Theorem caucvgprlemlol 7730

Description: Lemma for caucvgpr 7742. The lower cut of the putative limit is lower. (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
caucvgprlemlol	⊢ ((𝜑 ∧ 𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿)) → 𝑠 ∈ (1st ‘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 caucvgprlemlol

Dummy variables 𝑓 𝑔 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltrelnq 7425	. . . . 5 ⊢ <Q ⊆ (Q × Q)
2	1	brel 4711	. . . 4 ⊢ (𝑠 <Q 𝑟 → (𝑠 ∈ Q ∧ 𝑟 ∈ Q))
3	2	simpld 112	. . 3 ⊢ (𝑠 <Q 𝑟 → 𝑠 ∈ Q)
4	3	3ad2ant2 1021	. 2 ⊢ ((𝜑 ∧ 𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿)) → 𝑠 ∈ Q)
5		oveq1 5925	. . . . . . . 8 ⊢ (𝑙 = 𝑟 → (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )))
6	5	breq1d 4039	. . . . . . 7 ⊢ (𝑙 = 𝑟 → ((𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗) ↔ (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)))
7	6	rexbidv 2495	. . . . . 6 ⊢ (𝑙 = 𝑟 → (∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗) ↔ ∃𝑗 ∈ N (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)))
8		caucvgpr.lim	. . . . . . . 8 ⊢ 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}⟩
9	8	fveq2i 5557	. . . . . . 7 ⊢ (1st ‘𝐿) = (1st ‘⟨{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}⟩)
10		nqex 7423	. . . . . . . . 9 ⊢ Q ∈ V
11	10	rabex 4173	. . . . . . . 8 ⊢ {𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)} ∈ V
12	10	rabex 4173	. . . . . . . 8 ⊢ {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢} ∈ V
13	11, 12	op1st 6199	. . . . . . 7 ⊢ (1st ‘⟨{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}⟩) = {𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)}
14	9, 13	eqtri 2214	. . . . . 6 ⊢ (1st ‘𝐿) = {𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)}
15	7, 14	elrab2 2919	. . . . 5 ⊢ (𝑟 ∈ (1st ‘𝐿) ↔ (𝑟 ∈ Q ∧ ∃𝑗 ∈ N (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)))
16	15	simprbi 275	. . . 4 ⊢ (𝑟 ∈ (1st ‘𝐿) → ∃𝑗 ∈ N (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗))
17	16	3ad2ant3 1022	. . 3 ⊢ ((𝜑 ∧ 𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿)) → ∃𝑗 ∈ N (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗))
18		simpll2 1039	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿)) ∧ 𝑗 ∈ N) ∧ (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)) → 𝑠 <Q 𝑟)
19		ltanqg 7460	. . . . . . . . 9 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑓 <Q 𝑔 ↔ (ℎ +Q 𝑓) <Q (ℎ +Q 𝑔)))
20	19	adantl 277	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿)) ∧ 𝑗 ∈ N) ∧ (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)) ∧ (𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q)) → (𝑓 <Q 𝑔 ↔ (ℎ +Q 𝑓) <Q (ℎ +Q 𝑔)))
21	4	ad2antrr 488	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿)) ∧ 𝑗 ∈ N) ∧ (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)) → 𝑠 ∈ Q)
22	2	simprd 114	. . . . . . . . . 10 ⊢ (𝑠 <Q 𝑟 → 𝑟 ∈ Q)
23	22	3ad2ant2 1021	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿)) → 𝑟 ∈ Q)
24	23	ad2antrr 488	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿)) ∧ 𝑗 ∈ N) ∧ (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)) → 𝑟 ∈ Q)
25		simplr 528	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿)) ∧ 𝑗 ∈ N) ∧ (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)) → 𝑗 ∈ N)
26		nnnq 7482	. . . . . . . . 9 ⊢ (𝑗 ∈ N → [⟨𝑗, 1o⟩] ~Q ∈ Q)
27		recclnq 7452	. . . . . . . . 9 ⊢ ([⟨𝑗, 1o⟩] ~Q ∈ Q → (*Q‘[⟨𝑗, 1o⟩] ~Q ) ∈ Q)
28	25, 26, 27	3syl 17	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿)) ∧ 𝑗 ∈ N) ∧ (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)) → (*Q‘[⟨𝑗, 1o⟩] ~Q ) ∈ Q)
29		addcomnqg 7441	. . . . . . . . 9 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
30	29	adantl 277	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿)) ∧ 𝑗 ∈ N) ∧ (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)) ∧ (𝑓 ∈ Q ∧ 𝑔 ∈ Q)) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
31	20, 21, 24, 28, 30	caovord2d 6088	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿)) ∧ 𝑗 ∈ N) ∧ (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)) → (𝑠 <Q 𝑟 ↔ (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q ))))
32	18, 31	mpbid 147	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿)) ∧ 𝑗 ∈ N) ∧ (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)) → (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )))
33		ltsonq 7458	. . . . . . 7 ⊢ <Q Or Q
34	33, 1	sotri 5061	. . . . . 6 ⊢ (((𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) ∧ (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)) → (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗))
35	32, 34	sylancom 420	. . . . 5 ⊢ ((((𝜑 ∧ 𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿)) ∧ 𝑗 ∈ N) ∧ (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)) → (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗))
36	35	ex 115	. . . 4 ⊢ (((𝜑 ∧ 𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿)) ∧ 𝑗 ∈ N) → ((𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗) → (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)))
37	36	reximdva 2596	. . 3 ⊢ ((𝜑 ∧ 𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿)) → (∃𝑗 ∈ N (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗) → ∃𝑗 ∈ N (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)))
38	17, 37	mpd 13	. 2 ⊢ ((𝜑 ∧ 𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿)) → ∃𝑗 ∈ N (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗))
39		oveq1 5925	. . . . 5 ⊢ (𝑙 = 𝑠 → (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )))
40	39	breq1d 4039	. . . 4 ⊢ (𝑙 = 𝑠 → ((𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗) ↔ (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)))
41	40	rexbidv 2495	. . 3 ⊢ (𝑙 = 𝑠 → (∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗) ↔ ∃𝑗 ∈ N (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)))
42	41, 14	elrab2 2919	. 2 ⊢ (𝑠 ∈ (1st ‘𝐿) ↔ (𝑠 ∈ Q ∧ ∃𝑗 ∈ N (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)))
43	4, 38, 42	sylanbrc 417	1 ⊢ ((𝜑 ∧ 𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿)) → 𝑠 ∈ (1st ‘𝐿))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 980   = wceq 1364   ∈ wcel 2164  ∀wral 2472  ∃wrex 2473  {crab 2476  ⟨cop 3621   class class class wbr 4029  ⟶wf 5250  ‘cfv 5254  (class class class)co 5918  1^st c1st 6191  1_oc1o 6462  [cec 6585  Ncnpi 7332   <_N clti 7335   ~_Q ceq 7339  Qcnq 7340   +_Q cplq 7342  *_Qcrq 7344   <_Q cltq 7345

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 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620

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 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-eprel 4320  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-irdg 6423  df-1o 6469  df-oadd 6473  df-omul 6474  df-er 6587  df-ec 6589  df-qs 6593  df-ni 7364  df-pli 7365  df-mi 7366  df-lti 7367  df-plpq 7404  df-mpq 7405  df-enq 7407  df-nqqs 7408  df-plqqs 7409  df-mqqs 7410  df-1nqqs 7411  df-rq 7412  df-ltnqqs 7413

This theorem is referenced by:  caucvgprlemrnd  7733