Theorem cauappcvgprlemlol 7802

Description: Lemma for cauappcvgpr 7817. 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 7520	. . . . 5 ⊢ <Q ⊆ (Q × Q)
2	1	brel 4748	. . . 4 ⊢ (𝑠 <Q 𝑟 → (𝑠 ∈ Q ∧ 𝑟 ∈ Q))
3	2	simpld 112	. . 3 ⊢ (𝑠 <Q 𝑟 → 𝑠 ∈ Q)
4	3	3ad2ant2 1024	. 2 ⊢ ((𝜑 ∧ 𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿)) → 𝑠 ∈ Q)
5		oveq1 5981	. . . . . . . 8 ⊢ (𝑙 = 𝑟 → (𝑙 +Q 𝑞) = (𝑟 +Q 𝑞))
6	5	breq1d 4072	. . . . . . 7 ⊢ (𝑙 = 𝑟 → ((𝑙 +Q 𝑞) <Q (𝐹‘𝑞) ↔ (𝑟 +Q 𝑞) <Q (𝐹‘𝑞)))
7	6	rexbidv 2511	. . . . . 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 5606	. . . . . . 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 7518	. . . . . . . . 9 ⊢ Q ∈ V
11	10	rabex 4207	. . . . . . . 8 ⊢ {𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)} ∈ V
12	10	rabex 4207	. . . . . . . 8 ⊢ {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢} ∈ V
13	11, 12	op1st 6262	. . . . . . 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 2230	. . . . . 6 ⊢ (1st ‘𝐿) = {𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}
15	7, 14	elrab2 2942	. . . . 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 1025	. . 3 ⊢ ((𝜑 ∧ 𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿)) → ∃𝑞 ∈ Q (𝑟 +Q 𝑞) <Q (𝐹‘𝑞))
18		simpll2 1042	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿)) ∧ 𝑞 ∈ Q) ∧ (𝑟 +Q 𝑞) <Q (𝐹‘𝑞)) → 𝑠 <Q 𝑟)
19		ltanqg 7555	. . . . . . . . 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 488	. . . . . . . 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 1024	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿)) → 𝑟 ∈ Q)
24	23	ad2antrr 488	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿)) ∧ 𝑞 ∈ Q) ∧ (𝑟 +Q 𝑞) <Q (𝐹‘𝑞)) → 𝑟 ∈ Q)
25		simplr 528	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿)) ∧ 𝑞 ∈ Q) ∧ (𝑟 +Q 𝑞) <Q (𝐹‘𝑞)) → 𝑞 ∈ Q)
26		addcomnqg 7536	. . . . . . . . 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 6146	. . . . . . 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 7553	. . . . . . 7 ⊢ <Q Or Q
31	30, 1	sotri 5100	. . . . . 6 ⊢ (((𝑠 +Q 𝑞) <Q (𝑟 +Q 𝑞) ∧ (𝑟 +Q 𝑞) <Q (𝐹‘𝑞)) → (𝑠 +Q 𝑞) <Q (𝐹‘𝑞))
32	29, 31	sylancom 420	. . . . 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 2612	. . 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 5981	. . . . 5 ⊢ (𝑙 = 𝑠 → (𝑙 +Q 𝑞) = (𝑠 +Q 𝑞))
37	36	breq1d 4072	. . . 4 ⊢ (𝑙 = 𝑠 → ((𝑙 +Q 𝑞) <Q (𝐹‘𝑞) ↔ (𝑠 +Q 𝑞) <Q (𝐹‘𝑞)))
38	37	rexbidv 2511	. . 3 ⊢ (𝑙 = 𝑠 → (∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞) ↔ ∃𝑞 ∈ Q (𝑠 +Q 𝑞) <Q (𝐹‘𝑞)))
39	38, 14	elrab2 2942	. 2 ⊢ (𝑠 ∈ (1st ‘𝐿) ↔ (𝑠 ∈ Q ∧ ∃𝑞 ∈ Q (𝑠 +Q 𝑞) <Q (𝐹‘𝑞)))
40	4, 35, 39	sylanbrc 417	1 ⊢ ((𝜑 ∧ 𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿)) → 𝑠 ∈ (1st ‘𝐿))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 983   = wceq 1375   ∈ wcel 2180  ∀wral 2488  ∃wrex 2489  {crab 2492  ⟨cop 3649   class class class wbr 4062  ⟶wf 5290  ‘cfv 5294  (class class class)co 5974  1^st c1st 6254  Qcnq 7435   +_Q cplq 7437   <_Q cltq 7440

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 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-13 2182  ax-14 2183  ax-ext 2191  ax-coll 4178  ax-sep 4181  ax-nul 4189  ax-pow 4237  ax-pr 4272  ax-un 4501  ax-setind 4606  ax-iinf 4657

This theorem depends on definitions:  df-bi 117  df-dc 839  df-3or 984  df-3an 985  df-tru 1378  df-fal 1381  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ne 2381  df-ral 2493  df-rex 2494  df-reu 2495  df-rab 2497  df-v 2781  df-sbc 3009  df-csb 3105  df-dif 3179  df-un 3181  df-in 3183  df-ss 3190  df-nul 3472  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-uni 3868  df-int 3903  df-iun 3946  df-br 4063  df-opab 4125  df-mpt 4126  df-tr 4162  df-eprel 4357  df-id 4361  df-po 4364  df-iso 4365  df-iord 4434  df-on 4436  df-suc 4439  df-iom 4660  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-rn 4707  df-res 4708  df-ima 4709  df-iota 5254  df-fun 5296  df-fn 5297  df-f 5298  df-f1 5299  df-fo 5300  df-f1o 5301  df-fv 5302  df-ov 5977  df-oprab 5978  df-mpo 5979  df-1st 6256  df-2nd 6257  df-recs 6421  df-irdg 6486  df-oadd 6536  df-omul 6537  df-er 6650  df-ec 6652  df-qs 6656  df-ni 7459  df-pli 7460  df-mi 7461  df-lti 7462  df-plpq 7499  df-enq 7502  df-nqqs 7503  df-plqqs 7504  df-ltnqqs 7508

This theorem is referenced by:  cauappcvgprlemrnd  7805