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Theorem cauappcvgprlemm 6616
 Description: Lemma for cauappcvgpr 6633. The putative limit is inhabited. (Contributed by Jim Kingdon, 18-Jul-2020.)
Hypotheses
Ref Expression
cauappcvgpr.f (φ𝐹:QQ)
cauappcvgpr.app (φ𝑝 Q 𝑞 Q ((𝐹𝑝) <Q ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) (𝐹𝑞) <Q ((𝐹𝑝) +Q (𝑝 +Q 𝑞))))
cauappcvgpr.bnd (φ𝑝 Q A <Q (𝐹𝑝))
cauappcvgpr.lim 𝐿 = ⟨{𝑙 Q𝑞 Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {u Q𝑞 Q ((𝐹𝑞) +Q 𝑞) <Q u}⟩
Assertion
Ref Expression
cauappcvgprlemm (φ → (𝑠 Q 𝑠 (1st𝐿) 𝑟 Q 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 1nq 6350 . . . . . 6 1Q Q
2 cauappcvgpr.bnd . . . . . 6 (φ𝑝 Q A <Q (𝐹𝑝))
3 fveq2 5121 . . . . . . . 8 (𝑝 = 1Q → (𝐹𝑝) = (𝐹‘1Q))
43breq2d 3767 . . . . . . 7 (𝑝 = 1Q → (A <Q (𝐹𝑝) ↔ A <Q (𝐹‘1Q)))
54rspcv 2646 . . . . . 6 (1Q Q → (𝑝 Q A <Q (𝐹𝑝) → A <Q (𝐹‘1Q)))
61, 2, 5mpsyl 59 . . . . 5 (φA <Q (𝐹‘1Q))
7 ltrelnq 6349 . . . . . . 7 <Q ⊆ (Q × Q)
87brel 4335 . . . . . 6 (A <Q (𝐹‘1Q) → (A Q (𝐹‘1Q) Q))
98simpld 105 . . . . 5 (A <Q (𝐹‘1Q) → A Q)
106, 9syl 14 . . . 4 (φA Q)
11 halfnqq 6393 . . . 4 (A Q𝑠 Q (𝑠 +Q 𝑠) = A)
1210, 11syl 14 . . 3 (φ𝑠 Q (𝑠 +Q 𝑠) = A)
13 simplr 482 . . . . . 6 (((φ 𝑠 Q) (𝑠 +Q 𝑠) = A) → 𝑠 Q)
142ad2antrr 457 . . . . . . . . 9 (((φ 𝑠 Q) (𝑠 +Q 𝑠) = A) → 𝑝 Q A <Q (𝐹𝑝))
15 fveq2 5121 . . . . . . . . . . . 12 (𝑝 = 𝑠 → (𝐹𝑝) = (𝐹𝑠))
1615breq2d 3767 . . . . . . . . . . 11 (𝑝 = 𝑠 → (A <Q (𝐹𝑝) ↔ A <Q (𝐹𝑠)))
1716rspcv 2646 . . . . . . . . . 10 (𝑠 Q → (𝑝 Q A <Q (𝐹𝑝) → A <Q (𝐹𝑠)))
1817ad2antlr 458 . . . . . . . . 9 (((φ 𝑠 Q) (𝑠 +Q 𝑠) = A) → (𝑝 Q A <Q (𝐹𝑝) → A <Q (𝐹𝑠)))
1914, 18mpd 13 . . . . . . . 8 (((φ 𝑠 Q) (𝑠 +Q 𝑠) = A) → A <Q (𝐹𝑠))
20 breq1 3758 . . . . . . . . 9 ((𝑠 +Q 𝑠) = A → ((𝑠 +Q 𝑠) <Q (𝐹𝑠) ↔ A <Q (𝐹𝑠)))
2120adantl 262 . . . . . . . 8 (((φ 𝑠 Q) (𝑠 +Q 𝑠) = A) → ((𝑠 +Q 𝑠) <Q (𝐹𝑠) ↔ A <Q (𝐹𝑠)))
2219, 21mpbird 156 . . . . . . 7 (((φ 𝑠 Q) (𝑠 +Q 𝑠) = A) → (𝑠 +Q 𝑠) <Q (𝐹𝑠))
23 oveq2 5463 . . . . . . . . 9 (𝑞 = 𝑠 → (𝑠 +Q 𝑞) = (𝑠 +Q 𝑠))
24 fveq2 5121 . . . . . . . . 9 (𝑞 = 𝑠 → (𝐹𝑞) = (𝐹𝑠))
2523, 24breq12d 3768 . . . . . . . 8 (𝑞 = 𝑠 → ((𝑠 +Q 𝑞) <Q (𝐹𝑞) ↔ (𝑠 +Q 𝑠) <Q (𝐹𝑠)))
2625rspcev 2650 . . . . . . 7 ((𝑠 Q (𝑠 +Q 𝑠) <Q (𝐹𝑠)) → 𝑞 Q (𝑠 +Q 𝑞) <Q (𝐹𝑞))
2713, 22, 26syl2anc 391 . . . . . 6 (((φ 𝑠 Q) (𝑠 +Q 𝑠) = A) → 𝑞 Q (𝑠 +Q 𝑞) <Q (𝐹𝑞))
28 oveq1 5462 . . . . . . . . 9 (𝑙 = 𝑠 → (𝑙 +Q 𝑞) = (𝑠 +Q 𝑞))
2928breq1d 3765 . . . . . . . 8 (𝑙 = 𝑠 → ((𝑙 +Q 𝑞) <Q (𝐹𝑞) ↔ (𝑠 +Q 𝑞) <Q (𝐹𝑞)))
3029rexbidv 2321 . . . . . . 7 (𝑙 = 𝑠 → (𝑞 Q (𝑙 +Q 𝑞) <Q (𝐹𝑞) ↔ 𝑞 Q (𝑠 +Q 𝑞) <Q (𝐹𝑞)))
31 cauappcvgpr.lim . . . . . . . . 9 𝐿 = ⟨{𝑙 Q𝑞 Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {u Q𝑞 Q ((𝐹𝑞) +Q 𝑞) <Q u}⟩
3231fveq2i 5124 . . . . . . . 8 (1st𝐿) = (1st ‘⟨{𝑙 Q𝑞 Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {u Q𝑞 Q ((𝐹𝑞) +Q 𝑞) <Q u}⟩)
33 nqex 6347 . . . . . . . . . 10 Q V
3433rabex 3892 . . . . . . . . 9 {𝑙 Q𝑞 Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)} V
3533rabex 3892 . . . . . . . . 9 {u Q𝑞 Q ((𝐹𝑞) +Q 𝑞) <Q u} V
3634, 35op1st 5715 . . . . . . . 8 (1st ‘⟨{𝑙 Q𝑞 Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {u Q𝑞 Q ((𝐹𝑞) +Q 𝑞) <Q u}⟩) = {𝑙 Q𝑞 Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}
3732, 36eqtri 2057 . . . . . . 7 (1st𝐿) = {𝑙 Q𝑞 Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}
3830, 37elrab2 2694 . . . . . 6 (𝑠 (1st𝐿) ↔ (𝑠 Q 𝑞 Q (𝑠 +Q 𝑞) <Q (𝐹𝑞)))
3913, 27, 38sylanbrc 394 . . . . 5 (((φ 𝑠 Q) (𝑠 +Q 𝑠) = A) → 𝑠 (1st𝐿))
4039ex 108 . . . 4 ((φ 𝑠 Q) → ((𝑠 +Q 𝑠) = A𝑠 (1st𝐿)))
4140reximdva 2415 . . 3 (φ → (𝑠 Q (𝑠 +Q 𝑠) = A𝑠 Q 𝑠 (1st𝐿)))
4212, 41mpd 13 . 2 (φ𝑠 Q 𝑠 (1st𝐿))
43 cauappcvgpr.f . . . . . 6 (φ𝐹:QQ)
441a1i 9 . . . . . 6 (φ → 1Q Q)
4543, 44ffvelrnd 5246 . . . . 5 (φ → (𝐹‘1Q) Q)
46 addclnq 6359 . . . . 5 (((𝐹‘1Q) Q 1Q Q) → ((𝐹‘1Q) +Q 1Q) Q)
4745, 44, 46syl2anc 391 . . . 4 (φ → ((𝐹‘1Q) +Q 1Q) Q)
48 addclnq 6359 . . . 4 ((((𝐹‘1Q) +Q 1Q) Q 1Q Q) → (((𝐹‘1Q) +Q 1Q) +Q 1Q) Q)
4947, 44, 48syl2anc 391 . . 3 (φ → (((𝐹‘1Q) +Q 1Q) +Q 1Q) Q)
50 ltaddnq 6390 . . . . . 6 ((((𝐹‘1Q) +Q 1Q) Q 1Q Q) → ((𝐹‘1Q) +Q 1Q) <Q (((𝐹‘1Q) +Q 1Q) +Q 1Q))
5147, 44, 50syl2anc 391 . . . . 5 (φ → ((𝐹‘1Q) +Q 1Q) <Q (((𝐹‘1Q) +Q 1Q) +Q 1Q))
52 fveq2 5121 . . . . . . . 8 (𝑞 = 1Q → (𝐹𝑞) = (𝐹‘1Q))
53 id 19 . . . . . . . 8 (𝑞 = 1Q𝑞 = 1Q)
5452, 53oveq12d 5473 . . . . . . 7 (𝑞 = 1Q → ((𝐹𝑞) +Q 𝑞) = ((𝐹‘1Q) +Q 1Q))
5554breq1d 3765 . . . . . 6 (𝑞 = 1Q → (((𝐹𝑞) +Q 𝑞) <Q (((𝐹‘1Q) +Q 1Q) +Q 1Q) ↔ ((𝐹‘1Q) +Q 1Q) <Q (((𝐹‘1Q) +Q 1Q) +Q 1Q)))
5655rspcev 2650 . . . . 5 ((1Q Q ((𝐹‘1Q) +Q 1Q) <Q (((𝐹‘1Q) +Q 1Q) +Q 1Q)) → 𝑞 Q ((𝐹𝑞) +Q 𝑞) <Q (((𝐹‘1Q) +Q 1Q) +Q 1Q))
5744, 51, 56syl2anc 391 . . . 4 (φ𝑞 Q ((𝐹𝑞) +Q 𝑞) <Q (((𝐹‘1Q) +Q 1Q) +Q 1Q))
58 breq2 3759 . . . . . 6 (u = (((𝐹‘1Q) +Q 1Q) +Q 1Q) → (((𝐹𝑞) +Q 𝑞) <Q u ↔ ((𝐹𝑞) +Q 𝑞) <Q (((𝐹‘1Q) +Q 1Q) +Q 1Q)))
5958rexbidv 2321 . . . . 5 (u = (((𝐹‘1Q) +Q 1Q) +Q 1Q) → (𝑞 Q ((𝐹𝑞) +Q 𝑞) <Q u𝑞 Q ((𝐹𝑞) +Q 𝑞) <Q (((𝐹‘1Q) +Q 1Q) +Q 1Q)))
6031fveq2i 5124 . . . . . 6 (2nd𝐿) = (2nd ‘⟨{𝑙 Q𝑞 Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {u Q𝑞 Q ((𝐹𝑞) +Q 𝑞) <Q u}⟩)
6134, 35op2nd 5716 . . . . . 6 (2nd ‘⟨{𝑙 Q𝑞 Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {u Q𝑞 Q ((𝐹𝑞) +Q 𝑞) <Q u}⟩) = {u Q𝑞 Q ((𝐹𝑞) +Q 𝑞) <Q u}
6260, 61eqtri 2057 . . . . 5 (2nd𝐿) = {u Q𝑞 Q ((𝐹𝑞) +Q 𝑞) <Q u}
6359, 62elrab2 2694 . . . 4 ((((𝐹‘1Q) +Q 1Q) +Q 1Q) (2nd𝐿) ↔ ((((𝐹‘1Q) +Q 1Q) +Q 1Q) Q 𝑞 Q ((𝐹𝑞) +Q 𝑞) <Q (((𝐹‘1Q) +Q 1Q) +Q 1Q)))
6449, 57, 63sylanbrc 394 . . 3 (φ → (((𝐹‘1Q) +Q 1Q) +Q 1Q) (2nd𝐿))
65 eleq1 2097 . . . 4 (𝑟 = (((𝐹‘1Q) +Q 1Q) +Q 1Q) → (𝑟 (2nd𝐿) ↔ (((𝐹‘1Q) +Q 1Q) +Q 1Q) (2nd𝐿)))
6665rspcev 2650 . . 3 (((((𝐹‘1Q) +Q 1Q) +Q 1Q) Q (((𝐹‘1Q) +Q 1Q) +Q 1Q) (2nd𝐿)) → 𝑟 Q 𝑟 (2nd𝐿))
6749, 64, 66syl2anc 391 . 2 (φ𝑟 Q 𝑟 (2nd𝐿))
6842, 67jca 290 1 (φ → (𝑠 Q 𝑠 (1st𝐿) 𝑟 Q 𝑟 (2nd𝐿)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242   ∈ wcel 1390  ∀wral 2300  ∃wrex 2301  {crab 2304  ⟨cop 3370   class class class wbr 3755  ⟶wf 4841  ‘cfv 4845  (class class class)co 5455  1st c1st 5707  2nd c2nd 5708  Qcnq 6264  1Qc1q 6265   +Q cplq 6266
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