Theorem cauappcvgprlem2 7720

Description: Lemma for cauappcvgpr 7722. 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 7467	. . . . 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 5694	. . . 4 ⊢ (𝜑 → (𝐹‘𝑄) ∈ Q)
7		ltanqi 7462	. . . 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 7475	. . 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 5554	. . . . . . . . 9 ⊢ (𝑞 = 𝑄 → (𝐹‘𝑞) = (𝐹‘𝑄))
15		id 19	. . . . . . . . 9 ⊢ (𝑞 = 𝑄 → 𝑞 = 𝑄)
16	14, 15	oveq12d 5936	. . . . . . . 8 ⊢ (𝑞 = 𝑄 → ((𝐹‘𝑞) +Q 𝑞) = ((𝐹‘𝑄) +Q 𝑄))
17	16	breq1d 4039	. . . . . . 7 ⊢ (𝑞 = 𝑄 → (((𝐹‘𝑞) +Q 𝑞) <Q 𝑥 ↔ ((𝐹‘𝑄) +Q 𝑄) <Q 𝑥))
18	17	rspcev 2864	. . . . . 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 4033	. . . . . . 7 ⊢ (𝑢 = 𝑥 → (((𝐹‘𝑞) +Q 𝑞) <Q 𝑢 ↔ ((𝐹‘𝑞) +Q 𝑞) <Q 𝑥))
21	20	rexbidv 2495	. . . . . 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 5557	. . . . . . 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 7423	. . . . . . . . 9 ⊢ Q ∈ V
25	24	rabex 4173	. . . . . . . 8 ⊢ {𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)} ∈ V
26	24	rabex 4173	. . . . . . . 8 ⊢ {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢} ∈ V
27	25, 26	op2nd 6200	. . . . . . 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 2214	. . . . . 6 ⊢ (2nd ‘𝐿) = {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}
29	21, 28	elrab2 2919	. . . . 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 2763	. . . . . . 7 ⊢ 𝑥 ∈ V
33		breq1 4032	. . . . . . 7 ⊢ (𝑙 = 𝑥 → (𝑙 <Q ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅)) ↔ 𝑥 <Q ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅))))
34	32, 33	elab 2904	. . . . . 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 7609	. . . . . 6 ⊢ {𝑙 ∣ 𝑙 <Q ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅))} ∈ V
37		gtnqex 7610	. . . . . 6 ⊢ {𝑢 ∣ ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅)) <Q 𝑢} ∈ V
38	36, 37	op1st 6199	. . . . 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 2287	. . . 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 2543	. . . 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 1247	. . 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 7713	. . . . 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 7435	. . . . . . . 8 ⊢ ((𝑄 ∈ Q ∧ 𝑅 ∈ Q) → (𝑄 +Q 𝑅) ∈ Q)
47	1, 2, 46	syl2anc 411	. . . . . . 7 ⊢ (𝜑 → (𝑄 +Q 𝑅) ∈ Q)
48		addclnq 7435	. . . . . . 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 7607	. . . . . 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 7566	. . . 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 2616	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 1364   ∈ wcel 2164  {cab 2179  ∀wral 2472  ∃wrex 2473  {crab 2476  ⟨cop 3621   class class class wbr 4029  ⟶wf 5250  ‘cfv 5254  (class class class)co 5918  1^st c1st 6191  2^nd c2nd 6192  Qcnq 7340   +_Q cplq 7342   <_Q cltq 7345  Pcnp 7351  <_P cltp 7355

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-eprel 4320  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-irdg 6423  df-1o 6469  df-oadd 6473  df-omul 6474  df-er 6587  df-ec 6589  df-qs 6593  df-ni 7364  df-pli 7365  df-mi 7366  df-lti 7367  df-plpq 7404  df-mpq 7405  df-enq 7407  df-nqqs 7408  df-plqqs 7409  df-mqqs 7410  df-1nqqs 7411  df-rq 7412  df-ltnqqs 7413  df-inp 7526  df-iltp 7530

This theorem is referenced by:  cauappcvgprlemlim  7721