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 Description: Lemma for caucvgprpr 7484. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 5-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}⟩}⟩
caucvgprprlemaddq.ex (𝜑 → ∃𝑟N (𝑋 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑟) +P 𝑄))
Assertion
Ref Expression
Distinct variable groups:   𝐴,𝑚   𝑚,𝐹   𝐴,𝑟,𝑚   𝐹,𝑙,𝑟,𝑢,𝑘,𝑛   𝑘,𝐿   𝑄,𝑟   𝑋,𝑟   𝑝,𝑙,𝑞,𝑟,𝑢   𝜑,𝑟   𝑘,𝑝,𝑞
Allowed substitution hints:   𝜑(𝑢,𝑘,𝑚,𝑛,𝑞,𝑝,𝑙)   𝐴(𝑢,𝑘,𝑛,𝑞,𝑝,𝑙)   𝑄(𝑢,𝑘,𝑚,𝑛,𝑞,𝑝,𝑙)   𝐹(𝑞,𝑝)   𝐿(𝑢,𝑚,𝑛,𝑟,𝑞,𝑝,𝑙)   𝑋(𝑢,𝑘,𝑚,𝑛,𝑞,𝑝,𝑙)

Dummy variables 𝑏 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 caucvgprprlemaddq.ex . 2 (𝜑 → ∃𝑟N (𝑋 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑟) +P 𝑄))
2 nfv 1491 . . 3 𝑟𝜑
3 nfcv 2256 . . . 4 𝑟𝑋
4 nfcv 2256 . . . 4 𝑟<P
5 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}⟩}⟩
6 nfre1 2451 . . . . . . . 8 𝑟𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)
7 nfcv 2256 . . . . . . . 8 𝑟Q
86, 7nfrabxy 2586 . . . . . . 7 𝑟{𝑙Q ∣ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)}
9 nfre1 2451 . . . . . . . 8 𝑟𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩
109, 7nfrabxy 2586 . . . . . . 7 𝑟{𝑢Q ∣ ∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩}
118, 10nfop 3689 . . . . . 6 𝑟⟨{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}⟩}⟩
125, 11nfcxfr 2253 . . . . 5 𝑟𝐿
13 nfcv 2256 . . . . 5 𝑟 +P
14 nfcv 2256 . . . . 5 𝑟𝑄
1512, 13, 14nfov 5767 . . . 4 𝑟(𝐿 +P 𝑄)
163, 4, 15nfbr 3942 . . 3 𝑟 𝑋<P (𝐿 +P 𝑄)
17 caucvgprpr.f . . . . . . . . . . . 12 (𝜑𝐹:NP)
1817ad2antrr 477 . . . . . . . . . . 11 (((𝜑 ∧ (𝑟N ∧ (𝑋 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑟) +P 𝑄))) ∧ 𝑏N) → 𝐹:NP)
19 caucvgprpr.cau . . . . . . . . . . . 12 (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}⟩))))
2019ad2antrr 477 . . . . . . . . . . 11 (((𝜑 ∧ (𝑟N ∧ (𝑋 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑟) +P 𝑄))) ∧ 𝑏N) → ∀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}⟩))))
21 simpr 109 . . . . . . . . . . 11 (((𝜑 ∧ (𝑟N ∧ (𝑋 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑟) +P 𝑄))) ∧ 𝑏N) → 𝑏N)
22 simplrl 507 . . . . . . . . . . 11 (((𝜑 ∧ (𝑟N ∧ (𝑋 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑟) +P 𝑄))) ∧ 𝑏N) → 𝑟N)
2318, 20, 21, 22caucvgprprlemnbj 7465 . . . . . . . . . 10 (((𝜑 ∧ (𝑟N ∧ (𝑋 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑟) +P 𝑄))) ∧ 𝑏N) → ¬ (((𝐹𝑏) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑢}⟩) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑢}⟩)<P (𝐹𝑟))
2418, 21ffvelrnd 5522 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑟N ∧ (𝑋 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑟) +P 𝑄))) ∧ 𝑏N) → (𝐹𝑏) ∈ P)
25 recnnpr 7320 . . . . . . . . . . . . . . 15 (𝑏N → ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑢}⟩ ∈ P)
2625adantl 273 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑟N ∧ (𝑋 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑟) +P 𝑄))) ∧ 𝑏N) → ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑢}⟩ ∈ P)
27 addclpr 7309 . . . . . . . . . . . . . 14 (((𝐹𝑏) ∈ P ∧ ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑢}⟩ ∈ P) → ((𝐹𝑏) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑢}⟩) ∈ P)
2824, 26, 27syl2anc 406 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑟N ∧ (𝑋 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑟) +P 𝑄))) ∧ 𝑏N) → ((𝐹𝑏) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑢}⟩) ∈ P)
29 recnnpr 7320 . . . . . . . . . . . . . 14 (𝑟N → ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑢}⟩ ∈ P)
3022, 29syl 14 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑟N ∧ (𝑋 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑟) +P 𝑄))) ∧ 𝑏N) → ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑢}⟩ ∈ P)
31 caucvgprprlemaddq.q . . . . . . . . . . . . . 14 (𝜑𝑄P)
3231ad2antrr 477 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑟N ∧ (𝑋 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑟) +P 𝑄))) ∧ 𝑏N) → 𝑄P)
33 addassprg 7351 . . . . . . . . . . . . 13 ((((𝐹𝑏) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑢}⟩) ∈ P ∧ ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑢}⟩ ∈ P𝑄P) → ((((𝐹𝑏) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑢}⟩) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑢}⟩) +P 𝑄) = (((𝐹𝑏) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑢}⟩) +P (⟨{𝑙𝑙 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 𝑄)))
3428, 30, 32, 33syl3anc 1199 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑟N ∧ (𝑋 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑟) +P 𝑄))) ∧ 𝑏N) → ((((𝐹𝑏) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑢}⟩) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑢}⟩) +P 𝑄) = (((𝐹𝑏) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑢}⟩) +P (⟨{𝑙𝑙 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 𝑄)))
3534breq1d 3907 . . . . . . . . . . 11 (((𝜑 ∧ (𝑟N ∧ (𝑋 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑟) +P 𝑄))) ∧ 𝑏N) → (((((𝐹𝑏) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑢}⟩) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑢}⟩) +P 𝑄)<P ((𝐹𝑟) +P 𝑄) ↔ (((𝐹𝑏) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑢}⟩) +P (⟨{𝑙𝑙 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 𝑄))<P ((𝐹𝑟) +P 𝑄)))
36 ltaprg 7391 . . . . . . . . . . . . 13 ((𝑓P𝑔PP) → (𝑓<P 𝑔 ↔ ( +P 𝑓)<P ( +P 𝑔)))
3736adantl 273 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑟N ∧ (𝑋 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑟) +P 𝑄))) ∧ 𝑏N) ∧ (𝑓P𝑔PP)) → (𝑓<P 𝑔 ↔ ( +P 𝑓)<P ( +P 𝑔)))
38 addclpr 7309 . . . . . . . . . . . . 13 ((((𝐹𝑏) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑢}⟩) ∈ P ∧ ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑢}⟩ ∈ P) → (((𝐹𝑏) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑢}⟩) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑢}⟩) ∈ P)
3928, 30, 38syl2anc 406 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑟N ∧ (𝑋 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑟) +P 𝑄))) ∧ 𝑏N) → (((𝐹𝑏) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑢}⟩) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑢}⟩) ∈ P)
4018, 22ffvelrnd 5522 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑟N ∧ (𝑋 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑟) +P 𝑄))) ∧ 𝑏N) → (𝐹𝑟) ∈ P)
41 addcomprg 7350 . . . . . . . . . . . . 13 ((𝑓P𝑔P) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
4241adantl 273 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑟N ∧ (𝑋 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑟) +P 𝑄))) ∧ 𝑏N) ∧ (𝑓P𝑔P)) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
4337, 39, 40, 32, 42caovord2d 5906 . . . . . . . . . . 11 (((𝜑 ∧ (𝑟N ∧ (𝑋 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑟) +P 𝑄))) ∧ 𝑏N) → ((((𝐹𝑏) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑢}⟩) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑢}⟩)<P (𝐹𝑟) ↔ ((((𝐹𝑏) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑢}⟩) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑢}⟩) +P 𝑄)<P ((𝐹𝑟) +P 𝑄)))
44 addcomprg 7350 . . . . . . . . . . . . . 14 ((𝑄P ∧ ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑢}⟩ ∈ P) → (𝑄 +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑢}⟩) = (⟨{𝑙𝑙 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 𝑄))
4532, 30, 44syl2anc 406 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑟N ∧ (𝑋 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑟) +P 𝑄))) ∧ 𝑏N) → (𝑄 +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑢}⟩) = (⟨{𝑙𝑙 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 𝑄))
4645oveq2d 5756 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑟N ∧ (𝑋 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑟) +P 𝑄))) ∧ 𝑏N) → (((𝐹𝑏) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑢}⟩) +P (𝑄 +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑢}⟩)) = (((𝐹𝑏) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑢}⟩) +P (⟨{𝑙𝑙 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 𝑄)))
4746breq1d 3907 . . . . . . . . . . 11 (((𝜑 ∧ (𝑟N ∧ (𝑋 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑟) +P 𝑄))) ∧ 𝑏N) → ((((𝐹𝑏) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑢}⟩) +P (𝑄 +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑢}⟩))<P ((𝐹𝑟) +P 𝑄) ↔ (((𝐹𝑏) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑢}⟩) +P (⟨{𝑙𝑙 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 𝑄))<P ((𝐹𝑟) +P 𝑄)))
4835, 43, 473bitr4rd 220 . . . . . . . . . 10 (((𝜑 ∧ (𝑟N ∧ (𝑋 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑟) +P 𝑄))) ∧ 𝑏N) → ((((𝐹𝑏) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑢}⟩) +P (𝑄 +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑢}⟩))<P ((𝐹𝑟) +P 𝑄) ↔ (((𝐹𝑏) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑢}⟩) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑢}⟩)<P (𝐹𝑟)))
4923, 48mtbird 645 . . . . . . . . 9 (((𝜑 ∧ (𝑟N ∧ (𝑋 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑟) +P 𝑄))) ∧ 𝑏N) → ¬ (((𝐹𝑏) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑢}⟩) +P (𝑄 +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑢}⟩))<P ((𝐹𝑟) +P 𝑄))
5049nrexdv 2500 . . . . . . . 8 ((𝜑 ∧ (𝑟N ∧ (𝑋 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑟) +P 𝑄))) → ¬ ∃𝑏N (((𝐹𝑏) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑢}⟩) +P (𝑄 +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑢}⟩))<P ((𝐹𝑟) +P 𝑄))
51 breq1 3900 . . . . . . . . . . . . . 14 (𝑝 = 𝑙 → (𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q ) ↔ 𝑙 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )))
5251cbvabv 2239 . . . . . . . . . . . . 13 {𝑝𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )} = {𝑙𝑙 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}
53 breq2 3901 . . . . . . . . . . . . . 14 (𝑞 = 𝑢 → ((*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞 ↔ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑢))
5453cbvabv 2239 . . . . . . . . . . . . 13 {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞} = {𝑢 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑢}
5552, 54opeq12i 3678 . . . . . . . . . . . 12 ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩ = ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑢}⟩
5655oveq2i 5751 . . . . . . . . . . 11 ((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩) = ((𝐹𝑏) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑢}⟩)
57 breq1 3900 . . . . . . . . . . . . . 14 (𝑝 = 𝑙 → (𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q ) ↔ 𝑙 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )))
5857cbvabv 2239 . . . . . . . . . . . . 13 {𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )} = {𝑙𝑙 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}
59 breq2 3901 . . . . . . . . . . . . . 14 (𝑞 = 𝑢 → ((*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞 ↔ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑢))
6059cbvabv 2239 . . . . . . . . . . . . 13 {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞} = {𝑢 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑢}
6158, 60opeq12i 3678 . . . . . . . . . . . 12 ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩ = ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑢}⟩
6261oveq2i 5751 . . . . . . . . . . 11 (𝑄 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩) = (𝑄 +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑢}⟩)
6356, 62oveq12i 5752 . . . . . . . . . 10 (((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}⟩)) = (((𝐹𝑏) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑢}⟩) +P (𝑄 +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑢}⟩))
6463breq1i 3904 . . . . . . . . 9 ((((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) ↔ (((𝐹𝑏) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑢}⟩) +P (𝑄 +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑢}⟩))<P ((𝐹𝑟) +P 𝑄))
6564rexbii 2417 . . . . . . . 8 (∃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) ↔ ∃𝑏N (((𝐹𝑏) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑢}⟩) +P (𝑄 +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑢}⟩))<P ((𝐹𝑟) +P 𝑄))
6650, 65sylnibr 649 . . . . . . 7 ((𝜑 ∧ (𝑟N ∧ (𝑋 +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))
6717adantr 272 . . . . . . . 8 ((𝜑 ∧ (𝑟N ∧ (𝑋 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑟) +P 𝑄))) → 𝐹:NP)
6819adantr 272 . . . . . . . 8 ((𝜑 ∧ (𝑟N ∧ (𝑋 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑟) +P 𝑄))) → ∀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}⟩))))
69 caucvgprpr.bnd . . . . . . . . 9 (𝜑 → ∀𝑚N 𝐴<P (𝐹𝑚))
7069adantr 272 . . . . . . . 8 ((𝜑 ∧ (𝑟N ∧ (𝑋 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑟) +P 𝑄))) → ∀𝑚N 𝐴<P (𝐹𝑚))
7131adantr 272 . . . . . . . 8 ((𝜑 ∧ (𝑟N ∧ (𝑋 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑟) +P 𝑄))) → 𝑄P)
72 simprl 503 . . . . . . . 8 ((𝜑 ∧ (𝑟N ∧ (𝑋 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑟) +P 𝑄))) → 𝑟N)
7367, 68, 70, 5, 71, 72caucvgprprlemexb 7479 . . . . . . 7 ((𝜑 ∧ (𝑟N ∧ (𝑋 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑟) +P 𝑄))) → (((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)))
7466, 73mtod 635 . . . . . 6 ((𝜑 ∧ (𝑟N ∧ (𝑋 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑟) +P 𝑄))) → ¬ ((𝐿 +P 𝑄) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑟) +P 𝑄))
75 simprr 504 . . . . . . 7 ((𝜑 ∧ (𝑟N ∧ (𝑋 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑟) +P 𝑄))) → (𝑋 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑟) +P 𝑄))
76 caucvgprprlemaddq.x . . . . . . . . . 10 (𝜑𝑋P)
7776adantr 272 . . . . . . . . 9 ((𝜑 ∧ (𝑟N ∧ (𝑋 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑟) +P 𝑄))) → 𝑋P)
78 recnnpr 7320 . . . . . . . . . 10 (𝑟N → ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩ ∈ P)
7972, 78syl 14 . . . . . . . . 9 ((𝜑 ∧ (𝑟N ∧ (𝑋 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑟) +P 𝑄))) → ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩ ∈ P)
80 addclpr 7309 . . . . . . . . 9 ((𝑋P ∧ ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩ ∈ P) → (𝑋 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩) ∈ P)
8177, 79, 80syl2anc 406 . . . . . . . 8 ((𝜑 ∧ (𝑟N ∧ (𝑋 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑟) +P 𝑄))) → (𝑋 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩) ∈ P)
8267, 72ffvelrnd 5522 . . . . . . . . 9 ((𝜑 ∧ (𝑟N ∧ (𝑋 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑟) +P 𝑄))) → (𝐹𝑟) ∈ P)
83 addclpr 7309 . . . . . . . . 9 (((𝐹𝑟) ∈ P𝑄P) → ((𝐹𝑟) +P 𝑄) ∈ P)
8482, 71, 83syl2anc 406 . . . . . . . 8 ((𝜑 ∧ (𝑟N ∧ (𝑋 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑟) +P 𝑄))) → ((𝐹𝑟) +P 𝑄) ∈ P)
8517, 19, 69, 5caucvgprprlemcl 7476 . . . . . . . . . . 11 (𝜑𝐿P)
8685adantr 272 . . . . . . . . . 10 ((𝜑 ∧ (𝑟N ∧ (𝑋 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑟) +P 𝑄))) → 𝐿P)
87 addclpr 7309 . . . . . . . . . 10 ((𝐿P𝑄P) → (𝐿 +P 𝑄) ∈ P)
8886, 71, 87syl2anc 406 . . . . . . . . 9 ((𝜑 ∧ (𝑟N ∧ (𝑋 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑟) +P 𝑄))) → (𝐿 +P 𝑄) ∈ P)
89 addclpr 7309 . . . . . . . . 9 (((𝐿 +P 𝑄) ∈ P ∧ ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩ ∈ P) → ((𝐿 +P 𝑄) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩) ∈ P)
9088, 79, 89syl2anc 406 . . . . . . . 8 ((𝜑 ∧ (𝑟N ∧ (𝑋 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑟) +P 𝑄))) → ((𝐿 +P 𝑄) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩) ∈ P)
91 ltsopr 7368 . . . . . . . . 9 <P Or P
92 sowlin 4210 . . . . . . . . 9 ((<P Or P ∧ ((𝑋 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩) ∈ P ∧ ((𝐹𝑟) +P 𝑄) ∈ P ∧ ((𝐿 +P 𝑄) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩) ∈ P)) → ((𝑋 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑟) +P 𝑄) → ((𝑋 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐿 +P 𝑄) +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}⟩)<P ((𝐹𝑟) +P 𝑄))))
9391, 92mpan 418 . . . . . . . 8 (((𝑋 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩) ∈ P ∧ ((𝐹𝑟) +P 𝑄) ∈ P ∧ ((𝐿 +P 𝑄) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩) ∈ P) → ((𝑋 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑟) +P 𝑄) → ((𝑋 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐿 +P 𝑄) +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}⟩)<P ((𝐹𝑟) +P 𝑄))))
9481, 84, 90, 93syl3anc 1199 . . . . . . 7 ((𝜑 ∧ (𝑟N ∧ (𝑋 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑟) +P 𝑄))) → ((𝑋 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑟) +P 𝑄) → ((𝑋 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐿 +P 𝑄) +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}⟩)<P ((𝐹𝑟) +P 𝑄))))
9575, 94mpd 13 . . . . . 6 ((𝜑 ∧ (𝑟N ∧ (𝑋 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑟) +P 𝑄))) → ((𝑋 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐿 +P 𝑄) +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}⟩)<P ((𝐹𝑟) +P 𝑄)))
9674, 95ecased 1310 . . . . 5 ((𝜑 ∧ (𝑟N ∧ (𝑋 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑟) +P 𝑄))) → (𝑋 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐿 +P 𝑄) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩))
9736adantl 273 . . . . . 6 (((𝜑 ∧ (𝑟N ∧ (𝑋 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑟) +P 𝑄))) ∧ (𝑓P𝑔PP)) → (𝑓<P 𝑔 ↔ ( +P 𝑓)<P ( +P 𝑔)))
9841adantl 273 . . . . . 6 (((𝜑 ∧ (𝑟N ∧ (𝑋 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑟) +P 𝑄))) ∧ (𝑓P𝑔P)) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
9997, 77, 88, 79, 98caovord2d 5906 . . . . 5 ((𝜑 ∧ (𝑟N ∧ (𝑋 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑟) +P 𝑄))) → (𝑋<P (𝐿 +P 𝑄) ↔ (𝑋 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐿 +P 𝑄) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)))
10096, 99mpbird 166 . . . 4 ((𝜑 ∧ (𝑟N ∧ (𝑋 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑟) +P 𝑄))) → 𝑋<P (𝐿 +P 𝑄))
101100exp32 360 . . 3 (𝜑 → (𝑟N → ((𝑋 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑟) +P 𝑄) → 𝑋<P (𝐿 +P 𝑄))))
1022, 16, 101rexlimd 2521 . 2 (𝜑 → (∃𝑟N (𝑋 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑟) +P 𝑄) → 𝑋<P (𝐿 +P 𝑄)))
1031, 102mpd 13 1 (𝜑𝑋<P (𝐿 +P 𝑄))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 680   ∧ w3a 945   = wceq 1314   ∈ wcel 1463  {cab 2101  ∀wral 2391  ∃wrex 2392  {crab 2395  ⟨cop 3498   class class class wbr 3897   Or wor 4185  ⟶wf 5087  ‘cfv 5091  (class class class)co 5740  1oc1o 6272  [cec 6393  Ncnpi 7044
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