Theorem cauappcvgprlem2 7468

Description: Lemma for cauappcvgpr 7470. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 23-Jun-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}⟩
cauappcvgprlem.q	⊢ (𝜑 → 𝑄 ∈ Q)
cauappcvgprlem.r	⊢ (𝜑 → 𝑅 ∈ Q)

Assertion

Ref	Expression
cauappcvgprlem2	⊢ (𝜑 → 𝐿<P ⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅))}, {𝑢 ∣ ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅)) <Q 𝑢}⟩)

Distinct variable groups:   𝐴,𝑝   𝐿,𝑝,𝑞   𝜑,𝑝,𝑞   𝐹,𝑝,𝑞,𝑙,𝑢   𝑄,𝑝,𝑞,𝑙,𝑢   𝑅,𝑝,𝑞,𝑙,𝑢

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

Proof of Theorem cauappcvgprlem2

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cauappcvgprlem.q	. . . . 5 ⊢ (𝜑 → 𝑄 ∈ Q)
2		cauappcvgprlem.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Q)
3		ltaddnq 7215	. . . . 5 ⊢ ((𝑄 ∈ Q ∧ 𝑅 ∈ Q) → 𝑄 <Q (𝑄 +Q 𝑅))
4	1, 2, 3	syl2anc 408	. . . 4 ⊢ (𝜑 → 𝑄 <Q (𝑄 +Q 𝑅))
5		cauappcvgpr.f	. . . . 5 ⊢ (𝜑 → 𝐹:Q⟶Q)
6	5, 1	ffvelrnd 5556	. . . 4 ⊢ (𝜑 → (𝐹‘𝑄) ∈ Q)
7		ltanqi 7210	. . . 4 ⊢ ((𝑄 <Q (𝑄 +Q 𝑅) ∧ (𝐹‘𝑄) ∈ Q) → ((𝐹‘𝑄) +Q 𝑄) <Q ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅)))
8	4, 6, 7	syl2anc 408	. . 3 ⊢ (𝜑 → ((𝐹‘𝑄) +Q 𝑄) <Q ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅)))
9		ltbtwnnqq 7223	. . 3 ⊢ (((𝐹‘𝑄) +Q 𝑄) <Q ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅)) ↔ ∃𝑥 ∈ Q (((𝐹‘𝑄) +Q 𝑄) <Q 𝑥 ∧ 𝑥 <Q ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅))))
10	8, 9	sylib 121	. 2 ⊢ (𝜑 → ∃𝑥 ∈ Q (((𝐹‘𝑄) +Q 𝑄) <Q 𝑥 ∧ 𝑥 <Q ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅))))
11		simprl 520	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (((𝐹‘𝑄) +Q 𝑄) <Q 𝑥 ∧ 𝑥 <Q ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅))))) → 𝑥 ∈ Q)
12	1	adantr 274	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (((𝐹‘𝑄) +Q 𝑄) <Q 𝑥 ∧ 𝑥 <Q ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅))))) → 𝑄 ∈ Q)
13		simprrl 528	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (((𝐹‘𝑄) +Q 𝑄) <Q 𝑥 ∧ 𝑥 <Q ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅))))) → ((𝐹‘𝑄) +Q 𝑄) <Q 𝑥)
14		fveq2 5421	. . . . . . . . 9 ⊢ (𝑞 = 𝑄 → (𝐹‘𝑞) = (𝐹‘𝑄))
15		id 19	. . . . . . . . 9 ⊢ (𝑞 = 𝑄 → 𝑞 = 𝑄)
16	14, 15	oveq12d 5792	. . . . . . . 8 ⊢ (𝑞 = 𝑄 → ((𝐹‘𝑞) +Q 𝑞) = ((𝐹‘𝑄) +Q 𝑄))
17	16	breq1d 3939	. . . . . . 7 ⊢ (𝑞 = 𝑄 → (((𝐹‘𝑞) +Q 𝑞) <Q 𝑥 ↔ ((𝐹‘𝑄) +Q 𝑄) <Q 𝑥))
18	17	rspcev 2789	. . . . . 6 ⊢ ((𝑄 ∈ Q ∧ ((𝐹‘𝑄) +Q 𝑄) <Q 𝑥) → ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑥)
19	12, 13, 18	syl2anc 408	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (((𝐹‘𝑄) +Q 𝑄) <Q 𝑥 ∧ 𝑥 <Q ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅))))) → ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑥)
20		breq2 3933	. . . . . . 7 ⊢ (𝑢 = 𝑥 → (((𝐹‘𝑞) +Q 𝑞) <Q 𝑢 ↔ ((𝐹‘𝑞) +Q 𝑞) <Q 𝑥))
21	20	rexbidv 2438	. . . . . 6 ⊢ (𝑢 = 𝑥 → (∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢 ↔ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑥))
22		cauappcvgpr.lim	. . . . . . . 8 ⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}⟩
23	22	fveq2i 5424	. . . . . . 7 ⊢ (2nd ‘𝐿) = (2nd ‘⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}⟩)
24		nqex 7171	. . . . . . . . 9 ⊢ Q ∈ V
25	24	rabex 4072	. . . . . . . 8 ⊢ {𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)} ∈ V
26	24	rabex 4072	. . . . . . . 8 ⊢ {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢} ∈ V
27	25, 26	op2nd 6045	. . . . . . 7 ⊢ (2nd ‘⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}⟩) = {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}
28	23, 27	eqtri 2160	. . . . . 6 ⊢ (2nd ‘𝐿) = {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}
29	21, 28	elrab2 2843	. . . . 5 ⊢ (𝑥 ∈ (2nd ‘𝐿) ↔ (𝑥 ∈ Q ∧ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑥))
30	11, 19, 29	sylanbrc 413	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (((𝐹‘𝑄) +Q 𝑄) <Q 𝑥 ∧ 𝑥 <Q ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅))))) → 𝑥 ∈ (2nd ‘𝐿))
31		simprrr 529	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (((𝐹‘𝑄) +Q 𝑄) <Q 𝑥 ∧ 𝑥 <Q ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅))))) → 𝑥 <Q ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅)))
32		vex 2689	. . . . . . 7 ⊢ 𝑥 ∈ V
33		breq1 3932	. . . . . . 7 ⊢ (𝑙 = 𝑥 → (𝑙 <Q ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅)) ↔ 𝑥 <Q ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅))))
34	32, 33	elab 2828	. . . . . 6 ⊢ (𝑥 ∈ {𝑙 ∣ 𝑙 <Q ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅))} ↔ 𝑥 <Q ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅)))
35	31, 34	sylibr 133	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (((𝐹‘𝑄) +Q 𝑄) <Q 𝑥 ∧ 𝑥 <Q ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅))))) → 𝑥 ∈ {𝑙 ∣ 𝑙 <Q ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅))})
36		ltnqex 7357	. . . . . 6 ⊢ {𝑙 ∣ 𝑙 <Q ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅))} ∈ V
37		gtnqex 7358	. . . . . 6 ⊢ {𝑢 ∣ ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅)) <Q 𝑢} ∈ V
38	36, 37	op1st 6044	. . . . 5 ⊢ (1st ‘⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅))}, {𝑢 ∣ ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅)) <Q 𝑢}⟩) = {𝑙 ∣ 𝑙 <Q ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅))}
39	35, 38	eleqtrrdi 2233	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (((𝐹‘𝑄) +Q 𝑄) <Q 𝑥 ∧ 𝑥 <Q ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅))))) → 𝑥 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅))}, {𝑢 ∣ ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅)) <Q 𝑢}⟩))
40		rspe 2481	. . . 4 ⊢ ((𝑥 ∈ Q ∧ (𝑥 ∈ (2nd ‘𝐿) ∧ 𝑥 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅))}, {𝑢 ∣ ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅)) <Q 𝑢}⟩))) → ∃𝑥 ∈ Q (𝑥 ∈ (2nd ‘𝐿) ∧ 𝑥 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅))}, {𝑢 ∣ ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅)) <Q 𝑢}⟩)))
41	11, 30, 39, 40	syl12anc 1214	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (((𝐹‘𝑄) +Q 𝑄) <Q 𝑥 ∧ 𝑥 <Q ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅))))) → ∃𝑥 ∈ Q (𝑥 ∈ (2nd ‘𝐿) ∧ 𝑥 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅))}, {𝑢 ∣ ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅)) <Q 𝑢}⟩)))
42		cauappcvgpr.app	. . . . . 6 ⊢ (𝜑 → ∀𝑝 ∈ Q ∀𝑞 ∈ Q ((𝐹‘𝑝) <Q ((𝐹‘𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹‘𝑞) <Q ((𝐹‘𝑝) +Q (𝑝 +Q 𝑞))))
43		cauappcvgpr.bnd	. . . . . 6 ⊢ (𝜑 → ∀𝑝 ∈ Q 𝐴 <Q (𝐹‘𝑝))
44	5, 42, 43, 22	cauappcvgprlemcl 7461	. . . . 5 ⊢ (𝜑 → 𝐿 ∈ P)
45	44	adantr 274	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (((𝐹‘𝑄) +Q 𝑄) <Q 𝑥 ∧ 𝑥 <Q ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅))))) → 𝐿 ∈ P)
46		addclnq 7183	. . . . . . . 8 ⊢ ((𝑄 ∈ Q ∧ 𝑅 ∈ Q) → (𝑄 +Q 𝑅) ∈ Q)
47	1, 2, 46	syl2anc 408	. . . . . . 7 ⊢ (𝜑 → (𝑄 +Q 𝑅) ∈ Q)
48		addclnq 7183	. . . . . . 7 ⊢ (((𝐹‘𝑄) ∈ Q ∧ (𝑄 +Q 𝑅) ∈ Q) → ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅)) ∈ Q)
49	6, 47, 48	syl2anc 408	. . . . . 6 ⊢ (𝜑 → ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅)) ∈ Q)
50		nqprlu 7355	. . . . . 6 ⊢ (((𝐹‘𝑄) +Q (𝑄 +Q 𝑅)) ∈ Q → ⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅))}, {𝑢 ∣ ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅)) <Q 𝑢}⟩ ∈ P)
51	49, 50	syl 14	. . . . 5 ⊢ (𝜑 → ⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅))}, {𝑢 ∣ ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅)) <Q 𝑢}⟩ ∈ P)
52	51	adantr 274	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (((𝐹‘𝑄) +Q 𝑄) <Q 𝑥 ∧ 𝑥 <Q ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅))))) → ⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅))}, {𝑢 ∣ ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅)) <Q 𝑢}⟩ ∈ P)
53		ltdfpr 7314	. . . 4 ⊢ ((𝐿 ∈ P ∧ ⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅))}, {𝑢 ∣ ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅)) <Q 𝑢}⟩ ∈ P) → (𝐿<P ⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅))}, {𝑢 ∣ ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅)) <Q 𝑢}⟩ ↔ ∃𝑥 ∈ Q (𝑥 ∈ (2nd ‘𝐿) ∧ 𝑥 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅))}, {𝑢 ∣ ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅)) <Q 𝑢}⟩))))
54	45, 52, 53	syl2anc 408	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (((𝐹‘𝑄) +Q 𝑄) <Q 𝑥 ∧ 𝑥 <Q ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅))))) → (𝐿<P ⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅))}, {𝑢 ∣ ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅)) <Q 𝑢}⟩ ↔ ∃𝑥 ∈ Q (𝑥 ∈ (2nd ‘𝐿) ∧ 𝑥 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅))}, {𝑢 ∣ ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅)) <Q 𝑢}⟩))))
55	41, 54	mpbird 166	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (((𝐹‘𝑄) +Q 𝑄) <Q 𝑥 ∧ 𝑥 <Q ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅))))) → 𝐿<P ⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅))}, {𝑢 ∣ ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅)) <Q 𝑢}⟩)
56	10, 55	rexlimddv 2554	1 ⊢ (𝜑 → 𝐿<P ⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅))}, {𝑢 ∣ ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅)) <Q 𝑢}⟩)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1331   ∈ wcel 1480  {cab 2125  ∀wral 2416  ∃wrex 2417  {crab 2420  ⟨cop 3530   class class class wbr 3929  ⟶wf 5119  ‘cfv 5123  (class class class)co 5774  1^st c1st 6036  2^nd c2nd 6037  Qcnq 7088   +_Q cplq 7090   <_Q cltq 7093  Pcnp 7099  <_P cltp 7103

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7112  df-pli 7113  df-mi 7114  df-lti 7115  df-plpq 7152  df-mpq 7153  df-enq 7155  df-nqqs 7156  df-plqqs 7157  df-mqqs 7158  df-1nqqs 7159  df-rq 7160  df-ltnqqs 7161  df-inp 7274  df-iltp 7278

This theorem is referenced by:  cauappcvgprlemlim  7469