveq2i 5651	. . . . . 6 ⊢ (2nd ‘𝐿) = (2nd ‘⟨{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)}, {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}⟩)
5		nqex 7626	. . . . . . . 8 ⊢ Q ∈ V
6	5	rabex 4239	. . . . . . 7 ⊢ {𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)} ∈ V
7	5	rabex 4239	. . . . . . 7 ⊢ {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢} ∈ V
8	6, 7	op2nd 6319	. . . . . 6 ⊢ (2nd ‘⟨{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)}, {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}⟩) = {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}
9	4, 8	eqtri 2252	. . . . 5 ⊢ (2nd ‘𝐿) = {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}
10	2, 9	elrab2 2966	. . . 4 ⊢ (𝑟 ∈ (2nd ‘𝐿) ↔ (𝑟 ∈ Q ∧ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑟))
11	10	simprbi 275	. . 3 ⊢ (𝑟 ∈ (2nd ‘𝐿) → ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑟)
12	11	adantl 277	. 2 ⊢ ((𝜑 ∧ 𝑟 ∈ (2nd ‘𝐿)) → ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑟)
13		simprr 533	. . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ (2nd ‘𝐿)) ∧ (𝑗 ∈ N ∧ ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑟)) → ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑟)
14		ltbtwnnqq 7678	. . . 4 ⊢ (((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑟 ↔ ∃𝑠 ∈ Q (((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠 ∧ 𝑠 <Q 𝑟))
15	13, 14	sylib 122	. . 3 ⊢ (((𝜑 ∧ 𝑟 ∈ (2nd ‘𝐿)) ∧ (𝑗 ∈ N ∧ ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑟)) → ∃𝑠 ∈ Q (((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠 ∧ 𝑠 <Q 𝑟))
16		simprr 533	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑟 ∈ (2nd ‘𝐿)) ∧ (𝑗 ∈ N ∧ ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑟)) ∧ 𝑠 ∈ Q) ∧ (((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠 ∧ 𝑠 <Q 𝑟)) → 𝑠 <Q 𝑟)
17		simplr 529	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑟 ∈ (2nd ‘𝐿)) ∧ (𝑗 ∈ N ∧ ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑟)) ∧ 𝑠 ∈ Q) ∧ (((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠 ∧ 𝑠 <Q 𝑟)) → 𝑠 ∈ Q)
18		simplrl 537	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑟 ∈ (2nd ‘𝐿)) ∧ (𝑗 ∈ N ∧ ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑟)) ∧ 𝑠 ∈ Q) → 𝑗 ∈ N)
19	18	adantr 276	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑟 ∈ (2nd ‘𝐿)) ∧ (𝑗 ∈ N ∧ ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑟)) ∧ 𝑠 ∈ Q) ∧ (((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠 ∧ 𝑠 <Q 𝑟)) → 𝑗 ∈ N)
20		simprl 531	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑟 ∈ (2nd ‘𝐿)) ∧ (𝑗 ∈ N ∧ ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑟)) ∧ 𝑠 ∈ Q) ∧ (((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠 ∧ 𝑠 <Q 𝑟)) → ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠)
21		rspe 2582	. . . . . . . 8 ⊢ ((𝑗 ∈ N ∧ ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠) → ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠)
22	19, 20, 21	syl2anc 411	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑟 ∈ (2nd ‘𝐿)) ∧ (𝑗 ∈ N ∧ ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑟)) ∧ 𝑠 ∈ Q) ∧ (((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠 ∧ 𝑠 <Q 𝑟)) → ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠)
23		breq2 4097	. . . . . . . . 9 ⊢ (𝑢 = 𝑠 → (((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢 ↔ ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠))
24	23	rexbidv 2534	. . . . . . . 8 ⊢ (𝑢 = 𝑠 → (∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢 ↔ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠))
25	24, 9	elrab2 2966	. . . . . . 7 ⊢ (𝑠 ∈ (2nd ‘𝐿) ↔ (𝑠 ∈ Q ∧ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠))
26	17, 22, 25	sylanbrc 417	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑟 ∈ (2nd ‘𝐿)) ∧ (𝑗 ∈ N ∧ ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑟)) ∧ 𝑠 ∈ Q) ∧ (((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠 ∧ 𝑠 <Q 𝑟)) → 𝑠 ∈ (2nd ‘𝐿))
27	16, 26	jca 306	. . . . 5 ⊢ (((((𝜑 ∧ 𝑟 ∈ (2nd ‘𝐿)) ∧ (𝑗 ∈ N ∧ ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑟)) ∧ 𝑠 ∈ Q) ∧ (((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠 ∧ 𝑠 <Q 𝑟)) → (𝑠 <Q 𝑟 ∧ 𝑠 ∈ (2nd ‘𝐿)))
28	27	ex 115	. . . 4 ⊢ ((((𝜑 ∧ 𝑟 ∈ (2nd ‘𝐿)) ∧ (𝑗 ∈ N ∧ ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑟)) ∧ 𝑠 ∈ Q) → ((((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠 ∧ 𝑠 <Q 𝑟) → (𝑠 <Q 𝑟 ∧ 𝑠 ∈ (2nd ‘𝐿))))
29	28	reximdva 2635	. . 3 ⊢ (((𝜑 ∧ 𝑟 ∈ (2nd ‘𝐿)) ∧ (𝑗 ∈ N ∧ ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑟)) → (∃𝑠 ∈ Q (((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠 ∧ 𝑠 <Q 𝑟) → ∃𝑠 ∈ Q (𝑠 <Q 𝑟 ∧ 𝑠 ∈ (2nd ‘𝐿))))
30	15, 29	mpd 13	. 2 ⊢ (((𝜑 ∧ 𝑟 ∈ (2nd ‘𝐿)) ∧ (𝑗 ∈ N ∧ ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑟)) → ∃𝑠 ∈ Q (𝑠 <Q 𝑟 ∧ 𝑠 ∈ (2nd ‘𝐿)))
31	12, 30	rexlimddv 2656	1 ⊢ ((𝜑 ∧ 𝑟 ∈ (2nd ‘𝐿)) → ∃𝑠 ∈ Q (𝑠 <Q 𝑟 ∧ 𝑠 ∈ (2nd ‘𝐿)))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398   ∈ wcel 2202  ∀wral 2511  ∃wrex 2512  {crab 2515  ⟨cop 3676   class class class wbr 4093  ⟶wf 5329  ‘cfv 5333  (class class class)co 6028  2^nd c2nd 6311  1_oc1o 6618  [cec 6743  Ncnpi 7535   <_N clti 7538   ~_Q ceq 7542  Qcnq 7543   +_Q cplq 7545  *_Qcrq 7547   <_Q cltq 7548

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-eprel 4392  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-irdg 6579  df-1o 6625  df-oadd 6629  df-omul 6630  df-er 6745  df-ec 6747  df-qs 6751  df-ni 7567  df-pli 7568  df-mi 7569  df-lti 7570  df-plpq 7607  df-mpq 7608  df-enq 7610  df-nqqs 7611  df-plqqs 7612  df-mqqs 7613  df-1nqqs 7614  df-rq 7615  df-ltnqqs 7616

This theorem is referenced by:  caucvgprlemrnd  7936