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 Description: Lemma for cauappcvgprlemladd 6756. The forward subset relationship for the lower 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}⟩
Assertion
Ref Expression
cauappcvgprlemladdfl (𝜑 → (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩)) ⊆ (1st ‘⟨{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)

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 6751 . . . . . 6 (𝜑𝐿P)
6 cauappcvgprlemladd.s . . . . . . 7 (𝜑𝑆Q)
7 nqprlu 6645 . . . . . . 7 (𝑆Q → ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩ ∈ P)
86, 7syl 14 . . . . . 6 (𝜑 → ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩ ∈ P)
9 df-iplp 6566 . . . . . . 7 +P = (𝑥P, 𝑦P ↦ ⟨{𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (1st𝑥) ∧ ∈ (1st𝑦) ∧ 𝑓 = (𝑔 +Q ))}, {𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (2nd𝑥) ∧ ∈ (2nd𝑦) ∧ 𝑓 = (𝑔 +Q ))}⟩)
10 addclnq 6473 . . . . . . 7 ((𝑔QQ) → (𝑔 +Q ) ∈ Q)
119, 10genpelvl 6610 . . . . . 6 ((𝐿P ∧ ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩ ∈ P) → (𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩)) ↔ ∃𝑠 ∈ (1st𝐿)∃𝑡 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩)𝑟 = (𝑠 +Q 𝑡)))
125, 8, 11syl2anc 391 . . . . 5 (𝜑 → (𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩)) ↔ ∃𝑠 ∈ (1st𝐿)∃𝑡 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩)𝑟 = (𝑠 +Q 𝑡)))
1312biimpa 280 . . . 4 ((𝜑𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) → ∃𝑠 ∈ (1st𝐿)∃𝑡 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩)𝑟 = (𝑠 +Q 𝑡))
14 oveq1 5519 . . . . . . . . . . . . . . . 16 (𝑙 = 𝑠 → (𝑙 +Q 𝑞) = (𝑠 +Q 𝑞))
1514breq1d 3774 . . . . . . . . . . . . . . 15 (𝑙 = 𝑠 → ((𝑙 +Q 𝑞) <Q (𝐹𝑞) ↔ (𝑠 +Q 𝑞) <Q (𝐹𝑞)))
1615rexbidv 2327 . . . . . . . . . . . . . 14 (𝑙 = 𝑠 → (∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞) ↔ ∃𝑞Q (𝑠 +Q 𝑞) <Q (𝐹𝑞)))
174fveq2i 5181 . . . . . . . . . . . . . . 15 (1st𝐿) = (1st ‘⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}⟩)
18 nqex 6461 . . . . . . . . . . . . . . . . 17 Q ∈ V
1918rabex 3901 . . . . . . . . . . . . . . . 16 {𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)} ∈ V
2018rabex 3901 . . . . . . . . . . . . . . . 16 {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢} ∈ V
2119, 20op1st 5773 . . . . . . . . . . . . . . 15 (1st ‘⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}⟩) = {𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}
2217, 21eqtri 2060 . . . . . . . . . . . . . 14 (1st𝐿) = {𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}
2316, 22elrab2 2700 . . . . . . . . . . . . 13 (𝑠 ∈ (1st𝐿) ↔ (𝑠Q ∧ ∃𝑞Q (𝑠 +Q 𝑞) <Q (𝐹𝑞)))
2423biimpi 113 . . . . . . . . . . . 12 (𝑠 ∈ (1st𝐿) → (𝑠Q ∧ ∃𝑞Q (𝑠 +Q 𝑞) <Q (𝐹𝑞)))
2524ad2antrl 459 . . . . . . . . . . 11 (((𝜑𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st𝐿) ∧ 𝑡 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) → (𝑠Q ∧ ∃𝑞Q (𝑠 +Q 𝑞) <Q (𝐹𝑞)))
2625adantr 261 . . . . . . . . . 10 ((((𝜑𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st𝐿) ∧ 𝑡 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → (𝑠Q ∧ ∃𝑞Q (𝑠 +Q 𝑞) <Q (𝐹𝑞)))
2726simpld 105 . . . . . . . . 9 ((((𝜑𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st𝐿) ∧ 𝑡 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → 𝑠Q)
28 vex 2560 . . . . . . . . . . . . . . 15 𝑡 ∈ V
29 breq1 3767 . . . . . . . . . . . . . . 15 (𝑙 = 𝑡 → (𝑙 <Q 𝑆𝑡 <Q 𝑆))
30 ltnqex 6647 . . . . . . . . . . . . . . . 16 {𝑙𝑙 <Q 𝑆} ∈ V
31 gtnqex 6648 . . . . . . . . . . . . . . . 16 {𝑢𝑆 <Q 𝑢} ∈ V
3230, 31op1st 5773 . . . . . . . . . . . . . . 15 (1st ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩) = {𝑙𝑙 <Q 𝑆}
3328, 29, 32elab2 2690 . . . . . . . . . . . . . 14 (𝑡 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩) ↔ 𝑡 <Q 𝑆)
3433biimpi 113 . . . . . . . . . . . . 13 (𝑡 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩) → 𝑡 <Q 𝑆)
3534ad2antll 460 . . . . . . . . . . . 12 (((𝜑𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st𝐿) ∧ 𝑡 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) → 𝑡 <Q 𝑆)
3635adantr 261 . . . . . . . . . . 11 ((((𝜑𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st𝐿) ∧ 𝑡 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → 𝑡 <Q 𝑆)
37 ltrelnq 6463 . . . . . . . . . . . 12 <Q ⊆ (Q × Q)
3837brel 4392 . . . . . . . . . . 11 (𝑡 <Q 𝑆 → (𝑡Q𝑆Q))
3936, 38syl 14 . . . . . . . . . 10 ((((𝜑𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st𝐿) ∧ 𝑡 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → (𝑡Q𝑆Q))
4039simpld 105 . . . . . . . . 9 ((((𝜑𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st𝐿) ∧ 𝑡 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → 𝑡Q)
41 addclnq 6473 . . . . . . . . 9 ((𝑠Q𝑡Q) → (𝑠 +Q 𝑡) ∈ Q)
4227, 40, 41syl2anc 391 . . . . . . . 8 ((((𝜑𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st𝐿) ∧ 𝑡 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → (𝑠 +Q 𝑡) ∈ Q)
43 eleq1 2100 . . . . . . . . 9 (𝑟 = (𝑠 +Q 𝑡) → (𝑟Q ↔ (𝑠 +Q 𝑡) ∈ Q))
4443adantl 262 . . . . . . . 8 ((((𝜑𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st𝐿) ∧ 𝑡 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → (𝑟Q ↔ (𝑠 +Q 𝑡) ∈ Q))
4542, 44mpbird 156 . . . . . . 7 ((((𝜑𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st𝐿) ∧ 𝑡 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → 𝑟Q)
4626simprd 107 . . . . . . . 8 ((((𝜑𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st𝐿) ∧ 𝑡 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → ∃𝑞Q (𝑠 +Q 𝑞) <Q (𝐹𝑞))
4727ad2antrr 457 . . . . . . . . . . . . 13 ((((((𝜑𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st𝐿) ∧ 𝑡 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑞Q) ∧ (𝑠 +Q 𝑞) <Q (𝐹𝑞)) → 𝑠Q)
48 simplr 482 . . . . . . . . . . . . 13 ((((((𝜑𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st𝐿) ∧ 𝑡 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑞Q) ∧ (𝑠 +Q 𝑞) <Q (𝐹𝑞)) → 𝑞Q)
4940ad2antrr 457 . . . . . . . . . . . . 13 ((((((𝜑𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st𝐿) ∧ 𝑡 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑞Q) ∧ (𝑠 +Q 𝑞) <Q (𝐹𝑞)) → 𝑡Q)
50 addcomnqg 6479 . . . . . . . . . . . . . 14 ((𝑓Q𝑔Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
5150adantl 262 . . . . . . . . . . . . 13 (((((((𝜑𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st𝐿) ∧ 𝑡 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑞Q) ∧ (𝑠 +Q 𝑞) <Q (𝐹𝑞)) ∧ (𝑓Q𝑔Q)) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
52 addassnqg 6480 . . . . . . . . . . . . . 14 ((𝑓Q𝑔QQ) → ((𝑓 +Q 𝑔) +Q ) = (𝑓 +Q (𝑔 +Q )))
5352adantl 262 . . . . . . . . . . . . 13 (((((((𝜑𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st𝐿) ∧ 𝑡 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑞Q) ∧ (𝑠 +Q 𝑞) <Q (𝐹𝑞)) ∧ (𝑓Q𝑔QQ)) → ((𝑓 +Q 𝑔) +Q ) = (𝑓 +Q (𝑔 +Q )))
5447, 48, 49, 51, 53caov32d 5681 . . . . . . . . . . . 12 ((((((𝜑𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st𝐿) ∧ 𝑡 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑞Q) ∧ (𝑠 +Q 𝑞) <Q (𝐹𝑞)) → ((𝑠 +Q 𝑞) +Q 𝑡) = ((𝑠 +Q 𝑡) +Q 𝑞))
55 simpr 103 . . . . . . . . . . . . . 14 (((((𝜑𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st𝐿) ∧ 𝑡 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ (𝑠 +Q 𝑞) <Q (𝐹𝑞)) → (𝑠 +Q 𝑞) <Q (𝐹𝑞))
5635ad2antrr 457 . . . . . . . . . . . . . 14 (((((𝜑𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st𝐿) ∧ 𝑡 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ (𝑠 +Q 𝑞) <Q (𝐹𝑞)) → 𝑡 <Q 𝑆)
5737brel 4392 . . . . . . . . . . . . . . 15 ((𝑠 +Q 𝑞) <Q (𝐹𝑞) → ((𝑠 +Q 𝑞) ∈ Q ∧ (𝐹𝑞) ∈ Q))
58 lt2addnq 6502 . . . . . . . . . . . . . . 15 ((((𝑠 +Q 𝑞) ∈ Q ∧ (𝐹𝑞) ∈ Q) ∧ (𝑡Q𝑆Q)) → (((𝑠 +Q 𝑞) <Q (𝐹𝑞) ∧ 𝑡 <Q 𝑆) → ((𝑠 +Q 𝑞) +Q 𝑡) <Q ((𝐹𝑞) +Q 𝑆)))
5957, 39, 58syl2anr 274 . . . . . . . . . . . . . 14 (((((𝜑𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st𝐿) ∧ 𝑡 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ (𝑠 +Q 𝑞) <Q (𝐹𝑞)) → (((𝑠 +Q 𝑞) <Q (𝐹𝑞) ∧ 𝑡 <Q 𝑆) → ((𝑠 +Q 𝑞) +Q 𝑡) <Q ((𝐹𝑞) +Q 𝑆)))
6055, 56, 59mp2and 409 . . . . . . . . . . . . 13 (((((𝜑𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st𝐿) ∧ 𝑡 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ (𝑠 +Q 𝑞) <Q (𝐹𝑞)) → ((𝑠 +Q 𝑞) +Q 𝑡) <Q ((𝐹𝑞) +Q 𝑆))
6160adantlr 446 . . . . . . . . . . . 12 ((((((𝜑𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st𝐿) ∧ 𝑡 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑞Q) ∧ (𝑠 +Q 𝑞) <Q (𝐹𝑞)) → ((𝑠 +Q 𝑞) +Q 𝑡) <Q ((𝐹𝑞) +Q 𝑆))
6254, 61eqbrtrrd 3786 . . . . . . . . . . 11 ((((((𝜑𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st𝐿) ∧ 𝑡 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑞Q) ∧ (𝑠 +Q 𝑞) <Q (𝐹𝑞)) → ((𝑠 +Q 𝑡) +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆))
63 oveq1 5519 . . . . . . . . . . . . 13 (𝑟 = (𝑠 +Q 𝑡) → (𝑟 +Q 𝑞) = ((𝑠 +Q 𝑡) +Q 𝑞))
6463breq1d 3774 . . . . . . . . . . . 12 (𝑟 = (𝑠 +Q 𝑡) → ((𝑟 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆) ↔ ((𝑠 +Q 𝑡) +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆)))
6564ad3antlr 462 . . . . . . . . . . 11 ((((((𝜑𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st𝐿) ∧ 𝑡 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑞Q) ∧ (𝑠 +Q 𝑞) <Q (𝐹𝑞)) → ((𝑟 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆) ↔ ((𝑠 +Q 𝑡) +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆)))
6662, 65mpbird 156 . . . . . . . . . 10 ((((((𝜑𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st𝐿) ∧ 𝑡 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑞Q) ∧ (𝑠 +Q 𝑞) <Q (𝐹𝑞)) → (𝑟 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆))
6766ex 108 . . . . . . . . 9 (((((𝜑𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st𝐿) ∧ 𝑡 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑞Q) → ((𝑠 +Q 𝑞) <Q (𝐹𝑞) → (𝑟 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆)))
6867reximdva 2421 . . . . . . . 8 ((((𝜑𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st𝐿) ∧ 𝑡 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → (∃𝑞Q (𝑠 +Q 𝑞) <Q (𝐹𝑞) → ∃𝑞Q (𝑟 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆)))
6946, 68mpd 13 . . . . . . 7 ((((𝜑𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st𝐿) ∧ 𝑡 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → ∃𝑞Q (𝑟 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆))
70 oveq1 5519 . . . . . . . . . 10 (𝑙 = 𝑟 → (𝑙 +Q 𝑞) = (𝑟 +Q 𝑞))
7170breq1d 3774 . . . . . . . . 9 (𝑙 = 𝑟 → ((𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆) ↔ (𝑟 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆)))
7271rexbidv 2327 . . . . . . . 8 (𝑙 = 𝑟 → (∃𝑞Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆) ↔ ∃𝑞Q (𝑟 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆)))
7318rabex 3901 . . . . . . . . 9 {𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆)} ∈ V
7418rabex 3901 . . . . . . . . 9 {𝑢Q ∣ ∃𝑞Q (((𝐹𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑢} ∈ V
7573, 74op1st 5773 . . . . . . . 8 (1st ‘⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆)}, {𝑢Q ∣ ∃𝑞Q (((𝐹𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑢}⟩) = {𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆)}
7672, 75elrab2 2700 . . . . . . 7 (𝑟 ∈ (1st ‘⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆)}, {𝑢Q ∣ ∃𝑞Q (((𝐹𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑢}⟩) ↔ (𝑟Q ∧ ∃𝑞Q (𝑟 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆)))
7745, 69, 76sylanbrc 394 . . . . . 6 ((((𝜑𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st𝐿) ∧ 𝑡 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → 𝑟 ∈ (1st ‘⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆)}, {𝑢Q ∣ ∃𝑞Q (((𝐹𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑢}⟩))
7877ex 108 . . . . 5 (((𝜑𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st𝐿) ∧ 𝑡 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) → (𝑟 = (𝑠 +Q 𝑡) → 𝑟 ∈ (1st ‘⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆)}, {𝑢Q ∣ ∃𝑞Q (((𝐹𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑢}⟩)))
7978rexlimdvva 2440 . . . 4 ((𝜑𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) → (∃𝑠 ∈ (1st𝐿)∃𝑡 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩)𝑟 = (𝑠 +Q 𝑡) → 𝑟 ∈ (1st ‘⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆)}, {𝑢Q ∣ ∃𝑞Q (((𝐹𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑢}⟩)))
8013, 79mpd 13 . . 3 ((𝜑𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) → 𝑟 ∈ (1st ‘⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆)}, {𝑢Q ∣ ∃𝑞Q (((𝐹𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑢}⟩))
8180ex 108 . 2 (𝜑 → (𝑟 ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩)) → 𝑟 ∈ (1st ‘⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆)}, {𝑢Q ∣ ∃𝑞Q (((𝐹𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑢}⟩)))
8281ssrdv 2951 1 (𝜑 → (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩)) ⊆ (1st ‘⟨{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 97   ↔ wb 98   ∧ w3a 885   = wceq 1243   ∈ wcel 1393  {cab 2026  ∀wral 2306  ∃wrex 2307  {crab 2310   ⊆ wss 2917  ⟨cop 3378   class class class wbr 3764  ⟶wf 4898  ‘cfv 4902  (class class class)co 5512  1st c1st 5765  Qcnq 6378   +Q cplq 6380
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