Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  caucvgprlemladdfu GIF version

 Description: Lemma for caucvgpr 7513. 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 (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )))))
caucvgpr.bnd (𝜑 → ∀𝑗N 𝐴 <Q (𝐹𝑗))
caucvgpr.lim 𝐿 = ⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}⟩
Assertion
Ref Expression
caucvgprlemladdfu (𝜑 → (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩)) ⊆ {𝑢Q ∣ ∃𝑗N (((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)

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 (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )))))
3 caucvgpr.bnd . . . . . . 7 (𝜑 → ∀𝑗N 𝐴 <Q (𝐹𝑗))
4 caucvgpr.lim . . . . . . 7 𝐿 = ⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}⟩
51, 2, 3, 4caucvgprlemcl 7507 . . . . . 6 (𝜑𝐿P)
6 caucvgprlemladd.s . . . . . . 7 (𝜑𝑆Q)
7 nqprlu 7378 . . . . . . 7 (𝑆Q → ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩ ∈ P)
86, 7syl 14 . . . . . 6 (𝜑 → ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩ ∈ P)
9 df-iplp 7299 . . . . . . 7 +P = (𝑥P, 𝑦P ↦ ⟨{𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (1st𝑥) ∧ ∈ (1st𝑦) ∧ 𝑓 = (𝑔 +Q ))}, {𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (2nd𝑥) ∧ ∈ (2nd𝑦) ∧ 𝑓 = (𝑔 +Q ))}⟩)
10 addclnq 7206 . . . . . . 7 ((𝑔QQ) → (𝑔 +Q ) ∈ Q)
119, 10genpelvu 7344 . . . . . 6 ((𝐿P ∧ ⟨{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 409 . . . . 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 294 . . . 4 ((𝜑𝑟 ∈ (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 3940 . . . . . . . . . . . . . . . 16 (𝑢 = 𝑠 → (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢 ↔ ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠))
1514rexbidv 2439 . . . . . . . . . . . . . . 15 (𝑢 = 𝑠 → (∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢 ↔ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠))
164fveq2i 5431 . . . . . . . . . . . . . . . 16 (2nd𝐿) = (2nd ‘⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}⟩)
17 nqex 7194 . . . . . . . . . . . . . . . . . 18 Q ∈ V
1817rabex 4079 . . . . . . . . . . . . . . . . 17 {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)} ∈ V
1917rabex 4079 . . . . . . . . . . . . . . . . 17 {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢} ∈ V
2018, 19op2nd 6052 . . . . . . . . . . . . . . . 16 (2nd ‘⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}⟩) = {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}
2116, 20eqtri 2161 . . . . . . . . . . . . . . 15 (2nd𝐿) = {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}
2215, 21elrab2 2846 . . . . . . . . . . . . . 14 (𝑠 ∈ (2nd𝐿) ↔ (𝑠Q ∧ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠))
2322biimpi 119 . . . . . . . . . . . . 13 (𝑠 ∈ (2nd𝐿) → (𝑠Q ∧ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠))
2423adantr 274 . . . . . . . . . . . 12 ((𝑠 ∈ (2nd𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩)) → (𝑠Q ∧ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠))
2524adantl 275 . . . . . . . . . . 11 (((𝜑𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) → (𝑠Q ∧ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠))
2625adantr 274 . . . . . . . . . 10 ((((𝜑𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → (𝑠Q ∧ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠))
2726simpld 111 . . . . . . . . 9 ((((𝜑𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → 𝑠Q)
28 vex 2692 . . . . . . . . . . . . . 14 𝑡 ∈ V
29 breq2 3940 . . . . . . . . . . . . . 14 (𝑢 = 𝑡 → (𝑆 <Q 𝑢𝑆 <Q 𝑡))
30 ltnqex 7380 . . . . . . . . . . . . . . 15 {𝑙𝑙 <Q 𝑆} ∈ V
31 gtnqex 7381 . . . . . . . . . . . . . . 15 {𝑢𝑆 <Q 𝑢} ∈ V
3230, 31op2nd 6052 . . . . . . . . . . . . . 14 (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩) = {𝑢𝑆 <Q 𝑢}
3328, 29, 32elab2 2835 . . . . . . . . . . . . 13 (𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩) ↔ 𝑆 <Q 𝑡)
34 ltrelnq 7196 . . . . . . . . . . . . . 14 <Q ⊆ (Q × Q)
3534brel 4598 . . . . . . . . . . . . 13 (𝑆 <Q 𝑡 → (𝑆Q𝑡Q))
3633, 35sylbi 120 . . . . . . . . . . . 12 (𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩) → (𝑆Q𝑡Q))
3736simprd 113 . . . . . . . . . . 11 (𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩) → 𝑡Q)
3837ad2antll 483 . . . . . . . . . 10 (((𝜑𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) → 𝑡Q)
3938adantr 274 . . . . . . . . 9 ((((𝜑𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → 𝑡Q)
40 addclnq 7206 . . . . . . . . 9 ((𝑠Q𝑡Q) → (𝑠 +Q 𝑡) ∈ Q)
4127, 39, 40syl2anc 409 . . . . . . . 8 ((((𝜑𝑟 ∈ (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 2203 . . . . . . . . 9 (𝑟 = (𝑠 +Q 𝑡) → (𝑟Q ↔ (𝑠 +Q 𝑡) ∈ Q))
4342adantl 275 . . . . . . . 8 ((((𝜑𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → (𝑟Q ↔ (𝑠 +Q 𝑡) ∈ Q))
4441, 43mpbird 166 . . . . . . 7 ((((𝜑𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → 𝑟Q)
4526simprd 113 . . . . . . . . . 10 ((((𝜑𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠)
46 fveq2 5428 . . . . . . . . . . . . 13 (𝑗 = 𝑚 → (𝐹𝑗) = (𝐹𝑚))
47 opeq1 3712 . . . . . . . . . . . . . . 15 (𝑗 = 𝑚 → ⟨𝑗, 1o⟩ = ⟨𝑚, 1o⟩)
4847eceq1d 6472 . . . . . . . . . . . . . 14 (𝑗 = 𝑚 → [⟨𝑗, 1o⟩] ~Q = [⟨𝑚, 1o⟩] ~Q )
4948fveq2d 5432 . . . . . . . . . . . . 13 (𝑗 = 𝑚 → (*Q‘[⟨𝑗, 1o⟩] ~Q ) = (*Q‘[⟨𝑚, 1o⟩] ~Q ))
5046, 49oveq12d 5799 . . . . . . . . . . . 12 (𝑗 = 𝑚 → ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )))
5150breq1d 3946 . . . . . . . . . . 11 (𝑗 = 𝑚 → (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠 ↔ ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) <Q 𝑠))
5251cbvrexv 2658 . . . . . . . . . 10 (∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠 ↔ ∃𝑚N ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) <Q 𝑠)
5345, 52sylib 121 . . . . . . . . 9 ((((𝜑𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → ∃𝑚N ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) <Q 𝑠)
5433biimpi 119 . . . . . . . . . . . . . . . . 17 (𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩) → 𝑆 <Q 𝑡)
5554ad2antll 483 . . . . . . . . . . . . . . . 16 (((𝜑𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) → 𝑆 <Q 𝑡)
5655adantr 274 . . . . . . . . . . . . . . 15 ((((𝜑𝑟 ∈ (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 480 . . . . . . . . . . . . . 14 ((((((𝜑𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑚N) ∧ ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) <Q 𝑠) → 𝑆 <Q 𝑡)
586ad5antr 488 . . . . . . . . . . . . . . 15 ((((((𝜑𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑚N) ∧ ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) <Q 𝑠) → 𝑆Q)
5939ad2antrr 480 . . . . . . . . . . . . . . 15 ((((((𝜑𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑚N) ∧ ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) <Q 𝑠) → 𝑡Q)
601ad5antr 488 . . . . . . . . . . . . . . . . 17 ((((((𝜑𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑚N) ∧ ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) <Q 𝑠) → 𝐹:NQ)
61 simplr 520 . . . . . . . . . . . . . . . . 17 ((((((𝜑𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑚N) ∧ ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) <Q 𝑠) → 𝑚N)
6260, 61ffvelrnd 5563 . . . . . . . . . . . . . . . 16 ((((((𝜑𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑚N) ∧ ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) <Q 𝑠) → (𝐹𝑚) ∈ Q)
63 nnnq 7253 . . . . . . . . . . . . . . . . 17 (𝑚N → [⟨𝑚, 1o⟩] ~QQ)
64 recclnq 7223 . . . . . . . . . . . . . . . . 17 ([⟨𝑚, 1o⟩] ~QQ → (*Q‘[⟨𝑚, 1o⟩] ~Q ) ∈ Q)
6561, 63, 643syl 17 . . . . . . . . . . . . . . . 16 ((((((𝜑𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑚N) ∧ ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) <Q 𝑠) → (*Q‘[⟨𝑚, 1o⟩] ~Q ) ∈ Q)
66 addclnq 7206 . . . . . . . . . . . . . . . 16 (((𝐹𝑚) ∈ Q ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) ∈ Q) → ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) ∈ Q)
6762, 65, 66syl2anc 409 . . . . . . . . . . . . . . 15 ((((((𝜑𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑚N) ∧ ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) <Q 𝑠) → ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) ∈ Q)
68 ltanqg 7231 . . . . . . . . . . . . . . 15 ((𝑆Q𝑡Q ∧ ((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 1217 . . . . . . . . . . . . . 14 ((((((𝜑𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑚N) ∧ ((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 146 . . . . . . . . . . . . 13 ((((((𝜑𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑚N) ∧ ((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 109 . . . . . . . . . . . . . 14 ((((((𝜑𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑚N) ∧ ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) <Q 𝑠) → ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) <Q 𝑠)
72 ltanqg 7231 . . . . . . . . . . . . . . . 16 ((𝑧Q𝑤Q𝑣Q) → (𝑧 <Q 𝑤 ↔ (𝑣 +Q 𝑧) <Q (𝑣 +Q 𝑤)))
7372adantl 275 . . . . . . . . . . . . . . 15 (((((((𝜑𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑚N) ∧ ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) <Q 𝑠) ∧ (𝑧Q𝑤Q𝑣Q)) → (𝑧 <Q 𝑤 ↔ (𝑣 +Q 𝑧) <Q (𝑣 +Q 𝑤)))
7427ad2antrr 480 . . . . . . . . . . . . . . 15 ((((((𝜑𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑚N) ∧ ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) <Q 𝑠) → 𝑠Q)
75 addcomnqg 7212 . . . . . . . . . . . . . . . 16 ((𝑧Q𝑤Q) → (𝑧 +Q 𝑤) = (𝑤 +Q 𝑧))
7675adantl 275 . . . . . . . . . . . . . . 15 (((((((𝜑𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑚N) ∧ ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) <Q 𝑠) ∧ (𝑧Q𝑤Q)) → (𝑧 +Q 𝑤) = (𝑤 +Q 𝑧))
7773, 67, 74, 59, 76caovord2d 5947 . . . . . . . . . . . . . 14 ((((((𝜑𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑚N) ∧ ((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 146 . . . . . . . . . . . . 13 ((((((𝜑𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑚N) ∧ ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) <Q 𝑠) → (((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) +Q 𝑡) <Q (𝑠 +Q 𝑡))
79 ltsonq 7229 . . . . . . . . . . . . . 14 <Q Or Q
8079, 34sotri 4941 . . . . . . . . . . . . 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 409 . . . . . . . . . . . 12 ((((((𝜑𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑚N) ∧ ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) <Q 𝑠) → (((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) +Q 𝑆) <Q (𝑠 +Q 𝑡))
82 simpllr 524 . . . . . . . . . . . 12 ((((((𝜑𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑚N) ∧ ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) <Q 𝑠) → 𝑟 = (𝑠 +Q 𝑡))
8381, 82breqtrrd 3963 . . . . . . . . . . 11 ((((((𝜑𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑚N) ∧ ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) <Q 𝑠) → (((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) +Q 𝑆) <Q 𝑟)
8483ex 114 . . . . . . . . . 10 (((((𝜑𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑚N) → (((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) <Q 𝑠 → (((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) +Q 𝑆) <Q 𝑟))
8584reximdva 2537 . . . . . . . . 9 ((((𝜑𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → (∃𝑚N ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) <Q 𝑠 → ∃𝑚N (((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) +Q 𝑆) <Q 𝑟))
8653, 85mpd 13 . . . . . . . 8 ((((𝜑𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → ∃𝑚N (((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) +Q 𝑆) <Q 𝑟)
8750oveq1d 5796 . . . . . . . . . 10 (𝑗 = 𝑚 → (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) +Q 𝑆) = (((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) +Q 𝑆))
8887breq1d 3946 . . . . . . . . 9 (𝑗 = 𝑚 → ((((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) +Q 𝑆) <Q 𝑟 ↔ (((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) +Q 𝑆) <Q 𝑟))
8988cbvrexv 2658 . . . . . . . 8 (∃𝑗N (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) +Q 𝑆) <Q 𝑟 ↔ ∃𝑚N (((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) +Q 𝑆) <Q 𝑟)
9086, 89sylibr 133 . . . . . . 7 ((((𝜑𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → ∃𝑗N (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) +Q 𝑆) <Q 𝑟)
91 breq2 3940 . . . . . . . . 9 (𝑢 = 𝑟 → ((((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) +Q 𝑆) <Q 𝑢 ↔ (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) +Q 𝑆) <Q 𝑟))
9291rexbidv 2439 . . . . . . . 8 (𝑢 = 𝑟 → (∃𝑗N (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) +Q 𝑆) <Q 𝑢 ↔ ∃𝑗N (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) +Q 𝑆) <Q 𝑟))
9392elrab 2843 . . . . . . 7 (𝑟 ∈ {𝑢Q ∣ ∃𝑗N (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) +Q 𝑆) <Q 𝑢} ↔ (𝑟Q ∧ ∃𝑗N (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) +Q 𝑆) <Q 𝑟))
9444, 90, 93sylanbrc 414 . . . . . 6 ((((𝜑𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → 𝑟 ∈ {𝑢Q ∣ ∃𝑗N (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) +Q 𝑆) <Q 𝑢})
9594ex 114 . . . . 5 (((𝜑𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) → (𝑟 = (𝑠 +Q 𝑡) → 𝑟 ∈ {𝑢Q ∣ ∃𝑗N (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) +Q 𝑆) <Q 𝑢}))
9695rexlimdvva 2560 . . . 4 ((𝜑𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) → (∃𝑠 ∈ (2nd𝐿)∃𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩)𝑟 = (𝑠 +Q 𝑡) → 𝑟 ∈ {𝑢Q ∣ ∃𝑗N (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) +Q 𝑆) <Q 𝑢}))
9713, 96mpd 13 . . 3 ((𝜑𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) → 𝑟 ∈ {𝑢Q ∣ ∃𝑗N (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) +Q 𝑆) <Q 𝑢})
9897ex 114 . 2 (𝜑 → (𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩)) → 𝑟 ∈ {𝑢Q ∣ ∃𝑗N (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) +Q 𝑆) <Q 𝑢}))
9998ssrdv 3107 1 (𝜑 → (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩)) ⊆ {𝑢Q ∣ ∃𝑗N (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) +Q 𝑆) <Q 𝑢})
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∧ w3a 963   = wceq 1332   ∈ wcel 1481  {cab 2126  ∀wral 2417  ∃wrex 2418  {crab 2421   ⊆ wss 3075  ⟨cop 3534   class class class wbr 3936  ⟶wf 5126  ‘cfv 5130  (class class class)co 5781  2nd c2nd 6044  1oc1o 6313  [cec 6434  Ncnpi 7103
 Copyright terms: Public domain W3C validator