Theorem cauappcvgprlem2 7815

Description: Lemma for cauappcvgpr 7817. 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 7562	. . . . 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 5744	. . . 4 ⊢ (𝜑 → (𝐹‘𝑄) ∈ Q)
7		ltanqi 7557	. . . 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 7570	. . 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 5603	. . . . . . . . 9 ⊢ (𝑞 = 𝑄 → (𝐹‘𝑞) = (𝐹‘𝑄))
15		id 19	. . . . . . . . 9 ⊢ (𝑞 = 𝑄 → 𝑞 = 𝑄)
16	14, 15	oveq12d 5992	. . . . . . . 8 ⊢ (𝑞 = 𝑄 → ((𝐹‘𝑞) +Q 𝑞) = ((𝐹‘𝑄) +Q 𝑄))
17	16	breq1d 4072	. . . . . . 7 ⊢ (𝑞 = 𝑄 → (((𝐹‘𝑞) +Q 𝑞) <Q 𝑥 ↔ ((𝐹‘𝑄) +Q 𝑄) <Q 𝑥))
18	17	rspcev 2887	. . . . . 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 4066	. . . . . . 7 ⊢ (𝑢 = 𝑥 → (((𝐹‘𝑞) +Q 𝑞) <Q 𝑢 ↔ ((𝐹‘𝑞) +Q 𝑞) <Q 𝑥))
21	20	rexbidv 2511	. . . . . 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 5606	. . . . . . 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 7518	. . . . . . . . 9 ⊢ Q ∈ V
25	24	rabex 4207	. . . . . . . 8 ⊢ {𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)} ∈ V
26	24	rabex 4207	. . . . . . . 8 ⊢ {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢} ∈ V
27	25, 26	op2nd 6263	. . . . . . 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 2230	. . . . . 6 ⊢ (2nd ‘𝐿) = {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}
29	21, 28	elrab2 2942	. . . . 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 2782	. . . . . . 7 ⊢ 𝑥 ∈ V
33		breq1 4065	. . . . . . 7 ⊢ (𝑙 = 𝑥 → (𝑙 <Q ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅)) ↔ 𝑥 <Q ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅))))
34	32, 33	elab 2927	. . . . . 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 7704	. . . . . 6 ⊢ {𝑙 ∣ 𝑙 <Q ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅))} ∈ V
37		gtnqex 7705	. . . . . 6 ⊢ {𝑢 ∣ ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅)) <Q 𝑢} ∈ V
38	36, 37	op1st 6262	. . . . 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 2303	. . . 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 2559	. . . 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 1250	. . 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 7808	. . . . 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 7530	. . . . . . . 8 ⊢ ((𝑄 ∈ Q ∧ 𝑅 ∈ Q) → (𝑄 +Q 𝑅) ∈ Q)
47	1, 2, 46	syl2anc 411	. . . . . . 7 ⊢ (𝜑 → (𝑄 +Q 𝑅) ∈ Q)
48		addclnq 7530	. . . . . . 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 7702	. . . . . 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 7661	. . . 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 2633	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 1375   ∈ wcel 2180  {cab 2195  ∀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  2^nd c2nd 6255  Qcnq 7435   +_Q cplq 7437   <_Q cltq 7440  Pcnp 7446  <_P cltp 7450

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-1o 6532  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-mpq 7500  df-enq 7502  df-nqqs 7503  df-plqqs 7504  df-mqqs 7505  df-1nqqs 7506  df-rq 7507  df-ltnqqs 7508  df-inp 7621  df-iltp 7625

This theorem is referenced by:  cauappcvgprlemlim  7816