Theorem caucvgprlem1 7435

Description: Lemma for caucvgpr 7438. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 3-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}⟩
caucvgprlemlim.q	⊢ (𝜑 → 𝑄 ∈ Q)
caucvgprlemlim.jk	⊢ (𝜑 → 𝐽 <N 𝐾)
caucvgprlemlim.jkq	⊢ (𝜑 → (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑄)

Assertion

Ref	Expression
caucvgprlem1	⊢ (𝜑 → ⟨{𝑙 ∣ 𝑙 <Q (𝐹‘𝐾)}, {𝑢 ∣ (𝐹‘𝐾) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩))

Distinct variable groups:   𝐴,𝑗   𝑗,𝐹,𝑙,𝑢   𝑗,𝐾,𝑙,𝑢   𝑄,𝑗,𝑙,𝑢   𝑄,𝑘   𝑗,𝐿,𝑘   𝑢,𝑗   𝑘,𝐹,𝑛   𝑗,𝑘

Allowed substitution hints:   𝜑(𝑢,𝑗,𝑘,𝑛,𝑙)   𝐴(𝑢,𝑘,𝑛,𝑙)   𝑄(𝑛)   𝐽(𝑢,𝑗,𝑘,𝑛,𝑙)   𝐾(𝑘,𝑛)   𝐿(𝑢,𝑛,𝑙)

Proof of Theorem caucvgprlem1

Step	Hyp	Ref	Expression
1		caucvgprlemlim.jk	. . . . . 6 ⊢ (𝜑 → 𝐽 <N 𝐾)
2		ltrelpi 7080	. . . . . . 7 ⊢ <N ⊆ (N × N)
3	2	brel 4551	. . . . . 6 ⊢ (𝐽 <N 𝐾 → (𝐽 ∈ N ∧ 𝐾 ∈ N))
4	1, 3	syl 14	. . . . 5 ⊢ (𝜑 → (𝐽 ∈ N ∧ 𝐾 ∈ N))
5	4	simprd 113	. . . 4 ⊢ (𝜑 → 𝐾 ∈ N)
6		caucvgprlemlim.jkq	. . . . . 6 ⊢ (𝜑 → (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑄)
7	1, 6	caucvgprlemk 7421	. . . . 5 ⊢ (𝜑 → (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑄)
8		caucvgpr.f	. . . . . 6 ⊢ (𝜑 → 𝐹:N⟶Q)
9	8, 5	ffvelrnd 5510	. . . . 5 ⊢ (𝜑 → (𝐹‘𝐾) ∈ Q)
10		ltanqi 7158	. . . . 5 ⊢ (((*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑄 ∧ (𝐹‘𝐾) ∈ Q) → ((𝐹‘𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q ((𝐹‘𝐾) +Q 𝑄))
11	7, 9, 10	syl2anc 406	. . . 4 ⊢ (𝜑 → ((𝐹‘𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q ((𝐹‘𝐾) +Q 𝑄))
12		opeq1 3671	. . . . . . . . 9 ⊢ (𝑗 = 𝐾 → ⟨𝑗, 1o⟩ = ⟨𝐾, 1o⟩)
13	12	eceq1d 6419	. . . . . . . 8 ⊢ (𝑗 = 𝐾 → [⟨𝑗, 1o⟩] ~Q = [⟨𝐾, 1o⟩] ~Q )
14	13	fveq2d 5379	. . . . . . 7 ⊢ (𝑗 = 𝐾 → (*Q‘[⟨𝑗, 1o⟩] ~Q ) = (*Q‘[⟨𝐾, 1o⟩] ~Q ))
15	14	oveq2d 5744	. . . . . 6 ⊢ (𝑗 = 𝐾 → ((𝐹‘𝐾) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = ((𝐹‘𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )))
16		fveq2 5375	. . . . . . 7 ⊢ (𝑗 = 𝐾 → (𝐹‘𝑗) = (𝐹‘𝐾))
17	16	oveq1d 5743	. . . . . 6 ⊢ (𝑗 = 𝐾 → ((𝐹‘𝑗) +Q 𝑄) = ((𝐹‘𝐾) +Q 𝑄))
18	15, 17	breq12d 3908	. . . . 5 ⊢ (𝑗 = 𝐾 → (((𝐹‘𝐾) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q ((𝐹‘𝑗) +Q 𝑄) ↔ ((𝐹‘𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q ((𝐹‘𝐾) +Q 𝑄)))
19	18	rspcev 2760	. . . 4 ⊢ ((𝐾 ∈ N ∧ ((𝐹‘𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q ((𝐹‘𝐾) +Q 𝑄)) → ∃𝑗 ∈ N ((𝐹‘𝐾) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q ((𝐹‘𝑗) +Q 𝑄))
20	5, 11, 19	syl2anc 406	. . 3 ⊢ (𝜑 → ∃𝑗 ∈ N ((𝐹‘𝐾) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q ((𝐹‘𝑗) +Q 𝑄))
21		oveq1 5735	. . . . . . . 8 ⊢ (𝑙 = (𝐹‘𝐾) → (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = ((𝐹‘𝐾) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )))
22	21	breq1d 3905	. . . . . . 7 ⊢ (𝑙 = (𝐹‘𝐾) → ((𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q ((𝐹‘𝑗) +Q 𝑄) ↔ ((𝐹‘𝐾) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q ((𝐹‘𝑗) +Q 𝑄)))
23	22	rexbidv 2412	. . . . . 6 ⊢ (𝑙 = (𝐹‘𝐾) → (∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q ((𝐹‘𝑗) +Q 𝑄) ↔ ∃𝑗 ∈ N ((𝐹‘𝐾) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q ((𝐹‘𝑗) +Q 𝑄)))
24	23	elrab3 2810	. . . . 5 ⊢ ((𝐹‘𝐾) ∈ Q → ((𝐹‘𝐾) ∈ {𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q ((𝐹‘𝑗) +Q 𝑄)} ↔ ∃𝑗 ∈ N ((𝐹‘𝐾) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q ((𝐹‘𝑗) +Q 𝑄)))
25	9, 24	syl 14	. . . 4 ⊢ (𝜑 → ((𝐹‘𝐾) ∈ {𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q ((𝐹‘𝑗) +Q 𝑄)} ↔ ∃𝑗 ∈ N ((𝐹‘𝐾) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q ((𝐹‘𝑗) +Q 𝑄)))
26		caucvgpr.cau	. . . . . 6 ⊢ (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 )))))
27		caucvgpr.bnd	. . . . . 6 ⊢ (𝜑 → ∀𝑗 ∈ N 𝐴 <Q (𝐹‘𝑗))
28		caucvgpr.lim	. . . . . 6 ⊢ 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}⟩
29		caucvgprlemlim.q	. . . . . 6 ⊢ (𝜑 → 𝑄 ∈ Q)
30	8, 26, 27, 28, 29	caucvgprlemladdrl 7434	. . . . 5 ⊢ (𝜑 → {𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q ((𝐹‘𝑗) +Q 𝑄)} ⊆ (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩)))
31	30	sseld 3062	. . . 4 ⊢ (𝜑 → ((𝐹‘𝐾) ∈ {𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q ((𝐹‘𝑗) +Q 𝑄)} → (𝐹‘𝐾) ∈ (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩))))
32	25, 31	sylbird 169	. . 3 ⊢ (𝜑 → (∃𝑗 ∈ N ((𝐹‘𝐾) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q ((𝐹‘𝑗) +Q 𝑄) → (𝐹‘𝐾) ∈ (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩))))
33	20, 32	mpd 13	. 2 ⊢ (𝜑 → (𝐹‘𝐾) ∈ (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩)))
34	8, 26, 27, 28	caucvgprlemcl 7432	. . . 4 ⊢ (𝜑 → 𝐿 ∈ P)
35		nqprlu 7303	. . . . 5 ⊢ (𝑄 ∈ Q → ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩ ∈ P)
36	29, 35	syl 14	. . . 4 ⊢ (𝜑 → ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩ ∈ P)
37		addclpr 7293	. . . 4 ⊢ ((𝐿 ∈ P ∧ ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩ ∈ P) → (𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩) ∈ P)
38	34, 36, 37	syl2anc 406	. . 3 ⊢ (𝜑 → (𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩) ∈ P)
39		nqprl 7307	. . 3 ⊢ (((𝐹‘𝐾) ∈ Q ∧ (𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩) ∈ P) → ((𝐹‘𝐾) ∈ (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩)) ↔ ⟨{𝑙 ∣ 𝑙 <Q (𝐹‘𝐾)}, {𝑢 ∣ (𝐹‘𝐾) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩)))
40	9, 38, 39	syl2anc 406	. 2 ⊢ (𝜑 → ((𝐹‘𝐾) ∈ (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩)) ↔ ⟨{𝑙 ∣ 𝑙 <Q (𝐹‘𝐾)}, {𝑢 ∣ (𝐹‘𝐾) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩)))
41	33, 40	mpbid 146	1 ⊢ (𝜑 → ⟨{𝑙 ∣ 𝑙 <Q (𝐹‘𝐾)}, {𝑢 ∣ (𝐹‘𝐾) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1314   ∈ wcel 1463  {cab 2101  ∀wral 2390  ∃wrex 2391  {crab 2394  ⟨cop 3496   class class class wbr 3895  ⟶wf 5077  ‘cfv 5081  (class class class)co 5728  1^st c1st 5990  1_oc1o 6260  [cec 6381  Ncnpi 7028   <_N clti 7031   ~_Q ceq 7035  Qcnq 7036   +_Q cplq 7038  *_Qcrq 7040   <_Q cltq 7041  Pcnp 7047   +_P cpp 7049  <_P cltp 7051

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4003  ax-sep 4006  ax-nul 4014  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-setind 4412  ax-iinf 4462

This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-ral 2395  df-rex 2396  df-reu 2397  df-rab 2399  df-v 2659  df-sbc 2879  df-csb 2972  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-nul 3330  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-int 3738  df-iun 3781  df-br 3896  df-opab 3950  df-mpt 3951  df-tr 3987  df-eprel 4171  df-id 4175  df-po 4178  df-iso 4179  df-iord 4248  df-on 4250  df-suc 4253  df-iom 4465  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-f1 5086  df-fo 5087  df-f1o 5088  df-fv 5089  df-ov 5731  df-oprab 5732  df-mpo 5733  df-1st 5992  df-2nd 5993  df-recs 6156  df-irdg 6221  df-1o 6267  df-2o 6268  df-oadd 6271  df-omul 6272  df-er 6383  df-ec 6385  df-qs 6389  df-ni 7060  df-pli 7061  df-mi 7062  df-lti 7063  df-plpq 7100  df-mpq 7101  df-enq 7103  df-nqqs 7104  df-plqqs 7105  df-mqqs 7106  df-1nqqs 7107  df-rq 7108  df-ltnqqs 7109  df-enq0 7180  df-nq0 7181  df-0nq0 7182  df-plq0 7183  df-mq0 7184  df-inp 7222  df-iplp 7224  df-iltp 7226

This theorem is referenced by:  caucvgprlemlim  7437