Theorem cauappcvgprlem2 7855

Description: Lemma for cauappcvgpr 7857. 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 7602	. . . . 5 ⊢ ((𝑄 ∈ Q ∧ 𝑅 ∈ Q) → 𝑄 <Q (𝑄 +Q 𝑅))
4	1, 2, 3	syl2anc 411	. . . 4 ⊢ (𝜑 → 𝑄 <Q (𝑄 +Q 𝑅))
5		cauappcvgpr.f	. . . . 5 ⊢ (𝜑 → 𝐹:Q⟶Q)
6	5, 1	ffvelcdmd 5773	. . . 4 ⊢ (𝜑 → (𝐹‘𝑄) ∈ Q)
7		ltanqi 7597	. . . 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 411	. . 3 ⊢ (𝜑 → ((𝐹‘𝑄) +Q 𝑄) <Q ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅)))
9		ltbtwnnqq 7610	. . 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 122	. 2 ⊢ (𝜑 → ∃𝑥 ∈ Q (((𝐹‘𝑄) +Q 𝑄) <Q 𝑥 ∧ 𝑥 <Q ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅))))
11		simprl 529	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (((𝐹‘𝑄) +Q 𝑄) <Q 𝑥 ∧ 𝑥 <Q ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅))))) → 𝑥 ∈ Q)
12	1	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (((𝐹‘𝑄) +Q 𝑄) <Q 𝑥 ∧ 𝑥 <Q ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅))))) → 𝑄 ∈ Q)
13		simprrl 539	. . . . . 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 5629	. . . . . . . . 9 ⊢ (𝑞 = 𝑄 → (𝐹‘𝑞) = (𝐹‘𝑄))
15		id 19	. . . . . . . . 9 ⊢ (𝑞 = 𝑄 → 𝑞 = 𝑄)
16	14, 15	oveq12d 6025	. . . . . . . 8 ⊢ (𝑞 = 𝑄 → ((𝐹‘𝑞) +Q 𝑞) = ((𝐹‘𝑄) +Q 𝑄))
17	16	breq1d 4093	. . . . . . 7 ⊢ (𝑞 = 𝑄 → (((𝐹‘𝑞) +Q 𝑞) <Q 𝑥 ↔ ((𝐹‘𝑄) +Q 𝑄) <Q 𝑥))
18	17	rspcev 2907	. . . . . 6 ⊢ ((𝑄 ∈ Q ∧ ((𝐹‘𝑄) +Q 𝑄) <Q 𝑥) → ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑥)
19	12, 13, 18	syl2anc 411	. . . . 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 4087	. . . . . . 7 ⊢ (𝑢 = 𝑥 → (((𝐹‘𝑞) +Q 𝑞) <Q 𝑢 ↔ ((𝐹‘𝑞) +Q 𝑞) <Q 𝑥))
21	20	rexbidv 2531	. . . . . 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 5632	. . . . . . 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 7558	. . . . . . . . 9 ⊢ Q ∈ V
25	24	rabex 4228	. . . . . . . 8 ⊢ {𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)} ∈ V
26	24	rabex 4228	. . . . . . . 8 ⊢ {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢} ∈ V
27	25, 26	op2nd 6299	. . . . . . 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 2250	. . . . . 6 ⊢ (2nd ‘𝐿) = {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}
29	21, 28	elrab2 2962	. . . . 5 ⊢ (𝑥 ∈ (2nd ‘𝐿) ↔ (𝑥 ∈ Q ∧ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑥))
30	11, 19, 29	sylanbrc 417	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (((𝐹‘𝑄) +Q 𝑄) <Q 𝑥 ∧ 𝑥 <Q ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅))))) → 𝑥 ∈ (2nd ‘𝐿))
31		simprrr 540	. . . . . 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 2802	. . . . . . 7 ⊢ 𝑥 ∈ V
33		breq1 4086	. . . . . . 7 ⊢ (𝑙 = 𝑥 → (𝑙 <Q ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅)) ↔ 𝑥 <Q ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅))))
34	32, 33	elab 2947	. . . . . 6 ⊢ (𝑥 ∈ {𝑙 ∣ 𝑙 <Q ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅))} ↔ 𝑥 <Q ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅)))
35	31, 34	sylibr 134	. . . . 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 7744	. . . . . 6 ⊢ {𝑙 ∣ 𝑙 <Q ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅))} ∈ V
37		gtnqex 7745	. . . . . 6 ⊢ {𝑢 ∣ ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅)) <Q 𝑢} ∈ V
38	36, 37	op1st 6298	. . . . 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 2323	. . . 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 2579	. . . 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 1269	. . 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 7848	. . . . 5 ⊢ (𝜑 → 𝐿 ∈ P)
45	44	adantr 276	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (((𝐹‘𝑄) +Q 𝑄) <Q 𝑥 ∧ 𝑥 <Q ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅))))) → 𝐿 ∈ P)
46		addclnq 7570	. . . . . . . 8 ⊢ ((𝑄 ∈ Q ∧ 𝑅 ∈ Q) → (𝑄 +Q 𝑅) ∈ Q)
47	1, 2, 46	syl2anc 411	. . . . . . 7 ⊢ (𝜑 → (𝑄 +Q 𝑅) ∈ Q)
48		addclnq 7570	. . . . . . 7 ⊢ (((𝐹‘𝑄) ∈ Q ∧ (𝑄 +Q 𝑅) ∈ Q) → ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅)) ∈ Q)
49	6, 47, 48	syl2anc 411	. . . . . 6 ⊢ (𝜑 → ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅)) ∈ Q)
50		nqprlu 7742	. . . . . 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 276	. . . 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 7701	. . . 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 411	. . 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 167	. 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 2653	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 104   ↔ wb 105   = wceq 1395   ∈ wcel 2200  {cab 2215  ∀wral 2508  ∃wrex 2509  {crab 2512  ⟨cop 3669   class class class wbr 4083  ⟶wf 5314  ‘cfv 5318  (class class class)co 6007  1^st c1st 6290  2^nd c2nd 6291  Qcnq 7475   +_Q cplq 7477   <_Q cltq 7480  Pcnp 7486  <_P cltp 7490

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-eprel 4380  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-irdg 6522  df-1o 6568  df-oadd 6572  df-omul 6573  df-er 6688  df-ec 6690  df-qs 6694  df-ni 7499  df-pli 7500  df-mi 7501  df-lti 7502  df-plpq 7539  df-mpq 7540  df-enq 7542  df-nqqs 7543  df-plqqs 7544  df-mqqs 7545  df-1nqqs 7546  df-rq 7547  df-ltnqqs 7548  df-inp 7661  df-iltp 7665

This theorem is referenced by:  cauappcvgprlemlim  7856