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 Description: Lemma for caucvgpr 6653. Adding 𝑆 after embedding in positive reals, or adding it as a rational. (Contributed by Jim Kingdon, 9-Oct-2020.)
Hypotheses
Ref Expression
caucvgpr.f (φ𝐹:NQ)
caucvgpr.cau (φ𝑛 N 𝑘 N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))
caucvgpr.bnd (φ𝑗 N A <Q (𝐹𝑗))
caucvgpr.lim 𝐿 = ⟨{𝑙 Q𝑗 N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {u Q𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q u}⟩
Assertion
Ref Expression
caucvgprlemladdfu (φ → (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {u𝑆 <Q u}⟩)) ⊆ {u Q𝑗 N (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) +Q 𝑆) <Q u})
Distinct variable groups:   A,𝑗   𝑗,𝐹,u,𝑙   𝑛,𝐹,𝑘   𝑘,𝐿,𝑗   𝑆,𝑙,u,𝑗   𝑗,𝑘
Allowed substitution hints:   φ(u,𝑗,𝑘,𝑛,𝑙)   A(u,𝑘,𝑛,𝑙)   𝑆(𝑘,𝑛)   𝐿(u,𝑛,𝑙)

Proof of Theorem caucvgprlemladdfu
Dummy variables 𝑚 𝑟 𝑠 𝑡 v w z f g x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 caucvgpr.f . . . . . . 7 (φ𝐹:NQ)
2 caucvgpr.cau . . . . . . 7 (φ𝑛 N 𝑘 N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))
3 caucvgpr.bnd . . . . . . 7 (φ𝑗 N A <Q (𝐹𝑗))
4 caucvgpr.lim . . . . . . 7 𝐿 = ⟨{𝑙 Q𝑗 N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {u Q𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q u}⟩
51, 2, 3, 4caucvgprlemcl 6647 . . . . . 6 (φ𝐿 P)
6 caucvgprlemladd.s . . . . . . 7 (φ𝑆 Q)
7 nqprlu 6530 . . . . . . 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 6451 . . . . . . 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 6496 . . . . . 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 → (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q u ↔ ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠))
1514rexbidv 2321 . . . . . . . . . . . . . . 15 (u = 𝑠 → (𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q u𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠))
164fveq2i 5124 . . . . . . . . . . . . . . . 16 (2nd𝐿) = (2nd ‘⟨{𝑙 Q𝑗 N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {u Q𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q u}⟩)
17 nqex 6347 . . . . . . . . . . . . . . . . . 18 Q V
1817rabex 3892 . . . . . . . . . . . . . . . . 17 {𝑙 Q𝑗 N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)} V
1917rabex 3892 . . . . . . . . . . . . . . . . 17 {u Q𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q u} V
2018, 19op2nd 5716 . . . . . . . . . . . . . . . 16 (2nd ‘⟨{𝑙 Q𝑗 N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {u Q𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q u}⟩) = {u Q𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q u}
2116, 20eqtri 2057 . . . . . . . . . . . . . . 15 (2nd𝐿) = {u Q𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q u}
2215, 21elrab2 2694 . . . . . . . . . . . . . 14 (𝑠 (2nd𝐿) ↔ (𝑠 Q 𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠))
2322biimpi 113 . . . . . . . . . . . . 13 (𝑠 (2nd𝐿) → (𝑠 Q 𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠))
2423adantr 261 . . . . . . . . . . . 12 ((𝑠 (2nd𝐿) 𝑡 (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {u𝑆 <Q u}⟩)) → (𝑠 Q 𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~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 𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~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 𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~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 6531 . . . . . . . . . . . . . . 15 {𝑙𝑙 <Q 𝑆} V
31 gtnqex 6532 . . . . . . . . . . . . . . 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 . . . . . . . . . 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)) → 𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠)
46 fveq2 5121 . . . . . . . . . . . . 13 (𝑗 = 𝑚 → (𝐹𝑗) = (𝐹𝑚))
47 opeq1 3540 . . . . . . . . . . . . . . 15 (𝑗 = 𝑚 → ⟨𝑗, 1𝑜⟩ = ⟨𝑚, 1𝑜⟩)
4847eceq1d 6078 . . . . . . . . . . . . . 14 (𝑗 = 𝑚 → [⟨𝑗, 1𝑜⟩] ~Q = [⟨𝑚, 1𝑜⟩] ~Q )
4948fveq2d 5125 . . . . . . . . . . . . 13 (𝑗 = 𝑚 → (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) = (*Q‘[⟨𝑚, 1𝑜⟩] ~Q ))
5046, 49oveq12d 5473 . . . . . . . . . . . 12 (𝑗 = 𝑚 → ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) = ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )))
5150breq1d 3765 . . . . . . . . . . 11 (𝑗 = 𝑚 → (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠 ↔ ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) <Q 𝑠))
5251cbvrexv 2528 . . . . . . . . . 10 (𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠𝑚 N ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) <Q 𝑠)
5345, 52sylib 127 . . . . . . . . 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)) → 𝑚 N ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) <Q 𝑠)
5433biimpi 113 . . . . . . . . . . . . . . . . 17 (𝑡 (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {u𝑆 <Q u}⟩) → 𝑆 <Q 𝑡)
5554ad2antll 460 . . . . . . . . . . . . . . . 16 (((φ 𝑟 (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {u𝑆 <Q u}⟩))) (𝑠 (2nd𝐿) 𝑡 (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {u𝑆 <Q u}⟩))) → 𝑆 <Q 𝑡)
5655adantr 261 . . . . . . . . . . . . . . 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)) → 𝑆 <Q 𝑡)
5756ad2antrr 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)) 𝑚 N) ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) <Q 𝑠) → 𝑆 <Q 𝑡)
586ad5antr 465 . . . . . . . . . . . . . . 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)) 𝑚 N) ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) <Q 𝑠) → 𝑆 Q)
5939ad2antrr 457 . . . . . . . . . . . . . . 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)) 𝑚 N) ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) <Q 𝑠) → 𝑡 Q)
601ad5antr 465 . . . . . . . . . . . . . . . . 17 ((((((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)) 𝑚 N) ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) <Q 𝑠) → 𝐹:NQ)
61 simplr 482 . . . . . . . . . . . . . . . . 17 ((((((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)) 𝑚 N) ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) <Q 𝑠) → 𝑚 N)
6260, 61ffvelrnd 5246 . . . . . . . . . . . . . . . 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)) 𝑚 N) ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) <Q 𝑠) → (𝐹𝑚) Q)
63 nnnq 6405 . . . . . . . . . . . . . . . . 17 (𝑚 N → [⟨𝑚, 1𝑜⟩] ~Q Q)
64 recclnq 6376 . . . . . . . . . . . . . . . . 17 ([⟨𝑚, 1𝑜⟩] ~Q Q → (*Q‘[⟨𝑚, 1𝑜⟩] ~Q ) Q)
6561, 63, 643syl 17 . . . . . . . . . . . . . . . 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)) 𝑚 N) ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) <Q 𝑠) → (*Q‘[⟨𝑚, 1𝑜⟩] ~Q ) Q)
66 addclnq 6359 . . . . . . . . . . . . . . . 16 (((𝐹𝑚) Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q ) Q) → ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) Q)
6762, 65, 66syl2anc 391 . . . . . . . . . . . . . . 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)) 𝑚 N) ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) <Q 𝑠) → ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) Q)
68 ltanqg 6384 . . . . . . . . . . . . . . 15 ((𝑆 Q 𝑡 Q ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) Q) → (𝑆 <Q 𝑡 ↔ (((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) +Q 𝑆) <Q (((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) +Q 𝑡)))
6958, 59, 67, 68syl3anc 1134 . . . . . . . . . . . . . 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)) 𝑚 N) ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) <Q 𝑠) → (𝑆 <Q 𝑡 ↔ (((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) +Q 𝑆) <Q (((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) +Q 𝑡)))
7057, 69mpbid 135 . . . . . . . . . . . . 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)) 𝑚 N) ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) <Q 𝑠) → (((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) +Q 𝑆) <Q (((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) +Q 𝑡))
71 simpr 103 . . . . . . . . . . . . . 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)) 𝑚 N) ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) <Q 𝑠) → ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) <Q 𝑠)
72 ltanqg 6384 . . . . . . . . . . . . . . . 16 ((z Q w Q v Q) → (z <Q w ↔ (v +Q z) <Q (v +Q w)))
7372adantl 262 . . . . . . . . . . . . . . 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)) 𝑚 N) ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) <Q 𝑠) (z Q w Q v Q)) → (z <Q w ↔ (v +Q z) <Q (v +Q w)))
7427ad2antrr 457 . . . . . . . . . . . . . . 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)) 𝑚 N) ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) <Q 𝑠) → 𝑠 Q)
75 addcomnqg 6365 . . . . . . . . . . . . . . . 16 ((z Q w Q) → (z +Q w) = (w +Q z))
7675adantl 262 . . . . . . . . . . . . . . 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)) 𝑚 N) ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) <Q 𝑠) (z Q w Q)) → (z +Q w) = (w +Q z))
7773, 67, 74, 59, 76caovord2d 5612 . . . . . . . . . . . . . 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)) 𝑚 N) ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) <Q 𝑠) → (((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) <Q 𝑠 ↔ (((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) +Q 𝑡) <Q (𝑠 +Q 𝑡)))
7871, 77mpbid 135 . . . . . . . . . . . . 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)) 𝑚 N) ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) <Q 𝑠) → (((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) +Q 𝑡) <Q (𝑠 +Q 𝑡))
79 ltsonq 6382 . . . . . . . . . . . . . 14 <Q Or Q
8079, 34sotri 4663 . . . . . . . . . . . . 13 (((((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) +Q 𝑆) <Q (((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) +Q 𝑡) (((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) +Q 𝑡) <Q (𝑠 +Q 𝑡)) → (((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) +Q 𝑆) <Q (𝑠 +Q 𝑡))
8170, 78, 80syl2anc 391 . . . . . . . . . . . 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)) 𝑚 N) ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) <Q 𝑠) → (((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) +Q 𝑆) <Q (𝑠 +Q 𝑡))
82 simpllr 486 . . . . . . . . . . . 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)) 𝑚 N) ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) <Q 𝑠) → 𝑟 = (𝑠 +Q 𝑡))
8381, 82breqtrrd 3781 . . . . . . . . . . 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)) 𝑚 N) ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) <Q 𝑠) → (((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) +Q 𝑆) <Q 𝑟)
8483ex 108 . . . . . . . . . 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)) 𝑚 N) → (((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) <Q 𝑠 → (((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) +Q 𝑆) <Q 𝑟))
8584reximdva 2415 . . . . . . . . 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)) → (𝑚 N ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) <Q 𝑠𝑚 N (((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) +Q 𝑆) <Q 𝑟))
8653, 85mpd 13 . . . . . . . 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)) → 𝑚 N (((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) +Q 𝑆) <Q 𝑟)
8750oveq1d 5470 . . . . . . . . . 10 (𝑗 = 𝑚 → (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) +Q 𝑆) = (((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) +Q 𝑆))
8887breq1d 3765 . . . . . . . . 9 (𝑗 = 𝑚 → ((((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) +Q 𝑆) <Q 𝑟 ↔ (((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) +Q 𝑆) <Q 𝑟))
8988cbvrexv 2528 . . . . . . . 8 (𝑗 N (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) +Q 𝑆) <Q 𝑟𝑚 N (((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) +Q 𝑆) <Q 𝑟)
9086, 89sylibr 137 . . . . . . 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)) → 𝑗 N (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) +Q 𝑆) <Q 𝑟)
91 breq2 3759 . . . . . . . . 9 (u = 𝑟 → ((((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) +Q 𝑆) <Q u ↔ (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) +Q 𝑆) <Q 𝑟))
9291rexbidv 2321 . . . . . . . 8 (u = 𝑟 → (𝑗 N (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) +Q 𝑆) <Q u𝑗 N (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) +Q 𝑆) <Q 𝑟))
9392elrab 2692 . . . . . . 7 (𝑟 {u Q𝑗 N (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) +Q 𝑆) <Q u} ↔ (𝑟 Q 𝑗 N (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) +Q 𝑆) <Q 𝑟))
9444, 90, 93sylanbrc 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 {u Q𝑗 N (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) +Q 𝑆) <Q u})
9594ex 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 {u Q𝑗 N (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) +Q 𝑆) <Q u}))
9695rexlimdvva 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 {u Q𝑗 N (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) +Q 𝑆) <Q u}))
9713, 96mpd 13 . . 3 ((φ 𝑟 (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {u𝑆 <Q u}⟩))) → 𝑟 {u Q𝑗 N (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) +Q 𝑆) <Q u})
9897ex 108 . 2 (φ → (𝑟 (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {u𝑆 <Q u}⟩)) → 𝑟 {u Q𝑗 N (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) +Q 𝑆) <Q u}))
9998ssrdv 2945 1 (φ → (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {u𝑆 <Q u}⟩)) ⊆ {u Q𝑗 N (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~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  1𝑜c1o 5933  [cec 6040  Ncnpi 6256
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