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Theorem cauappcvgprlemladdfu 6625
Description: Lemma for cauappcvgprlemladd 6629. The forward subset relationship for the upper 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
cauappcvgprlemladdfu (φ → (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {u𝑆 <Q u}⟩)) ⊆ (2nd ‘⟨{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 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 (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, 10genpelvu 6495 . . . . . 6 ((𝐿 P ⟨{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 391 . . . . 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 280 . . . 4 ((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))
14 breq2 3759 . . . . . . . . . . . . . . . 16 (u = 𝑠 → (((𝐹𝑞) +Q 𝑞) <Q u ↔ ((𝐹𝑞) +Q 𝑞) <Q 𝑠))
1514rexbidv 2321 . . . . . . . . . . . . . . 15 (u = 𝑠 → (𝑞 Q ((𝐹𝑞) +Q 𝑞) <Q u𝑞 Q ((𝐹𝑞) +Q 𝑞) <Q 𝑠))
164fveq2i 5124 . . . . . . . . . . . . . . . 16 (2nd𝐿) = (2nd ‘⟨{𝑙 Q𝑞 Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {u Q𝑞 Q ((𝐹𝑞) +Q 𝑞) <Q u}⟩)
17 nqex 6347 . . . . . . . . . . . . . . . . . 18 Q V
1817rabex 3892 . . . . . . . . . . . . . . . . 17 {𝑙 Q𝑞 Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)} V
1917rabex 3892 . . . . . . . . . . . . . . . . 17 {u Q𝑞 Q ((𝐹𝑞) +Q 𝑞) <Q u} V
2018, 19op2nd 5716 . . . . . . . . . . . . . . . 16 (2nd ‘⟨{𝑙 Q𝑞 Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {u Q𝑞 Q ((𝐹𝑞) +Q 𝑞) <Q u}⟩) = {u Q𝑞 Q ((𝐹𝑞) +Q 𝑞) <Q u}
2116, 20eqtri 2057 . . . . . . . . . . . . . . 15 (2nd𝐿) = {u Q𝑞 Q ((𝐹𝑞) +Q 𝑞) <Q u}
2215, 21elrab2 2694 . . . . . . . . . . . . . 14 (𝑠 (2nd𝐿) ↔ (𝑠 Q 𝑞 Q ((𝐹𝑞) +Q 𝑞) <Q 𝑠))
2322biimpi 113 . . . . . . . . . . . . 13 (𝑠 (2nd𝐿) → (𝑠 Q 𝑞 Q ((𝐹𝑞) +Q 𝑞) <Q 𝑠))
2423adantr 261 . . . . . . . . . . . 12 ((𝑠 (2nd𝐿) 𝑡 (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {u𝑆 <Q u}⟩)) → (𝑠 Q 𝑞 Q ((𝐹𝑞) +Q 𝑞) <Q 𝑠))
2524adantl 262 . . . . . . . . . . 11 (((φ 𝑟 (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {u𝑆 <Q u}⟩))) (𝑠 (2nd𝐿) 𝑡 (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {u𝑆 <Q u}⟩))) → (𝑠 Q 𝑞 Q ((𝐹𝑞) +Q 𝑞) <Q 𝑠))
2625adantr 261 . . . . . . . . . 10 ((((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)) → (𝑠 Q 𝑞 Q ((𝐹𝑞) +Q 𝑞) <Q 𝑠))
2726simpld 105 . . . . . . . . 9 ((((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)) → 𝑠 Q)
28 vex 2554 . . . . . . . . . . . . . 14 𝑡 V
29 breq2 3759 . . . . . . . . . . . . . 14 (u = 𝑡 → (𝑆 <Q u𝑆 <Q 𝑡))
30 ltnqex 6530 . . . . . . . . . . . . . . 15 {𝑙𝑙 <Q 𝑆} V
31 gtnqex 6531 . . . . . . . . . . . . . . 15 {u𝑆 <Q u} V
3230, 31op2nd 5716 . . . . . . . . . . . . . 14 (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {u𝑆 <Q u}⟩) = {u𝑆 <Q u}
3328, 29, 32elab2 2684 . . . . . . . . . . . . 13 (𝑡 (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {u𝑆 <Q u}⟩) ↔ 𝑆 <Q 𝑡)
34 ltrelnq 6349 . . . . . . . . . . . . . 14 <Q ⊆ (Q × Q)
3534brel 4335 . . . . . . . . . . . . 13 (𝑆 <Q 𝑡 → (𝑆 Q 𝑡 Q))
3633, 35sylbi 114 . . . . . . . . . . . 12 (𝑡 (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {u𝑆 <Q u}⟩) → (𝑆 Q 𝑡 Q))
3736simprd 107 . . . . . . . . . . 11 (𝑡 (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {u𝑆 <Q u}⟩) → 𝑡 Q)
3837ad2antll 460 . . . . . . . . . 10 (((φ 𝑟 (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {u𝑆 <Q u}⟩))) (𝑠 (2nd𝐿) 𝑡 (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {u𝑆 <Q u}⟩))) → 𝑡 Q)
3938adantr 261 . . . . . . . . 9 ((((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)) → 𝑡 Q)
40 addclnq 6359 . . . . . . . . 9 ((𝑠 Q 𝑡 Q) → (𝑠 +Q 𝑡) Q)
4127, 39, 40syl2anc 391 . . . . . . . 8 ((((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)) → (𝑠 +Q 𝑡) Q)
42 eleq1 2097 . . . . . . . . 9 (𝑟 = (𝑠 +Q 𝑡) → (𝑟 Q ↔ (𝑠 +Q 𝑡) Q))
4342adantl 262 . . . . . . . 8 ((((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)) → (𝑟 Q ↔ (𝑠 +Q 𝑡) Q))
4441, 43mpbird 156 . . . . . . 7 ((((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)) → 𝑟 Q)
4526simprd 107 . . . . . . . 8 ((((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)) → 𝑞 Q ((𝐹𝑞) +Q 𝑞) <Q 𝑠)
4633biimpi 113 . . . . . . . . . . . . . . . 16 (𝑡 (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {u𝑆 <Q u}⟩) → 𝑆 <Q 𝑡)
4746ad2antll 460 . . . . . . . . . . . . . . 15 (((φ 𝑟 (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {u𝑆 <Q u}⟩))) (𝑠 (2nd𝐿) 𝑡 (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {u𝑆 <Q u}⟩))) → 𝑆 <Q 𝑡)
4847adantr 261 . . . . . . . . . . . . . 14 ((((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)) → 𝑆 <Q 𝑡)
4948ad2antrr 457 . . . . . . . . . . . . 13 ((((((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)) 𝑞 Q) ((𝐹𝑞) +Q 𝑞) <Q 𝑠) → 𝑆 <Q 𝑡)
506ad5antr 465 . . . . . . . . . . . . . 14 ((((((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)) 𝑞 Q) ((𝐹𝑞) +Q 𝑞) <Q 𝑠) → 𝑆 Q)
5139ad2antrr 457 . . . . . . . . . . . . . 14 ((((((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)) 𝑞 Q) ((𝐹𝑞) +Q 𝑞) <Q 𝑠) → 𝑡 Q)
521ad5antr 465 . . . . . . . . . . . . . . . 16 ((((((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)) 𝑞 Q) ((𝐹𝑞) +Q 𝑞) <Q 𝑠) → 𝐹:QQ)
53 simplr 482 . . . . . . . . . . . . . . . 16 ((((((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)) 𝑞 Q) ((𝐹𝑞) +Q 𝑞) <Q 𝑠) → 𝑞 Q)
5452, 53ffvelrnd 5246 . . . . . . . . . . . . . . 15 ((((((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)) 𝑞 Q) ((𝐹𝑞) +Q 𝑞) <Q 𝑠) → (𝐹𝑞) Q)
55 addclnq 6359 . . . . . . . . . . . . . . 15 (((𝐹𝑞) Q 𝑞 Q) → ((𝐹𝑞) +Q 𝑞) Q)
5654, 53, 55syl2anc 391 . . . . . . . . . . . . . 14 ((((((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)) 𝑞 Q) ((𝐹𝑞) +Q 𝑞) <Q 𝑠) → ((𝐹𝑞) +Q 𝑞) Q)
57 ltanqg 6384 . . . . . . . . . . . . . 14 ((𝑆 Q 𝑡 Q ((𝐹𝑞) +Q 𝑞) Q) → (𝑆 <Q 𝑡 ↔ (((𝐹𝑞) +Q 𝑞) +Q 𝑆) <Q (((𝐹𝑞) +Q 𝑞) +Q 𝑡)))
5850, 51, 56, 57syl3anc 1134 . . . . . . . . . . . . 13 ((((((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)) 𝑞 Q) ((𝐹𝑞) +Q 𝑞) <Q 𝑠) → (𝑆 <Q 𝑡 ↔ (((𝐹𝑞) +Q 𝑞) +Q 𝑆) <Q (((𝐹𝑞) +Q 𝑞) +Q 𝑡)))
5949, 58mpbid 135 . . . . . . . . . . . 12 ((((((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)) 𝑞 Q) ((𝐹𝑞) +Q 𝑞) <Q 𝑠) → (((𝐹𝑞) +Q 𝑞) +Q 𝑆) <Q (((𝐹𝑞) +Q 𝑞) +Q 𝑡))
60 simpr 103 . . . . . . . . . . . . 13 ((((((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)) 𝑞 Q) ((𝐹𝑞) +Q 𝑞) <Q 𝑠) → ((𝐹𝑞) +Q 𝑞) <Q 𝑠)
61 ltanqg 6384 . . . . . . . . . . . . . . 15 ((f Q g Q Q) → (f <Q g ↔ ( +Q f) <Q ( +Q g)))
6261adantl 262 . . . . . . . . . . . . . 14 (((((((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)) 𝑞 Q) ((𝐹𝑞) +Q 𝑞) <Q 𝑠) (f Q g Q Q)) → (f <Q g ↔ ( +Q f) <Q ( +Q g)))
6327ad2antrr 457 . . . . . . . . . . . . . 14 ((((((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)) 𝑞 Q) ((𝐹𝑞) +Q 𝑞) <Q 𝑠) → 𝑠 Q)
64 addcomnqg 6365 . . . . . . . . . . . . . . 15 ((f Q g Q) → (f +Q g) = (g +Q f))
6564adantl 262 . . . . . . . . . . . . . 14 (((((((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)) 𝑞 Q) ((𝐹𝑞) +Q 𝑞) <Q 𝑠) (f Q g Q)) → (f +Q g) = (g +Q f))
6662, 56, 63, 51, 65caovord2d 5612 . . . . . . . . . . . . 13 ((((((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)) 𝑞 Q) ((𝐹𝑞) +Q 𝑞) <Q 𝑠) → (((𝐹𝑞) +Q 𝑞) <Q 𝑠 ↔ (((𝐹𝑞) +Q 𝑞) +Q 𝑡) <Q (𝑠 +Q 𝑡)))
6760, 66mpbid 135 . . . . . . . . . . . 12 ((((((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)) 𝑞 Q) ((𝐹𝑞) +Q 𝑞) <Q 𝑠) → (((𝐹𝑞) +Q 𝑞) +Q 𝑡) <Q (𝑠 +Q 𝑡))
68 ltsonq 6382 . . . . . . . . . . . . 13 <Q Or Q
6968, 34sotri 4663 . . . . . . . . . . . 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 391 . . . . . . . . . . 11 ((((((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)) 𝑞 Q) ((𝐹𝑞) +Q 𝑞) <Q 𝑠) → (((𝐹𝑞) +Q 𝑞) +Q 𝑆) <Q (𝑠 +Q 𝑡))
71 simpllr 486 . . . . . . . . . . 11 ((((((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)) 𝑞 Q) ((𝐹𝑞) +Q 𝑞) <Q 𝑠) → 𝑟 = (𝑠 +Q 𝑡))
7270, 71breqtrrd 3781 . . . . . . . . . 10 ((((((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)) 𝑞 Q) ((𝐹𝑞) +Q 𝑞) <Q 𝑠) → (((𝐹𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑟)
7372ex 108 . . . . . . . . 9 (((((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)) 𝑞 Q) → (((𝐹𝑞) +Q 𝑞) <Q 𝑠 → (((𝐹𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑟))
7473reximdva 2415 . . . . . . . 8 ((((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)) → (𝑞 Q ((𝐹𝑞) +Q 𝑞) <Q 𝑠𝑞 Q (((𝐹𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑟))
7545, 74mpd 13 . . . . . . 7 ((((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)) → 𝑞 Q (((𝐹𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑟)
76 breq2 3759 . . . . . . . . 9 (u = 𝑟 → ((((𝐹𝑞) +Q 𝑞) +Q 𝑆) <Q u ↔ (((𝐹𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑟))
7776rexbidv 2321 . . . . . . . 8 (u = 𝑟 → (𝑞 Q (((𝐹𝑞) +Q 𝑞) +Q 𝑆) <Q u𝑞 Q (((𝐹𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑟))
7817rabex 3892 . . . . . . . . 9 {𝑙 Q𝑞 Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆)} V
7917rabex 3892 . . . . . . . . 9 {u Q𝑞 Q (((𝐹𝑞) +Q 𝑞) +Q 𝑆) <Q u} V
8078, 79op2nd 5716 . . . . . . . 8 (2nd ‘⟨{𝑙 Q𝑞 Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆)}, {u Q𝑞 Q (((𝐹𝑞) +Q 𝑞) +Q 𝑆) <Q u}⟩) = {u Q𝑞 Q (((𝐹𝑞) +Q 𝑞) +Q 𝑆) <Q u}
8177, 80elrab2 2694 . . . . . . 7 (𝑟 (2nd ‘⟨{𝑙 Q𝑞 Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆)}, {u Q𝑞 Q (((𝐹𝑞) +Q 𝑞) +Q 𝑆) <Q u}⟩) ↔ (𝑟 Q 𝑞 Q (((𝐹𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑟))
8244, 75, 81sylanbrc 394 . . . . . 6 ((((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)) → 𝑟 (2nd ‘⟨{𝑙 Q𝑞 Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆)}, {u Q𝑞 Q (((𝐹𝑞) +Q 𝑞) +Q 𝑆) <Q u}⟩))
8382ex 108 . . . . 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) → 𝑟 (2nd ‘⟨{𝑙 Q𝑞 Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆)}, {u Q𝑞 Q (((𝐹𝑞) +Q 𝑞) +Q 𝑆) <Q u}⟩)))
8483rexlimdvva 2434 . . . 4 ((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) → 𝑟 (2nd ‘⟨{𝑙 Q𝑞 Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆)}, {u Q𝑞 Q (((𝐹𝑞) +Q 𝑞) +Q 𝑆) <Q u}⟩)))
8513, 84mpd 13 . . 3 ((φ 𝑟 (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {u𝑆 <Q u}⟩))) → 𝑟 (2nd ‘⟨{𝑙 Q𝑞 Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆)}, {u Q𝑞 Q (((𝐹𝑞) +Q 𝑞) +Q 𝑆) <Q u}⟩))
8685ex 108 . 2 (φ → (𝑟 (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {u𝑆 <Q u}⟩)) → 𝑟 (2nd ‘⟨{𝑙 Q𝑞 Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆)}, {u Q𝑞 Q (((𝐹𝑞) +Q 𝑞) +Q 𝑆) <Q u}⟩)))
8786ssrdv 2945 1 (φ → (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {u𝑆 <Q u}⟩)) ⊆ (2nd ‘⟨{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  2nd c2nd 5708  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:  cauappcvgprlemladdrl  6628  cauappcvgprlemladd  6629
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