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Theorem caucvgprlemladdfu 6981
Description: Lemma for caucvgpr 6986. 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}⟩
caucvgprlemladd.s (𝜑𝑆Q)
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)

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 (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 6980 . . . . . 6 (𝜑𝐿P)
6 caucvgprlemladd.s . . . . . . 7 (𝜑𝑆Q)
7 nqprlu 6851 . . . . . . 7 (𝑆Q → ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩ ∈ P)
86, 7syl 14 . . . . . 6 (𝜑 → ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩ ∈ P)
9 df-iplp 6772 . . . . . . 7 +P = (𝑥P, 𝑦P ↦ ⟨{𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (1st𝑥) ∧ ∈ (1st𝑦) ∧ 𝑓 = (𝑔 +Q ))}, {𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (2nd𝑥) ∧ ∈ (2nd𝑦) ∧ 𝑓 = (𝑔 +Q ))}⟩)
10 addclnq 6679 . . . . . . 7 ((𝑔QQ) → (𝑔 +Q ) ∈ Q)
119, 10genpelvu 6817 . . . . . 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 403 . . . . 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 290 . . . 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 3809 . . . . . . . . . . . . . . . 16 (𝑢 = 𝑠 → (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢 ↔ ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠))
1514rexbidv 2374 . . . . . . . . . . . . . . 15 (𝑢 = 𝑠 → (∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢 ↔ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠))
164fveq2i 5232 . . . . . . . . . . . . . . . 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 6667 . . . . . . . . . . . . . . . . . 18 Q ∈ V
1817rabex 3942 . . . . . . . . . . . . . . . . 17 {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)} ∈ V
1917rabex 3942 . . . . . . . . . . . . . . . . 17 {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢} ∈ V
2018, 19op2nd 5825 . . . . . . . . . . . . . . . 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 2103 . . . . . . . . . . . . . . 15 (2nd𝐿) = {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}
2215, 21elrab2 2760 . . . . . . . . . . . . . 14 (𝑠 ∈ (2nd𝐿) ↔ (𝑠Q ∧ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠))
2322biimpi 118 . . . . . . . . . . . . 13 (𝑠 ∈ (2nd𝐿) → (𝑠Q ∧ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠))
2423adantr 270 . . . . . . . . . . . 12 ((𝑠 ∈ (2nd𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩)) → (𝑠Q ∧ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠))
2524adantl 271 . . . . . . . . . . 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 270 . . . . . . . . . 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 110 . . . . . . . . 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 2613 . . . . . . . . . . . . . 14 𝑡 ∈ V
29 breq2 3809 . . . . . . . . . . . . . 14 (𝑢 = 𝑡 → (𝑆 <Q 𝑢𝑆 <Q 𝑡))
30 ltnqex 6853 . . . . . . . . . . . . . . 15 {𝑙𝑙 <Q 𝑆} ∈ V
31 gtnqex 6854 . . . . . . . . . . . . . . 15 {𝑢𝑆 <Q 𝑢} ∈ V
3230, 31op2nd 5825 . . . . . . . . . . . . . 14 (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩) = {𝑢𝑆 <Q 𝑢}
3328, 29, 32elab2 2749 . . . . . . . . . . . . 13 (𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩) ↔ 𝑆 <Q 𝑡)
34 ltrelnq 6669 . . . . . . . . . . . . . 14 <Q ⊆ (Q × Q)
3534brel 4438 . . . . . . . . . . . . 13 (𝑆 <Q 𝑡 → (𝑆Q𝑡Q))
3633, 35sylbi 119 . . . . . . . . . . . 12 (𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩) → (𝑆Q𝑡Q))
3736simprd 112 . . . . . . . . . . 11 (𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩) → 𝑡Q)
3837ad2antll 475 . . . . . . . . . 10 (((𝜑𝑟 ∈ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd𝐿) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))) → 𝑡Q)
3938adantr 270 . . . . . . . . 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 6679 . . . . . . . . 9 ((𝑠Q𝑡Q) → (𝑠 +Q 𝑡) ∈ Q)
4127, 39, 40syl2anc 403 . . . . . . . 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 2145 . . . . . . . . 9 (𝑟 = (𝑠 +Q 𝑡) → (𝑟Q ↔ (𝑠 +Q 𝑡) ∈ Q))
4342adantl 271 . . . . . . . 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 165 . . . . . . 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 112 . . . . . . . . . 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 5229 . . . . . . . . . . . . 13 (𝑗 = 𝑚 → (𝐹𝑗) = (𝐹𝑚))
47 opeq1 3590 . . . . . . . . . . . . . . 15 (𝑗 = 𝑚 → ⟨𝑗, 1𝑜⟩ = ⟨𝑚, 1𝑜⟩)
4847eceq1d 6229 . . . . . . . . . . . . . 14 (𝑗 = 𝑚 → [⟨𝑗, 1𝑜⟩] ~Q = [⟨𝑚, 1𝑜⟩] ~Q )
4948fveq2d 5233 . . . . . . . . . . . . 13 (𝑗 = 𝑚 → (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) = (*Q‘[⟨𝑚, 1𝑜⟩] ~Q ))
5046, 49oveq12d 5581 . . . . . . . . . . . 12 (𝑗 = 𝑚 → ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) = ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )))
5150breq1d 3815 . . . . . . . . . . 11 (𝑗 = 𝑚 → (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠 ↔ ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) <Q 𝑠))
5251cbvrexv 2583 . . . . . . . . . 10 (∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠 ↔ ∃𝑚N ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) <Q 𝑠)
5345, 52sylib 120 . . . . . . . . 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 118 . . . . . . . . . . . . . . . . 17 (𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩) → 𝑆 <Q 𝑡)
5554ad2antll 475 . . . . . . . . . . . . . . . 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 270 . . . . . . . . . . . . . . 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 472 . . . . . . . . . . . . . 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 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)
5939ad2antrr 472 . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . 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 497 . . . . . . . . . . . . . . . . 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 5355 . . . . . . . . . . . . . . . 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 6726 . . . . . . . . . . . . . . . . 17 (𝑚N → [⟨𝑚, 1𝑜⟩] ~QQ)
64 recclnq 6696 . . . . . . . . . . . . . . . . 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 6679 . . . . . . . . . . . . . . . 16 (((𝐹𝑚) ∈ Q ∧ (*Q‘[⟨𝑚, 1𝑜⟩] ~Q ) ∈ Q) → ((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) ∈ Q)
6762, 65, 66syl2anc 403 . . . . . . . . . . . . . . 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 6704 . . . . . . . . . . . . . . 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 1170 . . . . . . . . . . . . . 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 145 . . . . . . . . . . . . 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 108 . . . . . . . . . . . . . 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 6704 . . . . . . . . . . . . . . . 16 ((𝑧Q𝑤Q𝑣Q) → (𝑧 <Q 𝑤 ↔ (𝑣 +Q 𝑧) <Q (𝑣 +Q 𝑤)))
7372adantl 271 . . . . . . . . . . . . . . 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 472 . . . . . . . . . . . . . . 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 6685 . . . . . . . . . . . . . . . 16 ((𝑧Q𝑤Q) → (𝑧 +Q 𝑤) = (𝑤 +Q 𝑧))
7675adantl 271 . . . . . . . . . . . . . . 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 5721 . . . . . . . . . . . . . 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 145 . . . . . . . . . . . . 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 6702 . . . . . . . . . . . . . 14 <Q Or Q
8079, 34sotri 4770 . . . . . . . . . . . . 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 403 . . . . . . . . . . . 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 501 . . . . . . . . . . . 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 3831 . . . . . . . . . . 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 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)) ∧ 𝑚N) → (((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) <Q 𝑠 → (((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) +Q 𝑆) <Q 𝑟))
8584reximdva 2468 . . . . . . . . 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 5578 . . . . . . . . . 10 (𝑗 = 𝑚 → (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) +Q 𝑆) = (((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) +Q 𝑆))
8887breq1d 3815 . . . . . . . . 9 (𝑗 = 𝑚 → ((((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) +Q 𝑆) <Q 𝑟 ↔ (((𝐹𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) +Q 𝑆) <Q 𝑟))
8988cbvrexv 2583 . . . . . . . 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 132 . . . . . . 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 3809 . . . . . . . . 9 (𝑢 = 𝑟 → ((((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) +Q 𝑆) <Q 𝑢 ↔ (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) +Q 𝑆) <Q 𝑟))
9291rexbidv 2374 . . . . . . . 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 2757 . . . . . . 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 408 . . . . . 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 113 . . . . 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 2489 . . . 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 113 . 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 3014 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 102  wb 103  w3a 920   = wceq 1285  wcel 1434  {cab 2069  wral 2353  wrex 2354  {crab 2357  wss 2982  cop 3419   class class class wbr 3805  wf 4948  cfv 4952  (class class class)co 5563  2nd c2nd 5817  1𝑜c1o 6078  [cec 6191  Ncnpi 6576   <N clti 6579   ~Q ceq 6583  Qcnq 6584   +Q cplq 6586  *Qcrq 6588   <Q cltq 6589  Pcnp 6595   +P cpp 6597
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3913  ax-sep 3916  ax-nul 3924  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-iinf 4357
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-tr 3896  df-eprel 4072  df-id 4076  df-po 4079  df-iso 4080  df-iord 4149  df-on 4151  df-suc 4154  df-iom 4360  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-ov 5566  df-oprab 5567  df-mpt2 5568  df-1st 5818  df-2nd 5819  df-recs 5974  df-irdg 6039  df-1o 6085  df-oadd 6089  df-omul 6090  df-er 6193  df-ec 6195  df-qs 6199  df-ni 6608  df-pli 6609  df-mi 6610  df-lti 6611  df-plpq 6648  df-mpq 6649  df-enq 6651  df-nqqs 6652  df-plqqs 6653  df-mqqs 6654  df-1nqqs 6655  df-rq 6656  df-ltnqqs 6657  df-inp 6770  df-iplp 6772
This theorem is referenced by:  caucvgprlemladdrl  6982
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