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Theorem cauappcvgprlemm 7453
 Description: Lemma for cauappcvgpr 7470. The putative limit is inhabited. (Contributed by Jim Kingdon, 18-Jul-2020.)
Hypotheses
Ref Expression
cauappcvgpr.f (𝜑𝐹:QQ)
cauappcvgpr.app (𝜑 → ∀𝑝Q𝑞Q ((𝐹𝑝) <Q ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹𝑞) <Q ((𝐹𝑝) +Q (𝑝 +Q 𝑞))))
cauappcvgpr.bnd (𝜑 → ∀𝑝Q 𝐴 <Q (𝐹𝑝))
cauappcvgpr.lim 𝐿 = ⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}⟩
Assertion
Ref Expression
cauappcvgprlemm (𝜑 → (∃𝑠Q 𝑠 ∈ (1st𝐿) ∧ ∃𝑟Q 𝑟 ∈ (2nd𝐿)))
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 cauappcvgprlemm
StepHypRef Expression
1 fveq2 5421 . . . . . . 7 (𝑝 = 1Q → (𝐹𝑝) = (𝐹‘1Q))
21breq2d 3941 . . . . . 6 (𝑝 = 1Q → (𝐴 <Q (𝐹𝑝) ↔ 𝐴 <Q (𝐹‘1Q)))
3 cauappcvgpr.bnd . . . . . 6 (𝜑 → ∀𝑝Q 𝐴 <Q (𝐹𝑝))
4 1nq 7174 . . . . . . 7 1QQ
54a1i 9 . . . . . 6 (𝜑 → 1QQ)
62, 3, 5rspcdva 2794 . . . . 5 (𝜑𝐴 <Q (𝐹‘1Q))
7 ltrelnq 7173 . . . . . . 7 <Q ⊆ (Q × Q)
87brel 4591 . . . . . 6 (𝐴 <Q (𝐹‘1Q) → (𝐴Q ∧ (𝐹‘1Q) ∈ Q))
98simpld 111 . . . . 5 (𝐴 <Q (𝐹‘1Q) → 𝐴Q)
106, 9syl 14 . . . 4 (𝜑𝐴Q)
11 halfnqq 7218 . . . 4 (𝐴Q → ∃𝑠Q (𝑠 +Q 𝑠) = 𝐴)
1210, 11syl 14 . . 3 (𝜑 → ∃𝑠Q (𝑠 +Q 𝑠) = 𝐴)
13 simplr 519 . . . . . 6 (((𝜑𝑠Q) ∧ (𝑠 +Q 𝑠) = 𝐴) → 𝑠Q)
143ad2antrr 479 . . . . . . . . 9 (((𝜑𝑠Q) ∧ (𝑠 +Q 𝑠) = 𝐴) → ∀𝑝Q 𝐴 <Q (𝐹𝑝))
15 fveq2 5421 . . . . . . . . . . . 12 (𝑝 = 𝑠 → (𝐹𝑝) = (𝐹𝑠))
1615breq2d 3941 . . . . . . . . . . 11 (𝑝 = 𝑠 → (𝐴 <Q (𝐹𝑝) ↔ 𝐴 <Q (𝐹𝑠)))
1716rspcv 2785 . . . . . . . . . 10 (𝑠Q → (∀𝑝Q 𝐴 <Q (𝐹𝑝) → 𝐴 <Q (𝐹𝑠)))
1817ad2antlr 480 . . . . . . . . 9 (((𝜑𝑠Q) ∧ (𝑠 +Q 𝑠) = 𝐴) → (∀𝑝Q 𝐴 <Q (𝐹𝑝) → 𝐴 <Q (𝐹𝑠)))
1914, 18mpd 13 . . . . . . . 8 (((𝜑𝑠Q) ∧ (𝑠 +Q 𝑠) = 𝐴) → 𝐴 <Q (𝐹𝑠))
20 breq1 3932 . . . . . . . . 9 ((𝑠 +Q 𝑠) = 𝐴 → ((𝑠 +Q 𝑠) <Q (𝐹𝑠) ↔ 𝐴 <Q (𝐹𝑠)))
2120adantl 275 . . . . . . . 8 (((𝜑𝑠Q) ∧ (𝑠 +Q 𝑠) = 𝐴) → ((𝑠 +Q 𝑠) <Q (𝐹𝑠) ↔ 𝐴 <Q (𝐹𝑠)))
2219, 21mpbird 166 . . . . . . 7 (((𝜑𝑠Q) ∧ (𝑠 +Q 𝑠) = 𝐴) → (𝑠 +Q 𝑠) <Q (𝐹𝑠))
23 oveq2 5782 . . . . . . . . 9 (𝑞 = 𝑠 → (𝑠 +Q 𝑞) = (𝑠 +Q 𝑠))
24 fveq2 5421 . . . . . . . . 9 (𝑞 = 𝑠 → (𝐹𝑞) = (𝐹𝑠))
2523, 24breq12d 3942 . . . . . . . 8 (𝑞 = 𝑠 → ((𝑠 +Q 𝑞) <Q (𝐹𝑞) ↔ (𝑠 +Q 𝑠) <Q (𝐹𝑠)))
2625rspcev 2789 . . . . . . 7 ((𝑠Q ∧ (𝑠 +Q 𝑠) <Q (𝐹𝑠)) → ∃𝑞Q (𝑠 +Q 𝑞) <Q (𝐹𝑞))
2713, 22, 26syl2anc 408 . . . . . 6 (((𝜑𝑠Q) ∧ (𝑠 +Q 𝑠) = 𝐴) → ∃𝑞Q (𝑠 +Q 𝑞) <Q (𝐹𝑞))
28 oveq1 5781 . . . . . . . . 9 (𝑙 = 𝑠 → (𝑙 +Q 𝑞) = (𝑠 +Q 𝑞))
2928breq1d 3939 . . . . . . . 8 (𝑙 = 𝑠 → ((𝑙 +Q 𝑞) <Q (𝐹𝑞) ↔ (𝑠 +Q 𝑞) <Q (𝐹𝑞)))
3029rexbidv 2438 . . . . . . 7 (𝑙 = 𝑠 → (∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞) ↔ ∃𝑞Q (𝑠 +Q 𝑞) <Q (𝐹𝑞)))
31 cauappcvgpr.lim . . . . . . . . 9 𝐿 = ⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}⟩
3231fveq2i 5424 . . . . . . . 8 (1st𝐿) = (1st ‘⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}⟩)
33 nqex 7171 . . . . . . . . . 10 Q ∈ V
3433rabex 4072 . . . . . . . . 9 {𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)} ∈ V
3533rabex 4072 . . . . . . . . 9 {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢} ∈ V
3634, 35op1st 6044 . . . . . . . 8 (1st ‘⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}⟩) = {𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}
3732, 36eqtri 2160 . . . . . . 7 (1st𝐿) = {𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}
3830, 37elrab2 2843 . . . . . 6 (𝑠 ∈ (1st𝐿) ↔ (𝑠Q ∧ ∃𝑞Q (𝑠 +Q 𝑞) <Q (𝐹𝑞)))
3913, 27, 38sylanbrc 413 . . . . 5 (((𝜑𝑠Q) ∧ (𝑠 +Q 𝑠) = 𝐴) → 𝑠 ∈ (1st𝐿))
4039ex 114 . . . 4 ((𝜑𝑠Q) → ((𝑠 +Q 𝑠) = 𝐴𝑠 ∈ (1st𝐿)))
4140reximdva 2534 . . 3 (𝜑 → (∃𝑠Q (𝑠 +Q 𝑠) = 𝐴 → ∃𝑠Q 𝑠 ∈ (1st𝐿)))
4212, 41mpd 13 . 2 (𝜑 → ∃𝑠Q 𝑠 ∈ (1st𝐿))
43 cauappcvgpr.f . . . . . 6 (𝜑𝐹:QQ)
4443, 5ffvelrnd 5556 . . . . 5 (𝜑 → (𝐹‘1Q) ∈ Q)
45 addclnq 7183 . . . . 5 (((𝐹‘1Q) ∈ Q ∧ 1QQ) → ((𝐹‘1Q) +Q 1Q) ∈ Q)
4644, 5, 45syl2anc 408 . . . 4 (𝜑 → ((𝐹‘1Q) +Q 1Q) ∈ Q)
47 addclnq 7183 . . . 4 ((((𝐹‘1Q) +Q 1Q) ∈ Q ∧ 1QQ) → (((𝐹‘1Q) +Q 1Q) +Q 1Q) ∈ Q)
4846, 5, 47syl2anc 408 . . 3 (𝜑 → (((𝐹‘1Q) +Q 1Q) +Q 1Q) ∈ Q)
49 ltaddnq 7215 . . . . . 6 ((((𝐹‘1Q) +Q 1Q) ∈ Q ∧ 1QQ) → ((𝐹‘1Q) +Q 1Q) <Q (((𝐹‘1Q) +Q 1Q) +Q 1Q))
5046, 5, 49syl2anc 408 . . . . 5 (𝜑 → ((𝐹‘1Q) +Q 1Q) <Q (((𝐹‘1Q) +Q 1Q) +Q 1Q))
51 fveq2 5421 . . . . . . . 8 (𝑞 = 1Q → (𝐹𝑞) = (𝐹‘1Q))
52 id 19 . . . . . . . 8 (𝑞 = 1Q𝑞 = 1Q)
5351, 52oveq12d 5792 . . . . . . 7 (𝑞 = 1Q → ((𝐹𝑞) +Q 𝑞) = ((𝐹‘1Q) +Q 1Q))
5453breq1d 3939 . . . . . 6 (𝑞 = 1Q → (((𝐹𝑞) +Q 𝑞) <Q (((𝐹‘1Q) +Q 1Q) +Q 1Q) ↔ ((𝐹‘1Q) +Q 1Q) <Q (((𝐹‘1Q) +Q 1Q) +Q 1Q)))
5554rspcev 2789 . . . . 5 ((1QQ ∧ ((𝐹‘1Q) +Q 1Q) <Q (((𝐹‘1Q) +Q 1Q) +Q 1Q)) → ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q (((𝐹‘1Q) +Q 1Q) +Q 1Q))
565, 50, 55syl2anc 408 . . . 4 (𝜑 → ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q (((𝐹‘1Q) +Q 1Q) +Q 1Q))
57 breq2 3933 . . . . . 6 (𝑢 = (((𝐹‘1Q) +Q 1Q) +Q 1Q) → (((𝐹𝑞) +Q 𝑞) <Q 𝑢 ↔ ((𝐹𝑞) +Q 𝑞) <Q (((𝐹‘1Q) +Q 1Q) +Q 1Q)))
5857rexbidv 2438 . . . . 5 (𝑢 = (((𝐹‘1Q) +Q 1Q) +Q 1Q) → (∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢 ↔ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q (((𝐹‘1Q) +Q 1Q) +Q 1Q)))
5931fveq2i 5424 . . . . . 6 (2nd𝐿) = (2nd ‘⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}⟩)
6034, 35op2nd 6045 . . . . . 6 (2nd ‘⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}⟩) = {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}
6159, 60eqtri 2160 . . . . 5 (2nd𝐿) = {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}
6258, 61elrab2 2843 . . . 4 ((((𝐹‘1Q) +Q 1Q) +Q 1Q) ∈ (2nd𝐿) ↔ ((((𝐹‘1Q) +Q 1Q) +Q 1Q) ∈ Q ∧ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q (((𝐹‘1Q) +Q 1Q) +Q 1Q)))
6348, 56, 62sylanbrc 413 . . 3 (𝜑 → (((𝐹‘1Q) +Q 1Q) +Q 1Q) ∈ (2nd𝐿))
64 eleq1 2202 . . . 4 (𝑟 = (((𝐹‘1Q) +Q 1Q) +Q 1Q) → (𝑟 ∈ (2nd𝐿) ↔ (((𝐹‘1Q) +Q 1Q) +Q 1Q) ∈ (2nd𝐿)))
6564rspcev 2789 . . 3 (((((𝐹‘1Q) +Q 1Q) +Q 1Q) ∈ Q ∧ (((𝐹‘1Q) +Q 1Q) +Q 1Q) ∈ (2nd𝐿)) → ∃𝑟Q 𝑟 ∈ (2nd𝐿))
6648, 63, 65syl2anc 408 . 2 (𝜑 → ∃𝑟Q 𝑟 ∈ (2nd𝐿))
6742, 66jca 304 1 (𝜑 → (∃𝑠Q 𝑠 ∈ (1st𝐿) ∧ ∃𝑟Q 𝑟 ∈ (2nd𝐿)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1331   ∈ wcel 1480  ∀wral 2416  ∃wrex 2417  {crab 2420  ⟨cop 3530   class class class wbr 3929  ⟶wf 5119  ‘cfv 5123  (class class class)co 5774  1st c1st 6036  2nd c2nd 6037  Qcnq 7088  1Qc1q 7089   +Q cplq 7090
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