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Theorem caucvgprprlemaddq 7480
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.x (𝜑𝑋P)
caucvgprprlemaddq.q (𝜑𝑄P)
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
caucvgprprlemaddq (𝜑𝑋<P (𝐿 +P 𝑄))
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)

Proof of Theorem caucvgprprlemaddq
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   <N clti 7047   ~Q ceq 7051  Qcnq 7052   +Q cplq 7054  *Qcrq 7056   <Q cltq 7057  Pcnp 7063   +P cpp 7065  <P cltp 7067
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-eprel 4179  df-id 4183  df-po 4186  df-iso 4187  df-iord 4256  df-on 4258  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-irdg 6233  df-1o 6279  df-2o 6280  df-oadd 6283  df-omul 6284  df-er 6395  df-ec 6397  df-qs 6401  df-ni 7076  df-pli 7077  df-mi 7078  df-lti 7079  df-plpq 7116  df-mpq 7117  df-enq 7119  df-nqqs 7120  df-plqqs 7121  df-mqqs 7122  df-1nqqs 7123  df-rq 7124  df-ltnqqs 7125  df-enq0 7196  df-nq0 7197  df-0nq0 7198  df-plq0 7199  df-mq0 7200  df-inp 7238  df-iplp 7240  df-iltp 7242
This theorem is referenced by:  caucvgprprlem1  7481
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