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Theorem cauappcvgprlemladdfl 6626
Description: Lemma for cauappcvgprlemladd 6629. The forward subset relationship for the lower cut. (Contributed by Jim Kingdon, 11-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}⟩
cauappcvgprlemladd.s (φ𝑆 Q)
Assertion
Ref Expression
cauappcvgprlemladdfl (φ → (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {u𝑆 <Q u}⟩)) ⊆ (1st ‘⟨{𝑙 Q𝑞 Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆)}, {u Q𝑞 Q (((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 (φ𝑝 Q 𝑞 Q ((𝐹𝑝) <Q ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) (𝐹𝑞) <Q ((𝐹𝑝) +Q (𝑝 +Q 𝑞))))
3 cauappcvgpr.bnd . . . . . . 7 (φ𝑝 Q A <Q (𝐹𝑝))
4 cauappcvgpr.lim . . . . . . 7 𝐿 = ⟨{𝑙 Q𝑞 Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {u Q𝑞 Q ((𝐹𝑞) +Q 𝑞) <Q u}⟩
51, 2, 3, 4cauappcvgprlemcl 6624 . . . . . 6 (φ𝐿 P)
6 cauappcvgprlemladd.s . . . . . . 7 (φ𝑆 Q)
7 nqprlu 6529 . . . . . . 7 (𝑆 Q → ⟨{𝑙𝑙 <Q 𝑆}, {u𝑆 <Q u}⟩ P)
86, 7syl 14 . . . . . 6 (φ → ⟨{𝑙𝑙 <Q 𝑆}, {u𝑆 <Q u}⟩ P)
9 df-iplp 6450 . . . . . . 7 +P = (x P, y P ↦ ⟨{f Qg Q Q (g (1stx) (1sty) f = (g +Q ))}, {f Qg Q Q (g (2ndx) (2ndy) f = (g +Q ))}⟩)
10 addclnq 6359 . . . . . . 7 ((g Q Q) → (g +Q ) Q)
119, 10genpelvl 6494 . . . . . 6 ((𝐿 P ⟨{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 ((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))
14 oveq1 5462 . . . . . . . . . . . . . . . 16 (𝑙 = 𝑠 → (𝑙 +Q 𝑞) = (𝑠 +Q 𝑞))
1514breq1d 3765 . . . . . . . . . . . . . . 15 (𝑙 = 𝑠 → ((𝑙 +Q 𝑞) <Q (𝐹𝑞) ↔ (𝑠 +Q 𝑞) <Q (𝐹𝑞)))
1615rexbidv 2321 . . . . . . . . . . . . . 14 (𝑙 = 𝑠 → (𝑞 Q (𝑙 +Q 𝑞) <Q (𝐹𝑞) ↔ 𝑞 Q (𝑠 +Q 𝑞) <Q (𝐹𝑞)))
174fveq2i 5124 . . . . . . . . . . . . . . 15 (1st𝐿) = (1st ‘⟨{𝑙 Q𝑞 Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {u Q𝑞 Q ((𝐹𝑞) +Q 𝑞) <Q u}⟩)
18 nqex 6347 . . . . . . . . . . . . . . . . 17 Q V
1918rabex 3892 . . . . . . . . . . . . . . . 16 {𝑙 Q𝑞 Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)} V
2018rabex 3892 . . . . . . . . . . . . . . . 16 {u Q𝑞 Q ((𝐹𝑞) +Q 𝑞) <Q u} V
2119, 20op1st 5715 . . . . . . . . . . . . . . 15 (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)}
2217, 21eqtri 2057 . . . . . . . . . . . . . 14 (1st𝐿) = {𝑙 Q𝑞 Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}
2316, 22elrab2 2694 . . . . . . . . . . . . 13 (𝑠 (1st𝐿) ↔ (𝑠 Q 𝑞 Q (𝑠 +Q 𝑞) <Q (𝐹𝑞)))
2423biimpi 113 . . . . . . . . . . . 12 (𝑠 (1st𝐿) → (𝑠 Q 𝑞 Q (𝑠 +Q 𝑞) <Q (𝐹𝑞)))
2524ad2antrl 459 . . . . . . . . . . 11 (((φ 𝑟 (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {u𝑆 <Q u}⟩))) (𝑠 (1st𝐿) 𝑡 (1st ‘⟨{𝑙𝑙 <Q 𝑆}, {u𝑆 <Q u}⟩))) → (𝑠 Q 𝑞 Q (𝑠 +Q 𝑞) <Q (𝐹𝑞)))
2625adantr 261 . . . . . . . . . 10 ((((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)) → (𝑠 Q 𝑞 Q (𝑠 +Q 𝑞) <Q (𝐹𝑞)))
2726simpld 105 . . . . . . . . 9 ((((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)) → 𝑠 Q)
28 vex 2554 . . . . . . . . . . . . . . 15 𝑡 V
29 breq1 3758 . . . . . . . . . . . . . . 15 (𝑙 = 𝑡 → (𝑙 <Q 𝑆𝑡 <Q 𝑆))
30 ltnqex 6530 . . . . . . . . . . . . . . . 16 {𝑙𝑙 <Q 𝑆} V
31 gtnqex 6531 . . . . . . . . . . . . . . . 16 {u𝑆 <Q u} V
3230, 31op1st 5715 . . . . . . . . . . . . . . 15 (1st ‘⟨{𝑙𝑙 <Q 𝑆}, {u𝑆 <Q u}⟩) = {𝑙𝑙 <Q 𝑆}
3328, 29, 32elab2 2684 . . . . . . . . . . . . . 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 (((φ 𝑟 (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 ((((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)) → 𝑡 <Q 𝑆)
37 ltrelnq 6349 . . . . . . . . . . . 12 <Q ⊆ (Q × Q)
3837brel 4335 . . . . . . . . . . 11 (𝑡 <Q 𝑆 → (𝑡 Q 𝑆 Q))
3936, 38syl 14 . . . . . . . . . 10 ((((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)) → (𝑡 Q 𝑆 Q))
4039simpld 105 . . . . . . . . 9 ((((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)) → 𝑡 Q)
41 addclnq 6359 . . . . . . . . 9 ((𝑠 Q 𝑡 Q) → (𝑠 +Q 𝑡) Q)
4227, 40, 41syl2anc 391 . . . . . . . 8 ((((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)) → (𝑠 +Q 𝑡) Q)
43 eleq1 2097 . . . . . . . . 9 (𝑟 = (𝑠 +Q 𝑡) → (𝑟 Q ↔ (𝑠 +Q 𝑡) Q))
4443adantl 262 . . . . . . . 8 ((((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)) → (𝑟 Q ↔ (𝑠 +Q 𝑡) Q))
4542, 44mpbird 156 . . . . . . 7 ((((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)) → 𝑟 Q)
4626simprd 107 . . . . . . . 8 ((((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)) → 𝑞 Q (𝑠 +Q 𝑞) <Q (𝐹𝑞))
4727ad2antrr 457 . . . . . . . . . . . . 13 ((((((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)) 𝑞 Q) (𝑠 +Q 𝑞) <Q (𝐹𝑞)) → 𝑠 Q)
48 simplr 482 . . . . . . . . . . . . 13 ((((((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)) 𝑞 Q) (𝑠 +Q 𝑞) <Q (𝐹𝑞)) → 𝑞 Q)
4940ad2antrr 457 . . . . . . . . . . . . 13 ((((((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)) 𝑞 Q) (𝑠 +Q 𝑞) <Q (𝐹𝑞)) → 𝑡 Q)
50 addcomnqg 6365 . . . . . . . . . . . . . 14 ((f Q g Q) → (f +Q g) = (g +Q f))
5150adantl 262 . . . . . . . . . . . . 13 (((((((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)) 𝑞 Q) (𝑠 +Q 𝑞) <Q (𝐹𝑞)) (f Q g Q)) → (f +Q g) = (g +Q f))
52 addassnqg 6366 . . . . . . . . . . . . . 14 ((f Q g Q Q) → ((f +Q g) +Q ) = (f +Q (g +Q )))
5352adantl 262 . . . . . . . . . . . . 13 (((((((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)) 𝑞 Q) (𝑠 +Q 𝑞) <Q (𝐹𝑞)) (f Q g Q Q)) → ((f +Q g) +Q ) = (f +Q (g +Q )))
5447, 48, 49, 51, 53caov32d 5623 . . . . . . . . . . . 12 ((((((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)) 𝑞 Q) (𝑠 +Q 𝑞) <Q (𝐹𝑞)) → ((𝑠 +Q 𝑞) +Q 𝑡) = ((𝑠 +Q 𝑡) +Q 𝑞))
55 simpr 103 . . . . . . . . . . . . . 14 (((((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)) (𝑠 +Q 𝑞) <Q (𝐹𝑞)) → (𝑠 +Q 𝑞) <Q (𝐹𝑞))
5635ad2antrr 457 . . . . . . . . . . . . . 14 (((((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)) (𝑠 +Q 𝑞) <Q (𝐹𝑞)) → 𝑡 <Q 𝑆)
5737brel 4335 . . . . . . . . . . . . . . 15 ((𝑠 +Q 𝑞) <Q (𝐹𝑞) → ((𝑠 +Q 𝑞) Q (𝐹𝑞) Q))
58 lt2addnq 6388 . . . . . . . . . . . . . . 15 ((((𝑠 +Q 𝑞) Q (𝐹𝑞) Q) (𝑡 Q 𝑆 Q)) → (((𝑠 +Q 𝑞) <Q (𝐹𝑞) 𝑡 <Q 𝑆) → ((𝑠 +Q 𝑞) +Q 𝑡) <Q ((𝐹𝑞) +Q 𝑆)))
5957, 39, 58syl2anr 274 . . . . . . . . . . . . . 14 (((((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)) (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 (((((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)) (𝑠 +Q 𝑞) <Q (𝐹𝑞)) → ((𝑠 +Q 𝑞) +Q 𝑡) <Q ((𝐹𝑞) +Q 𝑆))
6160adantlr 446 . . . . . . . . . . . 12 ((((((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)) 𝑞 Q) (𝑠 +Q 𝑞) <Q (𝐹𝑞)) → ((𝑠 +Q 𝑞) +Q 𝑡) <Q ((𝐹𝑞) +Q 𝑆))
6254, 61eqbrtrrd 3777 . . . . . . . . . . 11 ((((((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)) 𝑞 Q) (𝑠 +Q 𝑞) <Q (𝐹𝑞)) → ((𝑠 +Q 𝑡) +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆))
63 oveq1 5462 . . . . . . . . . . . . 13 (𝑟 = (𝑠 +Q 𝑡) → (𝑟 +Q 𝑞) = ((𝑠 +Q 𝑡) +Q 𝑞))
6463breq1d 3765 . . . . . . . . . . . 12 (𝑟 = (𝑠 +Q 𝑡) → ((𝑟 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆) ↔ ((𝑠 +Q 𝑡) +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆)))
6564ad3antlr 462 . . . . . . . . . . 11 ((((((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)) 𝑞 Q) (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 ((((((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)) 𝑞 Q) (𝑠 +Q 𝑞) <Q (𝐹𝑞)) → (𝑟 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆))
6766ex 108 . . . . . . . . 9 (((((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)) 𝑞 Q) → ((𝑠 +Q 𝑞) <Q (𝐹𝑞) → (𝑟 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆)))
6867reximdva 2415 . . . . . . . 8 ((((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)) → (𝑞 Q (𝑠 +Q 𝑞) <Q (𝐹𝑞) → 𝑞 Q (𝑟 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆)))
6946, 68mpd 13 . . . . . . 7 ((((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)) → 𝑞 Q (𝑟 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆))
70 oveq1 5462 . . . . . . . . . 10 (𝑙 = 𝑟 → (𝑙 +Q 𝑞) = (𝑟 +Q 𝑞))
7170breq1d 3765 . . . . . . . . 9 (𝑙 = 𝑟 → ((𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆) ↔ (𝑟 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆)))
7271rexbidv 2321 . . . . . . . 8 (𝑙 = 𝑟 → (𝑞 Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆) ↔ 𝑞 Q (𝑟 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆)))
7318rabex 3892 . . . . . . . . 9 {𝑙 Q𝑞 Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆)} V
7418rabex 3892 . . . . . . . . 9 {u Q𝑞 Q (((𝐹𝑞) +Q 𝑞) +Q 𝑆) <Q u} V
7573, 74op1st 5715 . . . . . . . 8 (1st ‘⟨{𝑙 Q𝑞 Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆)}, {u Q𝑞 Q (((𝐹𝑞) +Q 𝑞) +Q 𝑆) <Q u}⟩) = {𝑙 Q𝑞 Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆)}
7672, 75elrab2 2694 . . . . . . 7 (𝑟 (1st ‘⟨{𝑙 Q𝑞 Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆)}, {u Q𝑞 Q (((𝐹𝑞) +Q 𝑞) +Q 𝑆) <Q u}⟩) ↔ (𝑟 Q 𝑞 Q (𝑟 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆)))
7745, 69, 76sylanbrc 394 . . . . . 6 ((((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)) → 𝑟 (1st ‘⟨{𝑙 Q𝑞 Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆)}, {u Q𝑞 Q (((𝐹𝑞) +Q 𝑞) +Q 𝑆) <Q u}⟩))
7877ex 108 . . . . 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) → 𝑟 (1st ‘⟨{𝑙 Q𝑞 Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆)}, {u Q𝑞 Q (((𝐹𝑞) +Q 𝑞) +Q 𝑆) <Q u}⟩)))
7978rexlimdvva 2434 . . . 4 ((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) → 𝑟 (1st ‘⟨{𝑙 Q𝑞 Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆)}, {u Q𝑞 Q (((𝐹𝑞) +Q 𝑞) +Q 𝑆) <Q u}⟩)))
8013, 79mpd 13 . . 3 ((φ 𝑟 (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {u𝑆 <Q u}⟩))) → 𝑟 (1st ‘⟨{𝑙 Q𝑞 Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆)}, {u Q𝑞 Q (((𝐹𝑞) +Q 𝑞) +Q 𝑆) <Q u}⟩))
8180ex 108 . 2 (φ → (𝑟 (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {u𝑆 <Q u}⟩)) → 𝑟 (1st ‘⟨{𝑙 Q𝑞 Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆)}, {u Q𝑞 Q (((𝐹𝑞) +Q 𝑞) +Q 𝑆) <Q u}⟩)))
8281ssrdv 2945 1 (φ → (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {u𝑆 <Q u}⟩)) ⊆ (1st ‘⟨{𝑙 Q𝑞 Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆)}, {u Q𝑞 Q (((𝐹𝑞) +Q 𝑞) +Q 𝑆) <Q u}⟩))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   w3a 884   = wceq 1242   wcel 1390  {cab 2023  wral 2300  wrex 2301  {crab 2304  wss 2911  cop 3370   class class class wbr 3755  wf 4841  cfv 4845  (class class class)co 5455  1st c1st 5707  Qcnq 6264   +Q cplq 6266   <Q cltq 6269  Pcnp 6275   +P cpp 6277
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254
This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-eprel 4017  df-id 4021  df-po 4024  df-iso 4025  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-1o 5940  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-pli 6289  df-mi 6290  df-lti 6291  df-plpq 6328  df-mpq 6329  df-enq 6331  df-nqqs 6332  df-plqqs 6333  df-mqqs 6334  df-1nqqs 6335  df-rq 6336  df-ltnqqs 6337  df-inp 6448  df-iplp 6450
This theorem is referenced by:  cauappcvgprlemladdru  6627  cauappcvgprlemladd  6629
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