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Theorem caucvgprprlemaddq 7211
Description: Lemma for caucvgprpr 7215. 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 1464 . . 3 𝑟𝜑
3 nfcv 2225 . . . 4 𝑟𝑋
4 nfcv 2225 . . . 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 2415 . . . . . . . 8 𝑟𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)
7 nfcv 2225 . . . . . . . 8 𝑟Q
86, 7nfrabxy 2543 . . . . . . 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 2415 . . . . . . . 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 2543 . . . . . . 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 3621 . . . . . 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 2222 . . . . 5 𝑟𝐿
13 nfcv 2225 . . . . 5 𝑟 +P
14 nfcv 2225 . . . . 5 𝑟𝑄
1512, 13, 14nfov 5636 . . . 4 𝑟(𝐿 +P 𝑄)
163, 4, 15nfbr 3864 . . 3 𝑟 𝑋<P (𝐿 +P 𝑄)
17 caucvgprpr.f . . . . . . . . . . . 12 (𝜑𝐹:NP)
1817ad2antrr 472 . . . . . . . . . . 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 472 . . . . . . . . . . 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 108 . . . . . . . . . . 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 502 . . . . . . . . . . 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 7196 . . . . . . . . . 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 5398 . . . . . . . . . . . . . 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 7051 . . . . . . . . . . . . . . 15 (𝑏N → ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ ∈ P)
2625adantl 271 . . . . . . . . . . . . . 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 7040 . . . . . . . . . . . . . 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 403 . . . . . . . . . . . . 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 7051 . . . . . . . . . . . . . 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 472 . . . . . . . . . . . . 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 7082 . . . . . . . . . . . . 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 1172 . . . . . . . . . . . 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 3830 . . . . . . . . . . 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 7122 . . . . . . . . . . . . 13 ((𝑓P𝑔PP) → (𝑓<P 𝑔 ↔ ( +P 𝑓)<P ( +P 𝑔)))
3736adantl 271 . . . . . . . . . . . 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 7040 . . . . . . . . . . . . 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 403 . . . . . . . . . . . 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 5398 . . . . . . . . . . . 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 7081 . . . . . . . . . . . . 13 ((𝑓P𝑔P) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
4241adantl 271 . . . . . . . . . . . 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 5771 . . . . . . . . . . 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 7081 . . . . . . . . . . . . . 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 403 . . . . . . . . . . . . 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 5629 . . . . . . . . . . . 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 3830 . . . . . . . . . . 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 219 . . . . . . . . . 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 631 . . . . . . . . 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 2462 . . . . . . . 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 3823 . . . . . . . . . . . . . 14 (𝑝 = 𝑙 → (𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) ↔ 𝑙 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )))
5251cbvabv 2208 . . . . . . . . . . . . 13 {𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )} = {𝑙𝑙 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}
53 breq2 3824 . . . . . . . . . . . . . 14 (𝑞 = 𝑢 → ((*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞 ↔ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑢))
5453cbvabv 2208 . . . . . . . . . . . . 13 {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞} = {𝑢 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑢}
5552, 54opeq12i 3610 . . . . . . . . . . . 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 5624 . . . . . . . . . . 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 3823 . . . . . . . . . . . . . 14 (𝑝 = 𝑙 → (𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) ↔ 𝑙 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )))
5857cbvabv 2208 . . . . . . . . . . . . 13 {𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )} = {𝑙𝑙 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}
59 breq2 3824 . . . . . . . . . . . . . 14 (𝑞 = 𝑢 → ((*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞 ↔ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑢))
6059cbvabv 2208 . . . . . . . . . . . . 13 {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞} = {𝑢 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑢}
6158, 60opeq12i 3610 . . . . . . . . . . . 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 5624 . . . . . . . . . . 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 5625 . . . . . . . . . 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 3827 . . . . . . . . 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 2381 . . . . . . . 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 635 . . . . . . 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 270 . . . . . . . 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 270 . . . . . . . 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 270 . . . . . . . 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 270 . . . . . . . 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 498 . . . . . . . 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 7210 . . . . . . 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 622 . . . . . 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 499 . . . . . . 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 270 . . . . . . . . 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 7051 . . . . . . . . . 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 7040 . . . . . . . . 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 403 . . . . . . . 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 5398 . . . . . . . . 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 7040 . . . . . . . . 9 (((𝐹𝑟) ∈ P𝑄P) → ((𝐹𝑟) +P 𝑄) ∈ P)
8482, 71, 83syl2anc 403 . . . . . . . 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 7207 . . . . . . . . . . 11 (𝜑𝐿P)
8685adantr 270 . . . . . . . . . 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 7040 . . . . . . . . . 10 ((𝐿P𝑄P) → (𝐿 +P 𝑄) ∈ P)
8886, 71, 87syl2anc 403 . . . . . . . . 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 7040 . . . . . . . . 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 403 . . . . . . . 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 7099 . . . . . . . . 9 <P Or P
92 sowlin 4121 . . . . . . . . 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 415 . . . . . . . 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 1172 . . . . . . 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 1283 . . . . 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 271 . . . . . 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 271 . . . . . 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 5771 . . . . 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 165 . . . 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 357 . . 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 2482 . 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 102  wb 103  wo 662  w3a 922   = wceq 1287  wcel 1436  {cab 2071  wral 2355  wrex 2356  {crab 2359  cop 3434   class class class wbr 3820   Or wor 4096  wf 4977  cfv 4981  (class class class)co 5613  1𝑜c1o 6128  [cec 6242  Ncnpi 6775   <N clti 6778   ~Q ceq 6782  Qcnq 6783   +Q cplq 6785  *Qcrq 6787   <Q cltq 6788  Pcnp 6794   +P cpp 6796  <P cltp 6798
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-coll 3929  ax-sep 3932  ax-nul 3940  ax-pow 3984  ax-pr 4010  ax-un 4234  ax-setind 4326  ax-iinf 4376
This theorem depends on definitions:  df-bi 115  df-dc 779  df-3or 923  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-ral 2360  df-rex 2361  df-reu 2362  df-rab 2364  df-v 2617  df-sbc 2830  df-csb 2923  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-nul 3276  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3637  df-int 3672  df-iun 3715  df-br 3821  df-opab 3875  df-mpt 3876  df-tr 3912  df-eprel 4090  df-id 4094  df-po 4097  df-iso 4098  df-iord 4167  df-on 4169  df-suc 4172  df-iom 4379  df-xp 4417  df-rel 4418  df-cnv 4419  df-co 4420  df-dm 4421  df-rn 4422  df-res 4423  df-ima 4424  df-iota 4946  df-fun 4983  df-fn 4984  df-f 4985  df-f1 4986  df-fo 4987  df-f1o 4988  df-fv 4989  df-ov 5616  df-oprab 5617  df-mpt2 5618  df-1st 5868  df-2nd 5869  df-recs 6024  df-irdg 6089  df-1o 6135  df-2o 6136  df-oadd 6139  df-omul 6140  df-er 6244  df-ec 6246  df-qs 6250  df-ni 6807  df-pli 6808  df-mi 6809  df-lti 6810  df-plpq 6847  df-mpq 6848  df-enq 6850  df-nqqs 6851  df-plqqs 6852  df-mqqs 6853  df-1nqqs 6854  df-rq 6855  df-ltnqqs 6856  df-enq0 6927  df-nq0 6928  df-0nq0 6929  df-plq0 6930  df-mq0 6931  df-inp 6969  df-iplp 6971  df-iltp 6973
This theorem is referenced by:  caucvgprprlem1  7212
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