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Theorem cauappcvgprlemladdfl 7475
Description: Lemma for cauappcvgprlemladd 7478. 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 7473 . . . . . 6 (𝜑𝐿P)
6 cauappcvgprlemladd.s . . . . . . 7 (𝜑𝑆Q)
7 nqprlu 7367 . . . . . . 7 (𝑆Q → ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩ ∈ P)
86, 7syl 14 . . . . . 6 (𝜑 → ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩ ∈ P)
9 df-iplp 7288 . . . . . . 7 +P = (𝑥P, 𝑦P ↦ ⟨{𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (1st𝑥) ∧ ∈ (1st𝑦) ∧ 𝑓 = (𝑔 +Q ))}, {𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (2nd𝑥) ∧ ∈ (2nd𝑦) ∧ 𝑓 = (𝑔 +Q ))}⟩)
10 addclnq 7195 . . . . . . 7 ((𝑔QQ) → (𝑔 +Q ) ∈ Q)
119, 10genpelvl 7332 . . . . . 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 408 . . . . 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 294 . . . 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 5781 . . . . . . . . . . . . . . . 16 (𝑙 = 𝑠 → (𝑙 +Q 𝑞) = (𝑠 +Q 𝑞))
1514breq1d 3939 . . . . . . . . . . . . . . 15 (𝑙 = 𝑠 → ((𝑙 +Q 𝑞) <Q (𝐹𝑞) ↔ (𝑠 +Q 𝑞) <Q (𝐹𝑞)))
1615rexbidv 2438 . . . . . . . . . . . . . 14 (𝑙 = 𝑠 → (∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞) ↔ ∃𝑞Q (𝑠 +Q 𝑞) <Q (𝐹𝑞)))
174fveq2i 5424 . . . . . . . . . . . . . . 15 (1st𝐿) = (1st ‘⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}⟩)
18 nqex 7183 . . . . . . . . . . . . . . . . 17 Q ∈ V
1918rabex 4072 . . . . . . . . . . . . . . . 16 {𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)} ∈ V
2018rabex 4072 . . . . . . . . . . . . . . . 16 {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢} ∈ V
2119, 20op1st 6044 . . . . . . . . . . . . . . 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 2160 . . . . . . . . . . . . . 14 (1st𝐿) = {𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}
2316, 22elrab2 2843 . . . . . . . . . . . . 13 (𝑠 ∈ (1st𝐿) ↔ (𝑠Q ∧ ∃𝑞Q (𝑠 +Q 𝑞) <Q (𝐹𝑞)))
2423biimpi 119 . . . . . . . . . . . 12 (𝑠 ∈ (1st𝐿) → (𝑠Q ∧ ∃𝑞Q (𝑠 +Q 𝑞) <Q (𝐹𝑞)))
2524ad2antrl 481 . . . . . . . . . . 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 274 . . . . . . . . . 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 111 . . . . . . . . 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 2689 . . . . . . . . . . . . . . 15 𝑡 ∈ V
29 breq1 3932 . . . . . . . . . . . . . . 15 (𝑙 = 𝑡 → (𝑙 <Q 𝑆𝑡 <Q 𝑆))
30 ltnqex 7369 . . . . . . . . . . . . . . . 16 {𝑙𝑙 <Q 𝑆} ∈ V
31 gtnqex 7370 . . . . . . . . . . . . . . . 16 {𝑢𝑆 <Q 𝑢} ∈ V
3230, 31op1st 6044 . . . . . . . . . . . . . . 15 (1st ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩) = {𝑙𝑙 <Q 𝑆}
3328, 29, 32elab2 2832 . . . . . . . . . . . . . 14 (𝑡 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩) ↔ 𝑡 <Q 𝑆)
3433biimpi 119 . . . . . . . . . . . . 13 (𝑡 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩) → 𝑡 <Q 𝑆)
3534ad2antll 482 . . . . . . . . . . . 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 274 . . . . . . . . . . 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 7185 . . . . . . . . . . . 12 <Q ⊆ (Q × Q)
3837brel 4591 . . . . . . . . . . 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 111 . . . . . . . . 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 7195 . . . . . . . . 9 ((𝑠Q𝑡Q) → (𝑠 +Q 𝑡) ∈ Q)
4227, 40, 41syl2anc 408 . . . . . . . 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 2202 . . . . . . . . 9 (𝑟 = (𝑠 +Q 𝑡) → (𝑟Q ↔ (𝑠 +Q 𝑡) ∈ Q))
4443adantl 275 . . . . . . . 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 166 . . . . . . 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 113 . . . . . . . 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 479 . . . . . . . . . . . . 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 519 . . . . . . . . . . . . 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 479 . . . . . . . . . . . . 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 7201 . . . . . . . . . . . . . 14 ((𝑓Q𝑔Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
5150adantl 275 . . . . . . . . . . . . 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 7202 . . . . . . . . . . . . . 14 ((𝑓Q𝑔QQ) → ((𝑓 +Q 𝑔) +Q ) = (𝑓 +Q (𝑔 +Q )))
5352adantl 275 . . . . . . . . . . . . 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 5951 . . . . . . . . . . . 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 109 . . . . . . . . . . . . . 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 479 . . . . . . . . . . . . . 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 4591 . . . . . . . . . . . . . . 15 ((𝑠 +Q 𝑞) <Q (𝐹𝑞) → ((𝑠 +Q 𝑞) ∈ Q ∧ (𝐹𝑞) ∈ Q))
58 lt2addnq 7224 . . . . . . . . . . . . . . 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 288 . . . . . . . . . . . . . 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 429 . . . . . . . . . . . . 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 468 . . . . . . . . . . . 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 3952 . . . . . . . . . . 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 5781 . . . . . . . . . . . . 13 (𝑟 = (𝑠 +Q 𝑡) → (𝑟 +Q 𝑞) = ((𝑠 +Q 𝑡) +Q 𝑞))
6463breq1d 3939 . . . . . . . . . . . 12 (𝑟 = (𝑠 +Q 𝑡) → ((𝑟 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆) ↔ ((𝑠 +Q 𝑡) +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆)))
6564ad3antlr 484 . . . . . . . . . . 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 166 . . . . . . . . . 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 114 . . . . . . . . 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 2534 . . . . . . . 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 5781 . . . . . . . . . 10 (𝑙 = 𝑟 → (𝑙 +Q 𝑞) = (𝑟 +Q 𝑞))
7170breq1d 3939 . . . . . . . . 9 (𝑙 = 𝑟 → ((𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆) ↔ (𝑟 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆)))
7271rexbidv 2438 . . . . . . . 8 (𝑙 = 𝑟 → (∃𝑞Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆) ↔ ∃𝑞Q (𝑟 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆)))
7318rabex 4072 . . . . . . . . 9 {𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆)} ∈ V
7418rabex 4072 . . . . . . . . 9 {𝑢Q ∣ ∃𝑞Q (((𝐹𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑢} ∈ V
7573, 74op1st 6044 . . . . . . . 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 2843 . . . . . . 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 413 . . . . . 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 114 . . . . 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 2557 . . . 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 114 . 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 3103 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 103  wb 104  w3a 962   = wceq 1331  wcel 1480  {cab 2125  wral 2416  wrex 2417  {crab 2420  wss 3071  cop 3530   class class class wbr 3929  wf 5119  cfv 5123  (class class class)co 5774  1st c1st 6036  Qcnq 7100   +Q cplq 7102   <Q cltq 7105  Pcnp 7111   +P cpp 7113
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7124  df-pli 7125  df-mi 7126  df-lti 7127  df-plpq 7164  df-mpq 7165  df-enq 7167  df-nqqs 7168  df-plqqs 7169  df-mqqs 7170  df-1nqqs 7171  df-rq 7172  df-ltnqqs 7173  df-inp 7286  df-iplp 7288
This theorem is referenced by:  cauappcvgprlemladdru  7476  cauappcvgprlemladd  7478
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