Theorem cauappcvgprlemlol 8021

Description: Lemma for cauappcvgpr 8036. The lower cut of the putative limit is lower. (Contributed by Jim Kingdon, 4-Aug-2020.)

Hypotheses

Ref	Expression
cauappcvgpr.f	⊢ (𝜑 → 𝐹:Q⟶Q)
cauappcvgpr.app	⊢ (𝜑 → ∀𝑝 ∈ Q ∀𝑞 ∈ Q ((𝐹‘𝑝) <Q ((𝐹‘𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹‘𝑞) <Q ((𝐹‘𝑝) +Q (𝑝 +Q 𝑞))))
cauappcvgpr.bnd	⊢ (𝜑 → ∀𝑝 ∈ Q 𝐴 <Q (𝐹‘𝑝))
cauappcvgpr.lim	⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}⟩

Assertion

Ref	Expression
cauappcvgprlemlol	⊢ ((𝜑 ∧ 𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿)) → 𝑠 ∈ (1st ‘𝐿))

Distinct variable groups:   𝐴,𝑝   𝐿,𝑝,𝑞   𝜑,𝑝,𝑞   𝐿,𝑟,𝑠   𝐴,𝑠,𝑝   𝐹,𝑙,𝑢,𝑝,𝑞,𝑟,𝑠   𝜑,𝑟,𝑠

Allowed substitution hints:   𝜑(𝑢, 𝑙)   𝐴(𝑢, 𝑟, 𝑞, 𝑙)   𝐿(𝑢, 𝑙)

Proof of Theorem cauappcvgprlemlol

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

Step	Hyp	Ref	Expression
1		ltrelnq 7739	. . . . 5 ⊢ <Q ⊆ (Q × Q)
2	1	brel 4829	. . . 4 ⊢ (𝑠 <Q 𝑟 → (𝑠 ∈ Q ∧ 𝑟 ∈ Q))
3	2	simpld 112	. . 3 ⊢ (𝑠 <Q 𝑟 → 𝑠 ∈ Q)
4	3	3ad2ant2 1050	. 2 ⊢ ((𝜑 ∧ 𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿)) → 𝑠 ∈ Q)
5		oveq1 6095	. . . . . . . 8 ⊢ (𝑙 = 𝑟 → (𝑙 +Q 𝑞) = (𝑟 +Q 𝑞))
6	5	breq1d 4141	. . . . . . 7 ⊢ (𝑙 = 𝑟 → ((𝑙 +Q 𝑞) <Q (𝐹‘𝑞) ↔ (𝑟 +Q 𝑞) <Q (𝐹‘𝑞)))
7	6	rexbidv 2551	. . . . . 6 ⊢ (𝑙 = 𝑟 → (∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞) ↔ ∃𝑞 ∈ Q (𝑟 +Q 𝑞) <Q (𝐹‘𝑞)))
8		cauappcvgpr.lim	. . . . . . . 8 ⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}⟩
9	8	fveq2i 5701	. . . . . . 7 ⊢ (1st ‘𝐿) = (1st ‘⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}⟩)
10		nqex 7737	. . . . . . . . 9 ⊢ Q ∈ V
11	10	rabex 4282	. . . . . . . 8 ⊢ {𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)} ∈ V
12	10	rabex 4282	. . . . . . . 8 ⊢ {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢} ∈ V
13	11, 12	op1st 6384	. . . . . . 7 ⊢ (1st ‘⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}⟩) = {𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}
14	9, 13	eqtri 2259	. . . . . 6 ⊢ (1st ‘𝐿) = {𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}
15	7, 14	elrab2 2985	. . . . 5 ⊢ (𝑟 ∈ (1st ‘𝐿) ↔ (𝑟 ∈ Q ∧ ∃𝑞 ∈ Q (𝑟 +Q 𝑞) <Q (𝐹‘𝑞)))
16	15	simprbi 275	. . . 4 ⊢ (𝑟 ∈ (1st ‘𝐿) → ∃𝑞 ∈ Q (𝑟 +Q 𝑞) <Q (𝐹‘𝑞))
17	16	3ad2ant3 1051	. . 3 ⊢ ((𝜑 ∧ 𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿)) → ∃𝑞 ∈ Q (𝑟 +Q 𝑞) <Q (𝐹‘𝑞))
18		simpll2 1068	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿)) ∧ 𝑞 ∈ Q) ∧ (𝑟 +Q 𝑞) <Q (𝐹‘𝑞)) → 𝑠 <Q 𝑟)
19		ltanqg 7774	. . . . . . . . 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)) ∧ 𝑞 ∈ Q) ∧ (𝑟 +Q 𝑞) <Q (𝐹‘𝑞)) ∧ (𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q)) → (𝑓 <Q 𝑔 ↔ (ℎ +Q 𝑓) <Q (ℎ +Q 𝑔)))
21	4	ad2antrr 492	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿)) ∧ 𝑞 ∈ Q) ∧ (𝑟 +Q 𝑞) <Q (𝐹‘𝑞)) → 𝑠 ∈ Q)
22	2	simprd 114	. . . . . . . . . 10 ⊢ (𝑠 <Q 𝑟 → 𝑟 ∈ Q)
23	22	3ad2ant2 1050	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿)) → 𝑟 ∈ Q)
24	23	ad2antrr 492	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿)) ∧ 𝑞 ∈ Q) ∧ (𝑟 +Q 𝑞) <Q (𝐹‘𝑞)) → 𝑟 ∈ Q)
25		simplr 533	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿)) ∧ 𝑞 ∈ Q) ∧ (𝑟 +Q 𝑞) <Q (𝐹‘𝑞)) → 𝑞 ∈ Q)
26		addcomnqg 7755	. . . . . . . . 9 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
27	26	adantl 277	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿)) ∧ 𝑞 ∈ Q) ∧ (𝑟 +Q 𝑞) <Q (𝐹‘𝑞)) ∧ (𝑓 ∈ Q ∧ 𝑔 ∈ Q)) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
28	20, 21, 24, 25, 27	caovord2d 6262	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿)) ∧ 𝑞 ∈ Q) ∧ (𝑟 +Q 𝑞) <Q (𝐹‘𝑞)) → (𝑠 <Q 𝑟 ↔ (𝑠 +Q 𝑞) <Q (𝑟 +Q 𝑞)))
29	18, 28	mpbid 147	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿)) ∧ 𝑞 ∈ Q) ∧ (𝑟 +Q 𝑞) <Q (𝐹‘𝑞)) → (𝑠 +Q 𝑞) <Q (𝑟 +Q 𝑞))
30		ltsonq 7772	. . . . . . 7 ⊢ <Q Or Q
31	30, 1	sotri 5186	. . . . . 6 ⊢ (((𝑠 +Q 𝑞) <Q (𝑟 +Q 𝑞) ∧ (𝑟 +Q 𝑞) <Q (𝐹‘𝑞)) → (𝑠 +Q 𝑞) <Q (𝐹‘𝑞))
32	29, 31	sylancom 424	. . . . 5 ⊢ ((((𝜑 ∧ 𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿)) ∧ 𝑞 ∈ Q) ∧ (𝑟 +Q 𝑞) <Q (𝐹‘𝑞)) → (𝑠 +Q 𝑞) <Q (𝐹‘𝑞))
33	32	ex 115	. . . 4 ⊢ (((𝜑 ∧ 𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿)) ∧ 𝑞 ∈ Q) → ((𝑟 +Q 𝑞) <Q (𝐹‘𝑞) → (𝑠 +Q 𝑞) <Q (𝐹‘𝑞)))
34	33	reximdva 2652	. . 3 ⊢ ((𝜑 ∧ 𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿)) → (∃𝑞 ∈ Q (𝑟 +Q 𝑞) <Q (𝐹‘𝑞) → ∃𝑞 ∈ Q (𝑠 +Q 𝑞) <Q (𝐹‘𝑞)))
35	17, 34	mpd 13	. 2 ⊢ ((𝜑 ∧ 𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿)) → ∃𝑞 ∈ Q (𝑠 +Q 𝑞) <Q (𝐹‘𝑞))
36		oveq1 6095	. . . . 5 ⊢ (𝑙 = 𝑠 → (𝑙 +Q 𝑞) = (𝑠 +Q 𝑞))
37	36	breq1d 4141	. . . 4 ⊢ (𝑙 = 𝑠 → ((𝑙 +Q 𝑞) <Q (𝐹‘𝑞) ↔ (𝑠 +Q 𝑞) <Q (𝐹‘𝑞)))
38	37	rexbidv 2551	. . 3 ⊢ (𝑙 = 𝑠 → (∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞) ↔ ∃𝑞 ∈ Q (𝑠 +Q 𝑞) <Q (𝐹‘𝑞)))
39	38, 14	elrab2 2985	. 2 ⊢ (𝑠 ∈ (1st ‘𝐿) ↔ (𝑠 ∈ Q ∧ ∃𝑞 ∈ Q (𝑠 +Q 𝑞) <Q (𝐹‘𝑞)))
40	4, 35, 39	sylanbrc 421	1 ⊢ ((𝜑 ∧ 𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿)) → 𝑠 ∈ (1st ‘𝐿))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1009   = wceq 1402   ∈ wcel 2209  ∀wral 2528  ∃wrex 2529  {crab 2532  ⟨cop 3712   class class class wbr 4131  ⟶wf 5376  ‘cfv 5380  (class class class)co 6088  1^st c1st 6376  Qcnq 7654   +_Q cplq 7656   <_Q cltq 7659

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-eprel 4436  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-irdg 6645  df-oadd 6695  df-omul 6696  df-er 6811  df-ec 6813  df-qs 6817  df-ni 7678  df-pli 7679  df-mi 7680  df-lti 7681  df-plpq 7718  df-enq 7721  df-nqqs 7722  df-plqqs 7723  df-ltnqqs 7727

This theorem is used by:  cauappcvgprlemrnd  8024