Theorem caucvgprlem1 7889

Description: Lemma for caucvgpr 7892. 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 7534	. . . . . . 7 ⊢ <N ⊆ (N × N)
3	2	brel 4776	. . . . . 6 ⊢ (𝐽 <N 𝐾 → (𝐽 ∈ N ∧ 𝐾 ∈ N))
4	1, 3	syl 14	. . . . 5 ⊢ (𝜑 → (𝐽 ∈ N ∧ 𝐾 ∈ N))
5	4	simprd 114	. . . 4 ⊢ (𝜑 → 𝐾 ∈ N)
6		caucvgprlemlim.jkq	. . . . . 6 ⊢ (𝜑 → (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑄)
7	1, 6	caucvgprlemk 7875	. . . . 5 ⊢ (𝜑 → (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑄)
8		caucvgpr.f	. . . . . 6 ⊢ (𝜑 → 𝐹:N⟶Q)
9	8, 5	ffvelcdmd 5779	. . . . 5 ⊢ (𝜑 → (𝐹‘𝐾) ∈ Q)
10		ltanqi 7612	. . . . 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 411	. . . 4 ⊢ (𝜑 → ((𝐹‘𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q ((𝐹‘𝐾) +Q 𝑄))
12		opeq1 3860	. . . . . . . . 9 ⊢ (𝑗 = 𝐾 → ⟨𝑗, 1o⟩ = ⟨𝐾, 1o⟩)
13	12	eceq1d 6733	. . . . . . . 8 ⊢ (𝑗 = 𝐾 → [⟨𝑗, 1o⟩] ~Q = [⟨𝐾, 1o⟩] ~Q )
14	13	fveq2d 5639	. . . . . . 7 ⊢ (𝑗 = 𝐾 → (*Q‘[⟨𝑗, 1o⟩] ~Q ) = (*Q‘[⟨𝐾, 1o⟩] ~Q ))
15	14	oveq2d 6029	. . . . . 6 ⊢ (𝑗 = 𝐾 → ((𝐹‘𝐾) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = ((𝐹‘𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )))
16		fveq2 5635	. . . . . . 7 ⊢ (𝑗 = 𝐾 → (𝐹‘𝑗) = (𝐹‘𝐾))
17	16	oveq1d 6028	. . . . . 6 ⊢ (𝑗 = 𝐾 → ((𝐹‘𝑗) +Q 𝑄) = ((𝐹‘𝐾) +Q 𝑄))
18	15, 17	breq12d 4099	. . . . 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 2908	. . . 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 411	. . 3 ⊢ (𝜑 → ∃𝑗 ∈ N ((𝐹‘𝐾) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q ((𝐹‘𝑗) +Q 𝑄))
21		oveq1 6020	. . . . . . . 8 ⊢ (𝑙 = (𝐹‘𝐾) → (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = ((𝐹‘𝐾) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )))
22	21	breq1d 4096	. . . . . . 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 2531	. . . . . 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 2961	. . . . 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 7888	. . . . 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 3224	. . . 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 170	. . 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 7886	. . . 4 ⊢ (𝜑 → 𝐿 ∈ P)
35		nqprlu 7757	. . . . 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 7747	. . . 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 411	. . 3 ⊢ (𝜑 → (𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩) ∈ P)
39		nqprl 7761	. . 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 411	. 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 147	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 104   ↔ wb 105   = wceq 1395   ∈ wcel 2200  {cab 2215  ∀wral 2508  ∃wrex 2509  {crab 2512  ⟨cop 3670   class class class wbr 4086  ⟶wf 5320  ‘cfv 5324  (class class class)co 6013  1^st c1st 6296  1_oc1o 6570  [cec 6695  Ncnpi 7482   <_N clti 7485   ~_Q ceq 7489  Qcnq 7490   +_Q cplq 7492  *_Qcrq 7494   <_Q cltq 7495  Pcnp 7501   +_P cpp 7503  <_P cltp 7505

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-eprel 4384  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-irdg 6531  df-1o 6577  df-2o 6578  df-oadd 6581  df-omul 6582  df-er 6697  df-ec 6699  df-qs 6703  df-ni 7514  df-pli 7515  df-mi 7516  df-lti 7517  df-plpq 7554  df-mpq 7555  df-enq 7557  df-nqqs 7558  df-plqqs 7559  df-mqqs 7560  df-1nqqs 7561  df-rq 7562  df-ltnqqs 7563  df-enq0 7634  df-nq0 7635  df-0nq0 7636  df-plq0 7637  df-mq0 7638  df-inp 7676  df-iplp 7678  df-iltp 7680

This theorem is referenced by:  caucvgprlemlim  7891