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Theorem cauappcvgprlemladdfl 6810
Description: Lemma for cauappcvgprlemladd 6813. 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}⟩
cauappcvgprlemladd.s (𝜑𝑆Q)
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)

Proof of Theorem cauappcvgprlemladdfl
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 6808 . . . . . 6 (𝜑𝐿P)
6 cauappcvgprlemladd.s . . . . . . 7 (𝜑𝑆Q)
7 nqprlu 6702 . . . . . . 7 (𝑆Q → ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩ ∈ P)
86, 7syl 14 . . . . . 6 (𝜑 → ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩ ∈ P)
9 df-iplp 6623 . . . . . . 7 +P = (𝑥P, 𝑦P ↦ ⟨{𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (1st𝑥) ∧ ∈ (1st𝑦) ∧ 𝑓 = (𝑔 +Q ))}, {𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (2nd𝑥) ∧ ∈ (2nd𝑦) ∧ 𝑓 = (𝑔 +Q ))}⟩)
10 addclnq 6530 . . . . . . 7 ((𝑔QQ) → (𝑔 +Q ) ∈ Q)
119, 10genpelvl 6667 . . . . . 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 397 . . . . 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 284 . . . 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 5546 . . . . . . . . . . . . . . . 16 (𝑙 = 𝑠 → (𝑙 +Q 𝑞) = (𝑠 +Q 𝑞))
1514breq1d 3801 . . . . . . . . . . . . . . 15 (𝑙 = 𝑠 → ((𝑙 +Q 𝑞) <Q (𝐹𝑞) ↔ (𝑠 +Q 𝑞) <Q (𝐹𝑞)))
1615rexbidv 2344 . . . . . . . . . . . . . 14 (𝑙 = 𝑠 → (∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞) ↔ ∃𝑞Q (𝑠 +Q 𝑞) <Q (𝐹𝑞)))
174fveq2i 5208 . . . . . . . . . . . . . . 15 (1st𝐿) = (1st ‘⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}⟩)
18 nqex 6518 . . . . . . . . . . . . . . . . 17 Q ∈ V
1918rabex 3928 . . . . . . . . . . . . . . . 16 {𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)} ∈ V
2018rabex 3928 . . . . . . . . . . . . . . . 16 {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢} ∈ V
2119, 20op1st 5800 . . . . . . . . . . . . . . 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 2076 . . . . . . . . . . . . . 14 (1st𝐿) = {𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}
2316, 22elrab2 2722 . . . . . . . . . . . . 13 (𝑠 ∈ (1st𝐿) ↔ (𝑠Q ∧ ∃𝑞Q (𝑠 +Q 𝑞) <Q (𝐹𝑞)))
2423biimpi 117 . . . . . . . . . . . 12 (𝑠 ∈ (1st𝐿) → (𝑠Q ∧ ∃𝑞Q (𝑠 +Q 𝑞) <Q (𝐹𝑞)))
2524ad2antrl 467 . . . . . . . . . . 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 265 . . . . . . . . . 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 109 . . . . . . . . 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 2577 . . . . . . . . . . . . . . 15 𝑡 ∈ V
29 breq1 3794 . . . . . . . . . . . . . . 15 (𝑙 = 𝑡 → (𝑙 <Q 𝑆𝑡 <Q 𝑆))
30 ltnqex 6704 . . . . . . . . . . . . . . . 16 {𝑙𝑙 <Q 𝑆} ∈ V
31 gtnqex 6705 . . . . . . . . . . . . . . . 16 {𝑢𝑆 <Q 𝑢} ∈ V
3230, 31op1st 5800 . . . . . . . . . . . . . . 15 (1st ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩) = {𝑙𝑙 <Q 𝑆}
3328, 29, 32elab2 2712 . . . . . . . . . . . . . 14 (𝑡 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩) ↔ 𝑡 <Q 𝑆)
3433biimpi 117 . . . . . . . . . . . . 13 (𝑡 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩) → 𝑡 <Q 𝑆)
3534ad2antll 468 . . . . . . . . . . . 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 265 . . . . . . . . . . 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 6520 . . . . . . . . . . . 12 <Q ⊆ (Q × Q)
3837brel 4419 . . . . . . . . . . 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 109 . . . . . . . . 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 6530 . . . . . . . . 9 ((𝑠Q𝑡Q) → (𝑠 +Q 𝑡) ∈ Q)
4227, 40, 41syl2anc 397 . . . . . . . 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 2116 . . . . . . . . 9 (𝑟 = (𝑠 +Q 𝑡) → (𝑟Q ↔ (𝑠 +Q 𝑡) ∈ Q))
4443adantl 266 . . . . . . . 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 160 . . . . . . 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 111 . . . . . . . 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 465 . . . . . . . . . . . . 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 490 . . . . . . . . . . . . 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 465 . . . . . . . . . . . . 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 6536 . . . . . . . . . . . . . 14 ((𝑓Q𝑔Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
5150adantl 266 . . . . . . . . . . . . 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 6537 . . . . . . . . . . . . . 14 ((𝑓Q𝑔QQ) → ((𝑓 +Q 𝑔) +Q ) = (𝑓 +Q (𝑔 +Q )))
5352adantl 266 . . . . . . . . . . . . 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 5708 . . . . . . . . . . . 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 107 . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . . 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 4419 . . . . . . . . . . . . . . 15 ((𝑠 +Q 𝑞) <Q (𝐹𝑞) → ((𝑠 +Q 𝑞) ∈ Q ∧ (𝐹𝑞) ∈ Q))
58 lt2addnq 6559 . . . . . . . . . . . . . . 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 278 . . . . . . . . . . . . . 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 417 . . . . . . . . . . . . 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 454 . . . . . . . . . . . 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 3813 . . . . . . . . . . 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 5546 . . . . . . . . . . . . 13 (𝑟 = (𝑠 +Q 𝑡) → (𝑟 +Q 𝑞) = ((𝑠 +Q 𝑡) +Q 𝑞))
6463breq1d 3801 . . . . . . . . . . . 12 (𝑟 = (𝑠 +Q 𝑡) → ((𝑟 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆) ↔ ((𝑠 +Q 𝑡) +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆)))
6564ad3antlr 470 . . . . . . . . . . 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 160 . . . . . . . . . 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 112 . . . . . . . . 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 2438 . . . . . . . 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 5546 . . . . . . . . . 10 (𝑙 = 𝑟 → (𝑙 +Q 𝑞) = (𝑟 +Q 𝑞))
7170breq1d 3801 . . . . . . . . 9 (𝑙 = 𝑟 → ((𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆) ↔ (𝑟 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆)))
7271rexbidv 2344 . . . . . . . 8 (𝑙 = 𝑟 → (∃𝑞Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆) ↔ ∃𝑞Q (𝑟 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆)))
7318rabex 3928 . . . . . . . . 9 {𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆)} ∈ V
7418rabex 3928 . . . . . . . . 9 {𝑢Q ∣ ∃𝑞Q (((𝐹𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑢} ∈ V
7573, 74op1st 5800 . . . . . . . 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 2722 . . . . . . 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 402 . . . . . 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 112 . . . . 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 2457 . . . 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 112 . 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 2978 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 101  wb 102  w3a 896   = wceq 1259  wcel 1409  {cab 2042  wral 2323  wrex 2324  {crab 2327  wss 2944  cop 3405   class class class wbr 3791  wf 4925  cfv 4929  (class class class)co 5539  1st c1st 5792  Qcnq 6435   +Q cplq 6437   <Q cltq 6440  Pcnp 6446   +P cpp 6448
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3899  ax-sep 3902  ax-nul 3910  ax-pow 3954  ax-pr 3971  ax-un 4197  ax-setind 4289  ax-iinf 4338
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2787  df-csb 2880  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-int 3643  df-iun 3686  df-br 3792  df-opab 3846  df-mpt 3847  df-tr 3882  df-eprel 4053  df-id 4057  df-po 4060  df-iso 4061  df-iord 4130  df-on 4132  df-suc 4135  df-iom 4341  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-f1 4934  df-fo 4935  df-f1o 4936  df-fv 4937  df-ov 5542  df-oprab 5543  df-mpt2 5544  df-1st 5794  df-2nd 5795  df-recs 5950  df-irdg 5987  df-1o 6031  df-oadd 6035  df-omul 6036  df-er 6136  df-ec 6138  df-qs 6142  df-ni 6459  df-pli 6460  df-mi 6461  df-lti 6462  df-plpq 6499  df-mpq 6500  df-enq 6502  df-nqqs 6503  df-plqqs 6504  df-mqqs 6505  df-1nqqs 6506  df-rq 6507  df-ltnqqs 6508  df-inp 6621  df-iplp 6623
This theorem is referenced by:  cauappcvgprlemladdru  6811  cauappcvgprlemladd  6813
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