Theorem caucvgprlemopu 7612

Description: Lemma for caucvgpr 7623. 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 3986	. . . . . 6 ⊢ (𝑢 = 𝑟 → (((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢 ↔ ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑟))
2	1	rexbidv 2467	. . . . 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 5489	. . . . . 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 7304	. . . . . . . 8 ⊢ Q ∈ V
6	5	rabex 4126	. . . . . . 7 ⊢ {𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)} ∈ V
7	5	rabex 4126	. . . . . . 7 ⊢ {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢} ∈ V
8	6, 7	op2nd 6115	. . . . . 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 2186	. . . . 5 ⊢ (2nd ‘𝐿) = {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}
10	2, 9	elrab2 2885	. . . 4 ⊢ (𝑟 ∈ (2nd ‘𝐿) ↔ (𝑟 ∈ Q ∧ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑟))
11	10	simprbi 273	. . 3 ⊢ (𝑟 ∈ (2nd ‘𝐿) → ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑟)
12	11	adantl 275	. 2 ⊢ ((𝜑 ∧ 𝑟 ∈ (2nd ‘𝐿)) → ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑟)
13		simprr 522	. . . 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 7356	. . . 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 121	. . 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 522	. . . . . 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 520	. . . . . . 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 525	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑟 ∈ (2nd ‘𝐿)) ∧ (𝑗 ∈ N ∧ ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑟)) ∧ 𝑠 ∈ Q) → 𝑗 ∈ N)
19	18	adantr 274	. . . . . . . 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 521	. . . . . . . 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 2515	. . . . . . . 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 409	. . . . . . 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 3986	. . . . . . . . 9 ⊢ (𝑢 = 𝑠 → (((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢 ↔ ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠))
24	23	rexbidv 2467	. . . . . . . 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 2885	. . . . . . 7 ⊢ (𝑠 ∈ (2nd ‘𝐿) ↔ (𝑠 ∈ Q ∧ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠))
26	17, 22, 25	sylanbrc 414	. . . . . 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 304	. . . . 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 114	. . . 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 2568	. . 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 2588	1 ⊢ ((𝜑 ∧ 𝑟 ∈ (2nd ‘𝐿)) → ∃𝑠 ∈ Q (𝑠 <Q 𝑟 ∧ 𝑠 ∈ (2nd ‘𝐿)))

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1343   ∈ wcel 2136  ∀wral 2444  ∃wrex 2445  {crab 2448  ⟨cop 3579   class class class wbr 3982  ⟶wf 5184  ‘cfv 5188  (class class class)co 5842  2^nd c2nd 6107  1_oc1o 6377  [cec 6499  Ncnpi 7213   <_N clti 7216   ~_Q ceq 7220  Qcnq 7221   +_Q cplq 7223  *_Qcrq 7225   <_Q cltq 7226

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-eprel 4267  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-1o 6384  df-oadd 6388  df-omul 6389  df-er 6501  df-ec 6503  df-qs 6507  df-ni 7245  df-pli 7246  df-mi 7247  df-lti 7248  df-plpq 7285  df-mpq 7286  df-enq 7288  df-nqqs 7289  df-plqqs 7290  df-mqqs 7291  df-1nqqs 7292  df-rq 7293  df-ltnqqs 7294

This theorem is referenced by:  caucvgprlemrnd  7614