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Theorem caucvgprprlemexb 6803
Description: Lemma for caucvgprpr 6808. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 15-Jun-2021.)
Hypotheses
Ref Expression
caucvgprpr.f (𝜑𝐹:NP)
caucvgprpr.cau (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛)<P ((𝐹𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑘)<P ((𝐹𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))
caucvgprpr.bnd (𝜑 → ∀𝑚N 𝐴<P (𝐹𝑚))
caucvgprpr.lim 𝐿 = ⟨{𝑙Q ∣ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)}, {𝑢Q ∣ ∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩}⟩
caucvgprprlemexb.q (𝜑𝑄P)
caucvgprprlemexb.r (𝜑𝑅N)
Assertion
Ref Expression
caucvgprprlemexb (𝜑 → (((𝐿 +P 𝑄) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑅) +P 𝑄) → ∃𝑏N (((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P (𝑄 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))<P ((𝐹𝑅) +P 𝑄)))
Distinct variable groups:   𝐴,𝑚   𝑚,𝐹   𝐴,𝑟,𝑚   𝐹,𝑏   𝑘,𝐹,𝑙,𝑛,𝑢   𝐹,𝑟   𝐿,𝑏   𝑘,𝐿   𝑅,𝑏,𝑝,𝑞   𝜑,𝑏   𝑘,𝑝,𝑞,𝑟,𝑙,𝑢   𝑟,𝑏
Allowed substitution hints:   𝜑(𝑢,𝑘,𝑚,𝑛,𝑟,𝑞,𝑝,𝑙)   𝐴(𝑢,𝑘,𝑛,𝑞,𝑝,𝑏,𝑙)   𝑄(𝑢,𝑘,𝑚,𝑛,𝑟,𝑞,𝑝,𝑏,𝑙)   𝑅(𝑢,𝑘,𝑚,𝑛,𝑟,𝑙)   𝐹(𝑞,𝑝)   𝐿(𝑢,𝑚,𝑛,𝑟,𝑞,𝑝,𝑙)

Proof of Theorem caucvgprprlemexb
Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 caucvgprpr.f . . . . . 6 (𝜑𝐹:NP)
2 caucvgprpr.cau . . . . . 6 (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛)<P ((𝐹𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑘)<P ((𝐹𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))
3 caucvgprpr.bnd . . . . . 6 (𝜑 → ∀𝑚N 𝐴<P (𝐹𝑚))
4 caucvgprpr.lim . . . . . 6 𝐿 = ⟨{𝑙Q ∣ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)}, {𝑢Q ∣ ∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩}⟩
51, 2, 3, 4caucvgprprlemclphr 6801 . . . . 5 (𝜑𝐿P)
6 caucvgprprlemexb.r . . . . . 6 (𝜑𝑅N)
7 recnnpr 6644 . . . . . 6 (𝑅N → ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ ∈ P)
86, 7syl 14 . . . . 5 (𝜑 → ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ ∈ P)
9 addclpr 6633 . . . . 5 ((𝐿P ∧ ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ ∈ P) → (𝐿 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ∈ P)
105, 8, 9syl2anc 391 . . . 4 (𝜑 → (𝐿 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ∈ P)
111, 6ffvelrnd 5303 . . . 4 (𝜑 → (𝐹𝑅) ∈ P)
12 caucvgprprlemexb.q . . . 4 (𝜑𝑄P)
13 ltaprg 6715 . . . 4 (((𝐿 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ∈ P ∧ (𝐹𝑅) ∈ P𝑄P) → ((𝐿 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹𝑅) ↔ (𝑄 +P (𝐿 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))<P (𝑄 +P (𝐹𝑅))))
1410, 11, 12, 13syl3anc 1135 . . 3 (𝜑 → ((𝐿 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹𝑅) ↔ (𝑄 +P (𝐿 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))<P (𝑄 +P (𝐹𝑅))))
15 addassprg 6675 . . . . . 6 ((𝑄P𝐿P ∧ ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ ∈ P) → ((𝑄 +P 𝐿) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) = (𝑄 +P (𝐿 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)))
1612, 5, 8, 15syl3anc 1135 . . . . 5 (𝜑 → ((𝑄 +P 𝐿) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) = (𝑄 +P (𝐿 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)))
17 addcomprg 6674 . . . . . . 7 ((𝑄P𝐿P) → (𝑄 +P 𝐿) = (𝐿 +P 𝑄))
1812, 5, 17syl2anc 391 . . . . . 6 (𝜑 → (𝑄 +P 𝐿) = (𝐿 +P 𝑄))
1918oveq1d 5527 . . . . 5 (𝜑 → ((𝑄 +P 𝐿) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) = ((𝐿 +P 𝑄) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))
2016, 19eqtr3d 2074 . . . 4 (𝜑 → (𝑄 +P (𝐿 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)) = ((𝐿 +P 𝑄) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))
21 addcomprg 6674 . . . . 5 ((𝑄P ∧ (𝐹𝑅) ∈ P) → (𝑄 +P (𝐹𝑅)) = ((𝐹𝑅) +P 𝑄))
2212, 11, 21syl2anc 391 . . . 4 (𝜑 → (𝑄 +P (𝐹𝑅)) = ((𝐹𝑅) +P 𝑄))
2320, 22breq12d 3777 . . 3 (𝜑 → ((𝑄 +P (𝐿 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))<P (𝑄 +P (𝐹𝑅)) ↔ ((𝐿 +P 𝑄) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑅) +P 𝑄)))
2414, 23bitrd 177 . 2 (𝜑 → ((𝐿 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹𝑅) ↔ ((𝐿 +P 𝑄) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑅) +P 𝑄)))
251adantr 261 . . . . 5 ((𝜑 ∧ (𝐿 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹𝑅)) → 𝐹:NP)
262adantr 261 . . . . 5 ((𝜑 ∧ (𝐿 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹𝑅)) → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛)<P ((𝐹𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑘)<P ((𝐹𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))
273adantr 261 . . . . 5 ((𝜑 ∧ (𝐿 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹𝑅)) → ∀𝑚N 𝐴<P (𝐹𝑚))
28 nnnq 6518 . . . . . . 7 (𝑅N → [⟨𝑅, 1𝑜⟩] ~QQ)
29 recclnq 6488 . . . . . . 7 ([⟨𝑅, 1𝑜⟩] ~QQ → (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) ∈ Q)
306, 28, 293syl 17 . . . . . 6 (𝜑 → (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) ∈ Q)
3130adantr 261 . . . . 5 ((𝜑 ∧ (𝐿 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹𝑅)) → (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) ∈ Q)
3211adantr 261 . . . . 5 ((𝜑 ∧ (𝐿 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹𝑅)) → (𝐹𝑅) ∈ P)
33 simpr 103 . . . . 5 ((𝜑 ∧ (𝐿 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹𝑅)) → (𝐿 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹𝑅))
3425, 26, 27, 4, 31, 32, 33caucvgprprlemexbt 6802 . . . 4 ((𝜑 ∧ (𝐿 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹𝑅)) → ∃𝑏N (((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹𝑅))
35 ltaprg 6715 . . . . . . . 8 ((𝑓P𝑔PP) → (𝑓<P 𝑔 ↔ ( +P 𝑓)<P ( +P 𝑔)))
3635adantl 262 . . . . . . 7 ((((𝜑 ∧ (𝐿 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹𝑅)) ∧ 𝑏N) ∧ (𝑓P𝑔PP)) → (𝑓<P 𝑔 ↔ ( +P 𝑓)<P ( +P 𝑔)))
3725ffvelrnda 5302 . . . . . . . . 9 (((𝜑 ∧ (𝐿 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹𝑅)) ∧ 𝑏N) → (𝐹𝑏) ∈ P)
38 recnnpr 6644 . . . . . . . . . 10 (𝑏N → ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ ∈ P)
3938adantl 262 . . . . . . . . 9 (((𝜑 ∧ (𝐿 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹𝑅)) ∧ 𝑏N) → ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ ∈ P)
40 addclpr 6633 . . . . . . . . 9 (((𝐹𝑏) ∈ P ∧ ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ ∈ P) → ((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ∈ P)
4137, 39, 40syl2anc 391 . . . . . . . 8 (((𝜑 ∧ (𝐿 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹𝑅)) ∧ 𝑏N) → ((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ∈ P)
426ad2antrr 457 . . . . . . . . 9 (((𝜑 ∧ (𝐿 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹𝑅)) ∧ 𝑏N) → 𝑅N)
4342, 7syl 14 . . . . . . . 8 (((𝜑 ∧ (𝐿 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹𝑅)) ∧ 𝑏N) → ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ ∈ P)
44 addclpr 6633 . . . . . . . 8 ((((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ∈ P ∧ ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ ∈ P) → (((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ∈ P)
4541, 43, 44syl2anc 391 . . . . . . 7 (((𝜑 ∧ (𝐿 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹𝑅)) ∧ 𝑏N) → (((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ∈ P)
4611ad2antrr 457 . . . . . . 7 (((𝜑 ∧ (𝐿 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹𝑅)) ∧ 𝑏N) → (𝐹𝑅) ∈ P)
4712ad2antrr 457 . . . . . . 7 (((𝜑 ∧ (𝐿 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹𝑅)) ∧ 𝑏N) → 𝑄P)
48 addcomprg 6674 . . . . . . . 8 ((𝑓P𝑔P) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
4948adantl 262 . . . . . . 7 ((((𝜑 ∧ (𝐿 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹𝑅)) ∧ 𝑏N) ∧ (𝑓P𝑔P)) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
5036, 45, 46, 47, 49caovord2d 5670 . . . . . 6 (((𝜑 ∧ (𝐿 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹𝑅)) ∧ 𝑏N) → ((((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹𝑅) ↔ ((((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P 𝑄)<P ((𝐹𝑅) +P 𝑄)))
51 addassprg 6675 . . . . . . . 8 ((((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ∈ P ∧ ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ ∈ P𝑄P) → ((((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P 𝑄) = (((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P (⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ +P 𝑄)))
5241, 43, 47, 51syl3anc 1135 . . . . . . 7 (((𝜑 ∧ (𝐿 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹𝑅)) ∧ 𝑏N) → ((((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P 𝑄) = (((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P (⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ +P 𝑄)))
5352breq1d 3774 . . . . . 6 (((𝜑 ∧ (𝐿 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹𝑅)) ∧ 𝑏N) → (((((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P 𝑄)<P ((𝐹𝑅) +P 𝑄) ↔ (((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P (⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ +P 𝑄))<P ((𝐹𝑅) +P 𝑄)))
54 addcomprg 6674 . . . . . . . . 9 ((⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ ∈ P𝑄P) → (⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ +P 𝑄) = (𝑄 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))
5543, 47, 54syl2anc 391 . . . . . . . 8 (((𝜑 ∧ (𝐿 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹𝑅)) ∧ 𝑏N) → (⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ +P 𝑄) = (𝑄 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))
5655oveq2d 5528 . . . . . . 7 (((𝜑 ∧ (𝐿 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹𝑅)) ∧ 𝑏N) → (((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P (⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ +P 𝑄)) = (((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P (𝑄 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)))
5756breq1d 3774 . . . . . 6 (((𝜑 ∧ (𝐿 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹𝑅)) ∧ 𝑏N) → ((((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P (⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ +P 𝑄))<P ((𝐹𝑅) +P 𝑄) ↔ (((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P (𝑄 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))<P ((𝐹𝑅) +P 𝑄)))
5850, 53, 573bitrd 203 . . . . 5 (((𝜑 ∧ (𝐿 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹𝑅)) ∧ 𝑏N) → ((((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹𝑅) ↔ (((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P (𝑄 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))<P ((𝐹𝑅) +P 𝑄)))
5958rexbidva 2323 . . . 4 ((𝜑 ∧ (𝐿 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹𝑅)) → (∃𝑏N (((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹𝑅) ↔ ∃𝑏N (((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P (𝑄 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))<P ((𝐹𝑅) +P 𝑄)))
6034, 59mpbid 135 . . 3 ((𝜑 ∧ (𝐿 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹𝑅)) → ∃𝑏N (((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P (𝑄 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))<P ((𝐹𝑅) +P 𝑄))
6160ex 108 . 2 (𝜑 → ((𝐿 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹𝑅) → ∃𝑏N (((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P (𝑄 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))<P ((𝐹𝑅) +P 𝑄)))
6224, 61sylbird 159 1 (𝜑 → (((𝐿 +P 𝑄) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑅) +P 𝑄) → ∃𝑏N (((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P (𝑄 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))<P ((𝐹𝑅) +P 𝑄)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wb 98  w3a 885   = wceq 1243  wcel 1393  {cab 2026  wral 2306  wrex 2307  {crab 2310  cop 3378   class class class wbr 3764  wf 4898  cfv 4902  (class class class)co 5512  1𝑜c1o 5994  [cec 6104  Ncnpi 6368   <N clti 6371   ~Q ceq 6375  Qcnq 6376   +Q cplq 6378  *Qcrq 6380   <Q cltq 6381  Pcnp 6387   +P cpp 6389  <P cltp 6391
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6400  df-pli 6401  df-mi 6402  df-lti 6403  df-plpq 6440  df-mpq 6441  df-enq 6443  df-nqqs 6444  df-plqqs 6445  df-mqqs 6446  df-1nqqs 6447  df-rq 6448  df-ltnqqs 6449  df-enq0 6520  df-nq0 6521  df-0nq0 6522  df-plq0 6523  df-mq0 6524  df-inp 6562  df-iplp 6564  df-iltp 6566
This theorem is referenced by:  caucvgprprlemaddq  6804
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