Theorem caucvgprlem1 7741

Description: Lemma for caucvgpr 7744. 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 7386	. . . . . . 7 ⊢ <N ⊆ (N × N)
3	2	brel 4712	. . . . . 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 7727	. . . . 5 ⊢ (𝜑 → (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑄)
8		caucvgpr.f	. . . . . 6 ⊢ (𝜑 → 𝐹:N⟶Q)
9	8, 5	ffvelcdmd 5695	. . . . 5 ⊢ (𝜑 → (𝐹‘𝐾) ∈ Q)
10		ltanqi 7464	. . . . 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 3805	. . . . . . . . 9 ⊢ (𝑗 = 𝐾 → ⟨𝑗, 1o⟩ = ⟨𝐾, 1o⟩)
13	12	eceq1d 6625	. . . . . . . 8 ⊢ (𝑗 = 𝐾 → [⟨𝑗, 1o⟩] ~Q = [⟨𝐾, 1o⟩] ~Q )
14	13	fveq2d 5559	. . . . . . 7 ⊢ (𝑗 = 𝐾 → (*Q‘[⟨𝑗, 1o⟩] ~Q ) = (*Q‘[⟨𝐾, 1o⟩] ~Q ))
15	14	oveq2d 5935	. . . . . 6 ⊢ (𝑗 = 𝐾 → ((𝐹‘𝐾) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = ((𝐹‘𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )))
16		fveq2 5555	. . . . . . 7 ⊢ (𝑗 = 𝐾 → (𝐹‘𝑗) = (𝐹‘𝐾))
17	16	oveq1d 5934	. . . . . 6 ⊢ (𝑗 = 𝐾 → ((𝐹‘𝑗) +Q 𝑄) = ((𝐹‘𝐾) +Q 𝑄))
18	15, 17	breq12d 4043	. . . . 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 2865	. . . 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 5926	. . . . . . . 8 ⊢ (𝑙 = (𝐹‘𝐾) → (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = ((𝐹‘𝐾) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )))
22	21	breq1d 4040	. . . . . . 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 2495	. . . . . 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 2918	. . . . 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 7740	. . . . 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 3179	. . . 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 7738	. . . 4 ⊢ (𝜑 → 𝐿 ∈ P)
35		nqprlu 7609	. . . . 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 7599	. . . 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 7613	. . 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 1364   ∈ wcel 2164  {cab 2179  ∀wral 2472  ∃wrex 2473  {crab 2476  ⟨cop 3622   class class class wbr 4030  ⟶wf 5251  ‘cfv 5255  (class class class)co 5919  1^st c1st 6193  1_oc1o 6464  [cec 6587  Ncnpi 7334   <_N clti 7337   ~_Q ceq 7341  Qcnq 7342   +_Q cplq 7344  *_Qcrq 7346   <_Q cltq 7347  Pcnp 7353   +_P cpp 7355  <_P cltp 7357

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-eprel 4321  df-id 4325  df-po 4328  df-iso 4329  df-iord 4398  df-on 4400  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-recs 6360  df-irdg 6425  df-1o 6471  df-2o 6472  df-oadd 6475  df-omul 6476  df-er 6589  df-ec 6591  df-qs 6595  df-ni 7366  df-pli 7367  df-mi 7368  df-lti 7369  df-plpq 7406  df-mpq 7407  df-enq 7409  df-nqqs 7410  df-plqqs 7411  df-mqqs 7412  df-1nqqs 7413  df-rq 7414  df-ltnqqs 7415  df-enq0 7486  df-nq0 7487  df-0nq0 7488  df-plq0 7489  df-mq0 7490  df-inp 7528  df-iplp 7530  df-iltp 7532

This theorem is referenced by:  caucvgprlemlim  7743