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Theorem cauappcvgprlemladdfu 7652
Description: Lemma for cauappcvgprlemladd 7656. The forward subset relationship for the upper cut. (Contributed by Jim Kingdon, 11-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}⟩
cauappcvgprlemladd.s (𝜑𝑆Q)
Assertion
Ref Expression
cauappcvgprlemladdfu (𝜑 → (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩)) ⊆ (2nd ‘⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆)}, {𝑢Q ∣ ∃𝑞Q (((𝐹𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑢}⟩))
Distinct variable groups:   𝐴,𝑝   𝐿,𝑝,𝑞   𝜑,𝑝,𝑞   𝐹,𝑙,𝑢,𝑝,𝑞   𝑆,𝑙,𝑞,𝑢
Allowed substitution hints:   𝜑(𝑢,𝑙)   𝐴(𝑢,𝑞,𝑙)   𝑆(𝑝)   𝐿(𝑢,𝑙)

Proof of Theorem cauappcvgprlemladdfu
Dummy variables 𝑓 𝑔 𝑟 𝑠 𝑡 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cauappcvgpr.f . . . . . . 7 (𝜑𝐹:QQ)
2 cauappcvgpr.app . . . . . . 7 (𝜑 → ∀𝑝Q𝑞Q ((𝐹𝑝) <Q ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹𝑞) <Q ((𝐹𝑝) +Q (𝑝 +Q 𝑞))))
3 cauappcvgpr.bnd . . . . . . 7 (𝜑 → ∀𝑝Q 𝐴 <Q (𝐹𝑝))
4 cauappcvgpr.lim . . . . . . 7 𝐿 = ⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}⟩
51, 2, 3, 4cauappcvgprlemcl 7651 . . . . . 6 (𝜑𝐿P)
6 cauappcvgprlemladd.s . . . . . . 7 (𝜑𝑆Q)
7 nqprlu 7545 . . . . . . 7 (𝑆Q → ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩ ∈ P)
86, 7syl 14 . . . . . 6 (𝜑 → ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩ ∈ P)
9 df-iplp 7466 . . . . . . 7 +P = (𝑥P, 𝑦P ↦ ⟨{𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (1st𝑥) ∧ ∈ (1st𝑦) ∧ 𝑓 = (𝑔 +Q ))}, {𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (2nd𝑥) ∧ ∈ (2nd𝑦) ∧ 𝑓 = (𝑔 +Q ))}⟩)
10 addclnq 7373 . . . . . . 7 ((𝑔QQ) → (𝑔 +Q ) ∈ Q)
119, 10genpelvu 7511 . . . . . 6 ((𝐿P ∧ ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩ ∈ P) → (𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩)) ↔ ∃𝑠 ∈ (2nd𝐿)∃𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩)𝑟 = (𝑠 +Q 𝑡)))
125, 8, 11syl2anc 411 . . . . 5 (𝜑 → (𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩)) ↔ ∃𝑠 ∈ (2nd𝐿)∃𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩)𝑟 = (𝑠 +Q 𝑡)))
1312biimpa 296 . . . 4 ((𝜑𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) → ∃𝑠 ∈ (2nd𝐿)∃𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩)𝑟 = (𝑠 +Q 𝑡))
14 breq2 4007 . . . . . . . . . . . . . . . 16 (𝑢 = 𝑠 → (((𝐹𝑞) +Q 𝑞) <Q 𝑢 ↔ ((𝐹𝑞) +Q 𝑞) <Q 𝑠))
1514rexbidv 2478 . . . . . . . . . . . . . . 15 (𝑢 = 𝑠 → (∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢 ↔ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑠))
164fveq2i 5518 . . . . . . . . . . . . . . . 16 (2nd𝐿) = (2nd ‘⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}⟩)
17 nqex 7361 . . . . . . . . . . . . . . . . . 18 Q ∈ V
1817rabex 4147 . . . . . . . . . . . . . . . . 17 {𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)} ∈ V
1917rabex 4147 . . . . . . . . . . . . . . . . 17 {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢} ∈ V
2018, 19op2nd 6147 . . . . . . . . . . . . . . . 16 (2nd ‘⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}⟩) = {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}
2116, 20eqtri 2198 . . . . . . . . . . . . . . 15 (2nd𝐿) = {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}
2215, 21elrab2 2896 . . . . . . . . . . . . . 14 (𝑠 ∈ (2nd𝐿) ↔ (𝑠Q ∧ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑠))
2322biimpi 120 . . . . . . . . . . . . 13 (𝑠 ∈ (2nd𝐿) → (𝑠Q ∧ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑠))
2423adantr 276 . . . . . . . . . . . 12 ((𝑠 ∈ (2nd𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩)) → (𝑠Q ∧ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑠))
2524adantl 277 . . . . . . . . . . 11 (((𝜑𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) → (𝑠Q ∧ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑠))
2625adantr 276 . . . . . . . . . 10 ((((𝜑𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → (𝑠Q ∧ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑠))
2726simpld 112 . . . . . . . . 9 ((((𝜑𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → 𝑠Q)
28 vex 2740 . . . . . . . . . . . . . 14 𝑡 ∈ V
29 breq2 4007 . . . . . . . . . . . . . 14 (𝑢 = 𝑡 → (𝑆 <Q 𝑢𝑆 <Q 𝑡))
30 ltnqex 7547 . . . . . . . . . . . . . . 15 {𝑙𝑙 <Q 𝑆} ∈ V
31 gtnqex 7548 . . . . . . . . . . . . . . 15 {𝑢𝑆 <Q 𝑢} ∈ V
3230, 31op2nd 6147 . . . . . . . . . . . . . 14 (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩) = {𝑢𝑆 <Q 𝑢}
3328, 29, 32elab2 2885 . . . . . . . . . . . . 13 (𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩) ↔ 𝑆 <Q 𝑡)
34 ltrelnq 7363 . . . . . . . . . . . . . 14 <Q ⊆ (Q × Q)
3534brel 4678 . . . . . . . . . . . . 13 (𝑆 <Q 𝑡 → (𝑆Q𝑡Q))
3633, 35sylbi 121 . . . . . . . . . . . 12 (𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩) → (𝑆Q𝑡Q))
3736simprd 114 . . . . . . . . . . 11 (𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩) → 𝑡Q)
3837ad2antll 491 . . . . . . . . . 10 (((𝜑𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) → 𝑡Q)
3938adantr 276 . . . . . . . . 9 ((((𝜑𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → 𝑡Q)
40 addclnq 7373 . . . . . . . . 9 ((𝑠Q𝑡Q) → (𝑠 +Q 𝑡) ∈ Q)
4127, 39, 40syl2anc 411 . . . . . . . 8 ((((𝜑𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → (𝑠 +Q 𝑡) ∈ Q)
42 eleq1 2240 . . . . . . . . 9 (𝑟 = (𝑠 +Q 𝑡) → (𝑟Q ↔ (𝑠 +Q 𝑡) ∈ Q))
4342adantl 277 . . . . . . . 8 ((((𝜑𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → (𝑟Q ↔ (𝑠 +Q 𝑡) ∈ Q))
4441, 43mpbird 167 . . . . . . 7 ((((𝜑𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → 𝑟Q)
4526simprd 114 . . . . . . . 8 ((((𝜑𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑠)
4633biimpi 120 . . . . . . . . . . . . . . . 16 (𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩) → 𝑆 <Q 𝑡)
4746ad2antll 491 . . . . . . . . . . . . . . 15 (((𝜑𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) → 𝑆 <Q 𝑡)
4847adantr 276 . . . . . . . . . . . . . 14 ((((𝜑𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → 𝑆 <Q 𝑡)
4948ad2antrr 488 . . . . . . . . . . . . 13 ((((((𝜑𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑞Q) ∧ ((𝐹𝑞) +Q 𝑞) <Q 𝑠) → 𝑆 <Q 𝑡)
506ad5antr 496 . . . . . . . . . . . . . 14 ((((((𝜑𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑞Q) ∧ ((𝐹𝑞) +Q 𝑞) <Q 𝑠) → 𝑆Q)
5139ad2antrr 488 . . . . . . . . . . . . . 14 ((((((𝜑𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑞Q) ∧ ((𝐹𝑞) +Q 𝑞) <Q 𝑠) → 𝑡Q)
521ad5antr 496 . . . . . . . . . . . . . . . 16 ((((((𝜑𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑞Q) ∧ ((𝐹𝑞) +Q 𝑞) <Q 𝑠) → 𝐹:QQ)
53 simplr 528 . . . . . . . . . . . . . . . 16 ((((((𝜑𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑞Q) ∧ ((𝐹𝑞) +Q 𝑞) <Q 𝑠) → 𝑞Q)
5452, 53ffvelcdmd 5652 . . . . . . . . . . . . . . 15 ((((((𝜑𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑞Q) ∧ ((𝐹𝑞) +Q 𝑞) <Q 𝑠) → (𝐹𝑞) ∈ Q)
55 addclnq 7373 . . . . . . . . . . . . . . 15 (((𝐹𝑞) ∈ Q𝑞Q) → ((𝐹𝑞) +Q 𝑞) ∈ Q)
5654, 53, 55syl2anc 411 . . . . . . . . . . . . . 14 ((((((𝜑𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑞Q) ∧ ((𝐹𝑞) +Q 𝑞) <Q 𝑠) → ((𝐹𝑞) +Q 𝑞) ∈ Q)
57 ltanqg 7398 . . . . . . . . . . . . . 14 ((𝑆Q𝑡Q ∧ ((𝐹𝑞) +Q 𝑞) ∈ Q) → (𝑆 <Q 𝑡 ↔ (((𝐹𝑞) +Q 𝑞) +Q 𝑆) <Q (((𝐹𝑞) +Q 𝑞) +Q 𝑡)))
5850, 51, 56, 57syl3anc 1238 . . . . . . . . . . . . 13 ((((((𝜑𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑞Q) ∧ ((𝐹𝑞) +Q 𝑞) <Q 𝑠) → (𝑆 <Q 𝑡 ↔ (((𝐹𝑞) +Q 𝑞) +Q 𝑆) <Q (((𝐹𝑞) +Q 𝑞) +Q 𝑡)))
5949, 58mpbid 147 . . . . . . . . . . . 12 ((((((𝜑𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑞Q) ∧ ((𝐹𝑞) +Q 𝑞) <Q 𝑠) → (((𝐹𝑞) +Q 𝑞) +Q 𝑆) <Q (((𝐹𝑞) +Q 𝑞) +Q 𝑡))
60 simpr 110 . . . . . . . . . . . . 13 ((((((𝜑𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑞Q) ∧ ((𝐹𝑞) +Q 𝑞) <Q 𝑠) → ((𝐹𝑞) +Q 𝑞) <Q 𝑠)
61 ltanqg 7398 . . . . . . . . . . . . . . 15 ((𝑓Q𝑔QQ) → (𝑓 <Q 𝑔 ↔ ( +Q 𝑓) <Q ( +Q 𝑔)))
6261adantl 277 . . . . . . . . . . . . . 14 (((((((𝜑𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑞Q) ∧ ((𝐹𝑞) +Q 𝑞) <Q 𝑠) ∧ (𝑓Q𝑔QQ)) → (𝑓 <Q 𝑔 ↔ ( +Q 𝑓) <Q ( +Q 𝑔)))
6327ad2antrr 488 . . . . . . . . . . . . . 14 ((((((𝜑𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑞Q) ∧ ((𝐹𝑞) +Q 𝑞) <Q 𝑠) → 𝑠Q)
64 addcomnqg 7379 . . . . . . . . . . . . . . 15 ((𝑓Q𝑔Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
6564adantl 277 . . . . . . . . . . . . . 14 (((((((𝜑𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑞Q) ∧ ((𝐹𝑞) +Q 𝑞) <Q 𝑠) ∧ (𝑓Q𝑔Q)) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
6662, 56, 63, 51, 65caovord2d 6043 . . . . . . . . . . . . 13 ((((((𝜑𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑞Q) ∧ ((𝐹𝑞) +Q 𝑞) <Q 𝑠) → (((𝐹𝑞) +Q 𝑞) <Q 𝑠 ↔ (((𝐹𝑞) +Q 𝑞) +Q 𝑡) <Q (𝑠 +Q 𝑡)))
6760, 66mpbid 147 . . . . . . . . . . . 12 ((((((𝜑𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑞Q) ∧ ((𝐹𝑞) +Q 𝑞) <Q 𝑠) → (((𝐹𝑞) +Q 𝑞) +Q 𝑡) <Q (𝑠 +Q 𝑡))
68 ltsonq 7396 . . . . . . . . . . . . 13 <Q Or Q
6968, 34sotri 5024 . . . . . . . . . . . 12 (((((𝐹𝑞) +Q 𝑞) +Q 𝑆) <Q (((𝐹𝑞) +Q 𝑞) +Q 𝑡) ∧ (((𝐹𝑞) +Q 𝑞) +Q 𝑡) <Q (𝑠 +Q 𝑡)) → (((𝐹𝑞) +Q 𝑞) +Q 𝑆) <Q (𝑠 +Q 𝑡))
7059, 67, 69syl2anc 411 . . . . . . . . . . 11 ((((((𝜑𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑞Q) ∧ ((𝐹𝑞) +Q 𝑞) <Q 𝑠) → (((𝐹𝑞) +Q 𝑞) +Q 𝑆) <Q (𝑠 +Q 𝑡))
71 simpllr 534 . . . . . . . . . . 11 ((((((𝜑𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑞Q) ∧ ((𝐹𝑞) +Q 𝑞) <Q 𝑠) → 𝑟 = (𝑠 +Q 𝑡))
7270, 71breqtrrd 4031 . . . . . . . . . 10 ((((((𝜑𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑞Q) ∧ ((𝐹𝑞) +Q 𝑞) <Q 𝑠) → (((𝐹𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑟)
7372ex 115 . . . . . . . . 9 (((((𝜑𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑞Q) → (((𝐹𝑞) +Q 𝑞) <Q 𝑠 → (((𝐹𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑟))
7473reximdva 2579 . . . . . . . 8 ((((𝜑𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → (∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑠 → ∃𝑞Q (((𝐹𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑟))
7545, 74mpd 13 . . . . . . 7 ((((𝜑𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → ∃𝑞Q (((𝐹𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑟)
76 breq2 4007 . . . . . . . . 9 (𝑢 = 𝑟 → ((((𝐹𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑢 ↔ (((𝐹𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑟))
7776rexbidv 2478 . . . . . . . 8 (𝑢 = 𝑟 → (∃𝑞Q (((𝐹𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑢 ↔ ∃𝑞Q (((𝐹𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑟))
7817rabex 4147 . . . . . . . . 9 {𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆)} ∈ V
7917rabex 4147 . . . . . . . . 9 {𝑢Q ∣ ∃𝑞Q (((𝐹𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑢} ∈ V
8078, 79op2nd 6147 . . . . . . . 8 (2nd ‘⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆)}, {𝑢Q ∣ ∃𝑞Q (((𝐹𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑢}⟩) = {𝑢Q ∣ ∃𝑞Q (((𝐹𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑢}
8177, 80elrab2 2896 . . . . . . 7 (𝑟 ∈ (2nd ‘⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆)}, {𝑢Q ∣ ∃𝑞Q (((𝐹𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑢}⟩) ↔ (𝑟Q ∧ ∃𝑞Q (((𝐹𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑟))
8244, 75, 81sylanbrc 417 . . . . . 6 ((((𝜑𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → 𝑟 ∈ (2nd ‘⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆)}, {𝑢Q ∣ ∃𝑞Q (((𝐹𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑢}⟩))
8382ex 115 . . . . 5 (((𝜑𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) → (𝑟 = (𝑠 +Q 𝑡) → 𝑟 ∈ (2nd ‘⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆)}, {𝑢Q ∣ ∃𝑞Q (((𝐹𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑢}⟩)))
8483rexlimdvva 2602 . . . 4 ((𝜑𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) → (∃𝑠 ∈ (2nd𝐿)∃𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩)𝑟 = (𝑠 +Q 𝑡) → 𝑟 ∈ (2nd ‘⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆)}, {𝑢Q ∣ ∃𝑞Q (((𝐹𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑢}⟩)))
8513, 84mpd 13 . . 3 ((𝜑𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) → 𝑟 ∈ (2nd ‘⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆)}, {𝑢Q ∣ ∃𝑞Q (((𝐹𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑢}⟩))
8685ex 115 . 2 (𝜑 → (𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩)) → 𝑟 ∈ (2nd ‘⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆)}, {𝑢Q ∣ ∃𝑞Q (((𝐹𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑢}⟩)))
8786ssrdv 3161 1 (𝜑 → (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩)) ⊆ (2nd ‘⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆)}, {𝑢Q ∣ ∃𝑞Q (((𝐹𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑢}⟩))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  w3a 978   = wceq 1353  wcel 2148  {cab 2163  wral 2455  wrex 2456  {crab 2459  wss 3129  cop 3595   class class class wbr 4003  wf 5212  cfv 5216  (class class class)co 5874  2nd c2nd 6139  Qcnq 7278   +Q cplq 7280   <Q cltq 7283  Pcnp 7289   +P cpp 7291
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-eprel 4289  df-id 4293  df-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-irdg 6370  df-1o 6416  df-oadd 6420  df-omul 6421  df-er 6534  df-ec 6536  df-qs 6540  df-ni 7302  df-pli 7303  df-mi 7304  df-lti 7305  df-plpq 7342  df-mpq 7343  df-enq 7345  df-nqqs 7346  df-plqqs 7347  df-mqqs 7348  df-1nqqs 7349  df-rq 7350  df-ltnqqs 7351  df-inp 7464  df-iplp 7466
This theorem is referenced by:  cauappcvgprlemladdrl  7655  cauappcvgprlemladd  7656
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