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Theorem caucvgprprlemaddq 7927

Description: Lemma for caucvgprpr 7931. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 5-Jun-2021.)

**Hypotheses**
Ref	Expression
caucvgprpr.f	⊢ (𝜑 → 𝐹:N⟶P)
caucvgprpr.cau	⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <_N 𝑘 → ((𝐹‘𝑛)<_P ((𝐹‘𝑘) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_o⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_o⟩] ~_Q ) <_Q 𝑢}⟩) ∧ (𝐹‘𝑘)<_P ((𝐹‘𝑛) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_o⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_o⟩] ~_Q ) <_Q 𝑢}⟩))))
caucvgprpr.bnd	⊢ (𝜑 → ∀𝑚 ∈ N 𝐴<_P (𝐹‘𝑚))
caucvgprpr.lim	⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <_Q (𝑙 +_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q ))}, {𝑞 ∣ (𝑙 +_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )) <_Q 𝑞}⟩<_P (𝐹‘𝑟)}, {𝑢 ∈ Q ∣ ∃𝑟 ∈ N ((𝐹‘𝑟) +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩)<_P ⟨{𝑝 ∣ 𝑝 <_Q 𝑢}, {𝑞 ∣ 𝑢 <_Q 𝑞}⟩}⟩
caucvgprprlemaddq.x	⊢ (𝜑 → 𝑋 ∈ P)
caucvgprprlemaddq.q	⊢ (𝜑 → 𝑄 ∈ P)
caucvgprprlemaddq.ex	⊢ (𝜑 → ∃𝑟 ∈ N (𝑋 +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩)<_P ((𝐹‘𝑟) +_P 𝑄))

**Assertion**
Ref	Expression
caucvgprprlemaddq	⊢ (𝜑 → 𝑋<_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 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		caucvgprprlemaddq.ex	. 2 ⊢ (𝜑 → ∃𝑟 ∈ N (𝑋 +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩)<_P ((𝐹‘𝑟) +_P 𝑄))
2		nfv 1576	. . 3 ⊢ Ⅎ𝑟𝜑
3		nfcv 2374	. . . 4 ⊢ Ⅎ𝑟𝑋
4		nfcv 2374	. . . 4 ⊢ Ⅎ𝑟<_P
5		caucvgprpr.lim	. . . . . 6 ⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <_Q (𝑙 +_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q ))}, {𝑞 ∣ (𝑙 +_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )) <_Q 𝑞}⟩<_P (𝐹‘𝑟)}, {𝑢 ∈ Q ∣ ∃𝑟 ∈ N ((𝐹‘𝑟) +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩)<_P ⟨{𝑝 ∣ 𝑝 <_Q 𝑢}, {𝑞 ∣ 𝑢 <_Q 𝑞}⟩}⟩
6		nfre1 2575	. . . . . . . 8 ⊢ Ⅎ𝑟∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <_Q (𝑙 +_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q ))}, {𝑞 ∣ (𝑙 +_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )) <_Q 𝑞}⟩<_P (𝐹‘𝑟)
7		nfcv 2374	. . . . . . . 8 ⊢ Ⅎ𝑟Q
8	6, 7	nfrabw 2714	. . . . . . 7 ⊢ Ⅎ𝑟{𝑙 ∈ Q ∣ ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <_Q (𝑙 +_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q ))}, {𝑞 ∣ (𝑙 +_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )) <_Q 𝑞}⟩<_P (𝐹‘𝑟)}
9		nfre1 2575	. . . . . . . 8 ⊢ Ⅎ𝑟∃𝑟 ∈ N ((𝐹‘𝑟) +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩)<_P ⟨{𝑝 ∣ 𝑝 <_Q 𝑢}, {𝑞 ∣ 𝑢 <_Q 𝑞}⟩
10	9, 7	nfrabw 2714	. . . . . . 7 ⊢ Ⅎ𝑟{𝑢 ∈ Q ∣ ∃𝑟 ∈ N ((𝐹‘𝑟) +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩)<_P ⟨{𝑝 ∣ 𝑝 <_Q 𝑢}, {𝑞 ∣ 𝑢 <_Q 𝑞}⟩}
11	8, 10	nfop 3878	. . . . . 6 ⊢ Ⅎ𝑟⟨{𝑙 ∈ Q ∣ ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <_Q (𝑙 +_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q ))}, {𝑞 ∣ (𝑙 +_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )) <_Q 𝑞}⟩<_P (𝐹‘𝑟)}, {𝑢 ∈ Q ∣ ∃𝑟 ∈ N ((𝐹‘𝑟) +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩)<_P ⟨{𝑝 ∣ 𝑝 <_Q 𝑢}, {𝑞 ∣ 𝑢 <_Q 𝑞}⟩}⟩
12	5, 11	nfcxfr 2371	. . . . 5 ⊢ Ⅎ𝑟𝐿
13		nfcv 2374	. . . . 5 ⊢ Ⅎ𝑟 +_P
14		nfcv 2374	. . . . 5 ⊢ Ⅎ𝑟𝑄
15	12, 13, 14	nfov 6047	. . . 4 ⊢ Ⅎ𝑟(𝐿 +_P 𝑄)
16	3, 4, 15	nfbr 4135	. . 3 ⊢ Ⅎ𝑟 𝑋<_P (𝐿 +_P 𝑄)
17		caucvgprpr.f	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:N⟶P)
18	17	ad2antrr 488	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑟 ∈ N ∧ (𝑋 +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩)<_P ((𝐹‘𝑟) +_P 𝑄))) ∧ 𝑏 ∈ N) → 𝐹:N⟶P)
19		caucvgprpr.cau	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <_N 𝑘 → ((𝐹‘𝑛)<_P ((𝐹‘𝑘) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_o⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_o⟩] ~_Q ) <_Q 𝑢}⟩) ∧ (𝐹‘𝑘)<_P ((𝐹‘𝑛) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_o⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_o⟩] ~_Q ) <_Q 𝑢}⟩))))
20	19	ad2antrr 488	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑟 ∈ N ∧ (𝑋 +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩)<_P ((𝐹‘𝑟) +_P 𝑄))) ∧ 𝑏 ∈ N) → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <_N 𝑘 → ((𝐹‘𝑛)<_P ((𝐹‘𝑘) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_o⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_o⟩] ~_Q ) <_Q 𝑢}⟩) ∧ (𝐹‘𝑘)<_P ((𝐹‘𝑛) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_o⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_o⟩] ~_Q ) <_Q 𝑢}⟩))))
21		simpr 110	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑟 ∈ N ∧ (𝑋 +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩)<_P ((𝐹‘𝑟) +_P 𝑄))) ∧ 𝑏 ∈ N) → 𝑏 ∈ N)
22		simplrl 537	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑟 ∈ N ∧ (𝑋 +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩)<_P ((𝐹‘𝑟) +_P 𝑄))) ∧ 𝑏 ∈ N) → 𝑟 ∈ N)
23	18, 20, 21, 22	caucvgprprlemnbj 7912	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑟 ∈ N ∧ (𝑋 +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩)<_P ((𝐹‘𝑟) +_P 𝑄))) ∧ 𝑏 ∈ N) → ¬ (((𝐹‘𝑏) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑏, 1_o⟩] ~_Q ) <_Q 𝑢}⟩) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑢}⟩)<_P (𝐹‘𝑟))
24	18, 21	ffvelcdmd 5783	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑟 ∈ N ∧ (𝑋 +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩)<_P ((𝐹‘𝑟) +_P 𝑄))) ∧ 𝑏 ∈ N) → (𝐹‘𝑏) ∈ P)
25		recnnpr 7767	. . . . . . . . . . . . . . 15 ⊢ (𝑏 ∈ N → ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑏, 1_o⟩] ~_Q ) <_Q 𝑢}⟩ ∈ P)
26	25	adantl 277	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑟 ∈ N ∧ (𝑋 +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩)<_P ((𝐹‘𝑟) +_P 𝑄))) ∧ 𝑏 ∈ N) → ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑏, 1_o⟩] ~_Q ) <_Q 𝑢}⟩ ∈ P)
27		addclpr 7756	. . . . . . . . . . . . . 14 ⊢ (((𝐹‘𝑏) ∈ P ∧ ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑏, 1_o⟩] ~_Q ) <_Q 𝑢}⟩ ∈ P) → ((𝐹‘𝑏) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑏, 1_o⟩] ~_Q ) <_Q 𝑢}⟩) ∈ P)
28	24, 26, 27	syl2anc 411	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑟 ∈ N ∧ (𝑋 +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩)<_P ((𝐹‘𝑟) +_P 𝑄))) ∧ 𝑏 ∈ N) → ((𝐹‘𝑏) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑏, 1_o⟩] ~_Q ) <_Q 𝑢}⟩) ∈ P)
29		recnnpr 7767	. . . . . . . . . . . . . 14 ⊢ (𝑟 ∈ N → ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑢}⟩ ∈ P)
30	22, 29	syl 14	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑟 ∈ N ∧ (𝑋 +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩)<_P ((𝐹‘𝑟) +_P 𝑄))) ∧ 𝑏 ∈ N) → ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑢}⟩ ∈ P)
31		caucvgprprlemaddq.q	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑄 ∈ P)
32	31	ad2antrr 488	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑟 ∈ N ∧ (𝑋 +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩)<_P ((𝐹‘𝑟) +_P 𝑄))) ∧ 𝑏 ∈ N) → 𝑄 ∈ P)
33		addassprg 7798	. . . . . . . . . . . . 13 ⊢ ((((𝐹‘𝑏) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑏, 1_o⟩] ~_Q ) <_Q 𝑢}⟩) ∈ P ∧ ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑢}⟩ ∈ P ∧ 𝑄 ∈ P) → ((((𝐹‘𝑏) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑏, 1_o⟩] ~_Q ) <_Q 𝑢}⟩) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑢}⟩) +_P 𝑄) = (((𝐹‘𝑏) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑏, 1_o⟩] ~_Q ) <_Q 𝑢}⟩) +_P (⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑢}⟩ +_P 𝑄)))
34	28, 30, 32, 33	syl3anc 1273	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑟 ∈ N ∧ (𝑋 +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩)<_P ((𝐹‘𝑟) +_P 𝑄))) ∧ 𝑏 ∈ N) → ((((𝐹‘𝑏) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑏, 1_o⟩] ~_Q ) <_Q 𝑢}⟩) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑢}⟩) +_P 𝑄) = (((𝐹‘𝑏) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑏, 1_o⟩] ~_Q ) <_Q 𝑢}⟩) +_P (⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑢}⟩ +_P 𝑄)))
35	34	breq1d 4098	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑟 ∈ N ∧ (𝑋 +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩)<_P ((𝐹‘𝑟) +_P 𝑄))) ∧ 𝑏 ∈ N) → (((((𝐹‘𝑏) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑏, 1_o⟩] ~_Q ) <_Q 𝑢}⟩) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑢}⟩) +_P 𝑄)<_P ((𝐹‘𝑟) +_P 𝑄) ↔ (((𝐹‘𝑏) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑏, 1_o⟩] ~_Q ) <_Q 𝑢}⟩) +_P (⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑢}⟩ +_P 𝑄))<_P ((𝐹‘𝑟) +_P 𝑄)))
36		ltaprg 7838	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P ∧ ℎ ∈ P) → (𝑓<_P 𝑔 ↔ (ℎ +_P 𝑓)<_P (ℎ +_P 𝑔)))
37	36	adantl 277	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑟 ∈ N ∧ (𝑋 +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩)<_P ((𝐹‘𝑟) +_P 𝑄))) ∧ 𝑏 ∈ N) ∧ (𝑓 ∈ P ∧ 𝑔 ∈ P ∧ ℎ ∈ P)) → (𝑓<_P 𝑔 ↔ (ℎ +_P 𝑓)<_P (ℎ +_P 𝑔)))
38		addclpr 7756	. . . . . . . . . . . . 13 ⊢ ((((𝐹‘𝑏) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑏, 1_o⟩] ~_Q ) <_Q 𝑢}⟩) ∈ P ∧ ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑢}⟩ ∈ P) → (((𝐹‘𝑏) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑏, 1_o⟩] ~_Q ) <_Q 𝑢}⟩) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑢}⟩) ∈ P)
39	28, 30, 38	syl2anc 411	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑟 ∈ N ∧ (𝑋 +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩)<_P ((𝐹‘𝑟) +_P 𝑄))) ∧ 𝑏 ∈ N) → (((𝐹‘𝑏) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑏, 1_o⟩] ~_Q ) <_Q 𝑢}⟩) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑢}⟩) ∈ P)
40	18, 22	ffvelcdmd 5783	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑟 ∈ N ∧ (𝑋 +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩)<_P ((𝐹‘𝑟) +_P 𝑄))) ∧ 𝑏 ∈ N) → (𝐹‘𝑟) ∈ P)
41		addcomprg 7797	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P) → (𝑓 +_P 𝑔) = (𝑔 +_P 𝑓))
42	41	adantl 277	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑟 ∈ N ∧ (𝑋 +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩)<_P ((𝐹‘𝑟) +_P 𝑄))) ∧ 𝑏 ∈ N) ∧ (𝑓 ∈ P ∧ 𝑔 ∈ P)) → (𝑓 +_P 𝑔) = (𝑔 +_P 𝑓))
43	37, 39, 40, 32, 42	caovord2d 6191	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑟 ∈ N ∧ (𝑋 +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩)<_P ((𝐹‘𝑟) +_P 𝑄))) ∧ 𝑏 ∈ N) → ((((𝐹‘𝑏) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑏, 1_o⟩] ~_Q ) <_Q 𝑢}⟩) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑢}⟩)<_P (𝐹‘𝑟) ↔ ((((𝐹‘𝑏) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑏, 1_o⟩] ~_Q ) <_Q 𝑢}⟩) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑢}⟩) +_P 𝑄)<_P ((𝐹‘𝑟) +_P 𝑄)))
44		addcomprg 7797	. . . . . . . . . . . . . 14 ⊢ ((𝑄 ∈ P ∧ ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑢}⟩ ∈ P) → (𝑄 +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑢}⟩) = (⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑢}⟩ +_P 𝑄))
45	32, 30, 44	syl2anc 411	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑟 ∈ N ∧ (𝑋 +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩)<_P ((𝐹‘𝑟) +_P 𝑄))) ∧ 𝑏 ∈ N) → (𝑄 +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑢}⟩) = (⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑢}⟩ +_P 𝑄))
46	45	oveq2d 6033	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑟 ∈ N ∧ (𝑋 +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩)<_P ((𝐹‘𝑟) +_P 𝑄))) ∧ 𝑏 ∈ N) → (((𝐹‘𝑏) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑏, 1_o⟩] ~_Q ) <_Q 𝑢}⟩) +_P (𝑄 +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑢}⟩)) = (((𝐹‘𝑏) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑏, 1_o⟩] ~_Q ) <_Q 𝑢}⟩) +_P (⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑢}⟩ +_P 𝑄)))
47	46	breq1d 4098	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑟 ∈ N ∧ (𝑋 +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩)<_P ((𝐹‘𝑟) +_P 𝑄))) ∧ 𝑏 ∈ N) → ((((𝐹‘𝑏) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑏, 1_o⟩] ~_Q ) <_Q 𝑢}⟩) +_P (𝑄 +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑢}⟩))<_P ((𝐹‘𝑟) +_P 𝑄) ↔ (((𝐹‘𝑏) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑏, 1_o⟩] ~_Q ) <_Q 𝑢}⟩) +_P (⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑢}⟩ +_P 𝑄))<_P ((𝐹‘𝑟) +_P 𝑄)))
48	35, 43, 47	3bitr4rd 221	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑟 ∈ N ∧ (𝑋 +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩)<_P ((𝐹‘𝑟) +_P 𝑄))) ∧ 𝑏 ∈ N) → ((((𝐹‘𝑏) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑏, 1_o⟩] ~_Q ) <_Q 𝑢}⟩) +_P (𝑄 +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑢}⟩))<_P ((𝐹‘𝑟) +_P 𝑄) ↔ (((𝐹‘𝑏) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑏, 1_o⟩] ~_Q ) <_Q 𝑢}⟩) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑢}⟩)<_P (𝐹‘𝑟)))
49	23, 48	mtbird 679	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑟 ∈ N ∧ (𝑋 +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩)<_P ((𝐹‘𝑟) +_P 𝑄))) ∧ 𝑏 ∈ N) → ¬ (((𝐹‘𝑏) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑏, 1_o⟩] ~_Q ) <_Q 𝑢}⟩) +_P (𝑄 +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑢}⟩))<_P ((𝐹‘𝑟) +_P 𝑄))
50	49	nrexdv 2625	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑟 ∈ N ∧ (𝑋 +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩)<_P ((𝐹‘𝑟) +_P 𝑄))) → ¬ ∃𝑏 ∈ N (((𝐹‘𝑏) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑏, 1_o⟩] ~_Q ) <_Q 𝑢}⟩) +_P (𝑄 +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑢}⟩))<_P ((𝐹‘𝑟) +_P 𝑄))
51		breq1 4091	. . . . . . . . . . . . . 14 ⊢ (𝑝 = 𝑙 → (𝑝 <_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q ) ↔ 𝑙 <_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )))
52	51	cbvabv 2356	. . . . . . . . . . . . 13 ⊢ {𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )} = {𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )}
53		breq2 4092	. . . . . . . . . . . . . 14 ⊢ (𝑞 = 𝑢 → ((_Q‘[⟨𝑏, 1_o⟩] ~_Q ) <_Q 𝑞 ↔ (_Q‘[⟨𝑏, 1_o⟩] ~_Q ) <_Q 𝑢))
54	53	cbvabv 2356	. . . . . . . . . . . . 13 ⊢ {𝑞 ∣ (_Q‘[⟨𝑏, 1_o⟩] ~_Q ) <_Q 𝑞} = {𝑢 ∣ (_Q‘[⟨𝑏, 1_o⟩] ~_Q ) <_Q 𝑢}
55	52, 54	opeq12i 3867	. . . . . . . . . . . 12 ⊢ ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑏, 1_o⟩] ~_Q ) <_Q 𝑞}⟩ = ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑏, 1_o⟩] ~_Q ) <_Q 𝑢}⟩
56	55	oveq2i 6028	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑏) +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑏, 1_o⟩] ~_Q ) <_Q 𝑞}⟩) = ((𝐹‘𝑏) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑏, 1_o⟩] ~_Q ) <_Q 𝑢}⟩)
57		breq1 4091	. . . . . . . . . . . . . 14 ⊢ (𝑝 = 𝑙 → (𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) ↔ 𝑙 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )))
58	57	cbvabv 2356	. . . . . . . . . . . . 13 ⊢ {𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )} = {𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}
59		breq2 4092	. . . . . . . . . . . . . 14 ⊢ (𝑞 = 𝑢 → ((_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞 ↔ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑢))
60	59	cbvabv 2356	. . . . . . . . . . . . 13 ⊢ {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞} = {𝑢 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑢}
61	58, 60	opeq12i 3867	. . . . . . . . . . . 12 ⊢ ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩ = ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑢}⟩
62	61	oveq2i 6028	. . . . . . . . . . 11 ⊢ (𝑄 +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩) = (𝑄 +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑢}⟩)
63	56, 62	oveq12i 6029	. . . . . . . . . 10 ⊢ (((𝐹‘𝑏) +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑏, 1_o⟩] ~_Q ) <_Q 𝑞}⟩) +_P (𝑄 +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩)) = (((𝐹‘𝑏) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑏, 1_o⟩] ~_Q ) <_Q 𝑢}⟩) +_P (𝑄 +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑢}⟩))
64	63	breq1i 4095	. . . . . . . . 9 ⊢ ((((𝐹‘𝑏) +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑏, 1_o⟩] ~_Q ) <_Q 𝑞}⟩) +_P (𝑄 +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩))<_P ((𝐹‘𝑟) +_P 𝑄) ↔ (((𝐹‘𝑏) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑏, 1_o⟩] ~_Q ) <_Q 𝑢}⟩) +_P (𝑄 +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑢}⟩))<_P ((𝐹‘𝑟) +_P 𝑄))
65	64	rexbii 2539	. . . . . . . 8 ⊢ (∃𝑏 ∈ N (((𝐹‘𝑏) +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑏, 1_o⟩] ~_Q ) <_Q 𝑞}⟩) +_P (𝑄 +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩))<_P ((𝐹‘𝑟) +_P 𝑄) ↔ ∃𝑏 ∈ N (((𝐹‘𝑏) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑏, 1_o⟩] ~_Q ) <_Q 𝑢}⟩) +_P (𝑄 +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑢}⟩))<_P ((𝐹‘𝑟) +_P 𝑄))
66	50, 65	sylnibr 683	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑟 ∈ N ∧ (𝑋 +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩)<_P ((𝐹‘𝑟) +_P 𝑄))) → ¬ ∃𝑏 ∈ N (((𝐹‘𝑏) +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑏, 1_o⟩] ~_Q ) <_Q 𝑞}⟩) +_P (𝑄 +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩))<_P ((𝐹‘𝑟) +_P 𝑄))
67	17	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑟 ∈ N ∧ (𝑋 +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩)<_P ((𝐹‘𝑟) +_P 𝑄))) → 𝐹:N⟶P)
68	19	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑟 ∈ N ∧ (𝑋 +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩)<_P ((𝐹‘𝑟) +_P 𝑄))) → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <_N 𝑘 → ((𝐹‘𝑛)<_P ((𝐹‘𝑘) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_o⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_o⟩] ~_Q ) <_Q 𝑢}⟩) ∧ (𝐹‘𝑘)<_P ((𝐹‘𝑛) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_o⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_o⟩] ~_Q ) <_Q 𝑢}⟩))))
69		caucvgprpr.bnd	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑚 ∈ N 𝐴<_P (𝐹‘𝑚))
70	69	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑟 ∈ N ∧ (𝑋 +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩)<_P ((𝐹‘𝑟) +_P 𝑄))) → ∀𝑚 ∈ N 𝐴<_P (𝐹‘𝑚))
71	31	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑟 ∈ N ∧ (𝑋 +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩)<_P ((𝐹‘𝑟) +_P 𝑄))) → 𝑄 ∈ P)
72		simprl 531	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑟 ∈ N ∧ (𝑋 +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩)<_P ((𝐹‘𝑟) +_P 𝑄))) → 𝑟 ∈ N)
73	67, 68, 70, 5, 71, 72	caucvgprprlemexb 7926	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑟 ∈ N ∧ (𝑋 +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩)<_P ((𝐹‘𝑟) +_P 𝑄))) → (((𝐿 +_P 𝑄) +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩)<_P ((𝐹‘𝑟) +_P 𝑄) → ∃𝑏 ∈ N (((𝐹‘𝑏) +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑏, 1_o⟩] ~_Q ) <_Q 𝑞}⟩) +_P (𝑄 +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩))<_P ((𝐹‘𝑟) +_P 𝑄)))
74	66, 73	mtod 669	. . . . . 6 ⊢ ((𝜑 ∧ (𝑟 ∈ N ∧ (𝑋 +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩)<_P ((𝐹‘𝑟) +_P 𝑄))) → ¬ ((𝐿 +_P 𝑄) +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩)<_P ((𝐹‘𝑟) +_P 𝑄))
75		simprr 533	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑟 ∈ N ∧ (𝑋 +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩)<_P ((𝐹‘𝑟) +_P 𝑄))) → (𝑋 +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩)<_P ((𝐹‘𝑟) +_P 𝑄))
76		caucvgprprlemaddq.x	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ P)
77	76	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑟 ∈ N ∧ (𝑋 +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩)<_P ((𝐹‘𝑟) +_P 𝑄))) → 𝑋 ∈ P)
78		recnnpr 7767	. . . . . . . . . 10 ⊢ (𝑟 ∈ N → ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩ ∈ P)
79	72, 78	syl 14	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑟 ∈ N ∧ (𝑋 +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩)<_P ((𝐹‘𝑟) +_P 𝑄))) → ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩ ∈ P)
80		addclpr 7756	. . . . . . . . 9 ⊢ ((𝑋 ∈ P ∧ ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩ ∈ P) → (𝑋 +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩) ∈ P)
81	77, 79, 80	syl2anc 411	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑟 ∈ N ∧ (𝑋 +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩)<_P ((𝐹‘𝑟) +_P 𝑄))) → (𝑋 +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩) ∈ P)
82	67, 72	ffvelcdmd 5783	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑟 ∈ N ∧ (𝑋 +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩)<_P ((𝐹‘𝑟) +_P 𝑄))) → (𝐹‘𝑟) ∈ P)
83		addclpr 7756	. . . . . . . . 9 ⊢ (((𝐹‘𝑟) ∈ P ∧ 𝑄 ∈ P) → ((𝐹‘𝑟) +_P 𝑄) ∈ P)
84	82, 71, 83	syl2anc 411	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑟 ∈ N ∧ (𝑋 +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩)<_P ((𝐹‘𝑟) +_P 𝑄))) → ((𝐹‘𝑟) +_P 𝑄) ∈ P)
85	17, 19, 69, 5	caucvgprprlemcl 7923	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐿 ∈ P)
86	85	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑟 ∈ N ∧ (𝑋 +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩)<_P ((𝐹‘𝑟) +_P 𝑄))) → 𝐿 ∈ P)
87		addclpr 7756	. . . . . . . . . 10 ⊢ ((𝐿 ∈ P ∧ 𝑄 ∈ P) → (𝐿 +_P 𝑄) ∈ P)
88	86, 71, 87	syl2anc 411	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑟 ∈ N ∧ (𝑋 +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩)<_P ((𝐹‘𝑟) +_P 𝑄))) → (𝐿 +_P 𝑄) ∈ P)
89		addclpr 7756	. . . . . . . . 9 ⊢ (((𝐿 +_P 𝑄) ∈ P ∧ ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩ ∈ P) → ((𝐿 +_P 𝑄) +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩) ∈ P)
90	88, 79, 89	syl2anc 411	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑟 ∈ N ∧ (𝑋 +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩)<_P ((𝐹‘𝑟) +_P 𝑄))) → ((𝐿 +_P 𝑄) +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩) ∈ P)
91		ltsopr 7815	. . . . . . . . 9 ⊢ <_P Or P
92		sowlin 4417	. . . . . . . . 9 ⊢ ((<_P Or P ∧ ((𝑋 +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩) ∈ P ∧ ((𝐹‘𝑟) +_P 𝑄) ∈ P ∧ ((𝐿 +_P 𝑄) +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩) ∈ P)) → ((𝑋 +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩)<_P ((𝐹‘𝑟) +_P 𝑄) → ((𝑋 +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩)<_P ((𝐿 +_P 𝑄) +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩) ∨ ((𝐿 +_P 𝑄) +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩)<_P ((𝐹‘𝑟) +_P 𝑄))))
93	91, 92	mpan 424	. . . . . . . 8 ⊢ (((𝑋 +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩) ∈ P ∧ ((𝐹‘𝑟) +_P 𝑄) ∈ P ∧ ((𝐿 +_P 𝑄) +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩) ∈ P) → ((𝑋 +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩)<_P ((𝐹‘𝑟) +_P 𝑄) → ((𝑋 +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩)<_P ((𝐿 +_P 𝑄) +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩) ∨ ((𝐿 +_P 𝑄) +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩)<_P ((𝐹‘𝑟) +_P 𝑄))))
94	81, 84, 90, 93	syl3anc 1273	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑟 ∈ N ∧ (𝑋 +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩)<_P ((𝐹‘𝑟) +_P 𝑄))) → ((𝑋 +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩)<_P ((𝐹‘𝑟) +_P 𝑄) → ((𝑋 +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩)<_P ((𝐿 +_P 𝑄) +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩) ∨ ((𝐿 +_P 𝑄) +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩)<_P ((𝐹‘𝑟) +_P 𝑄))))
95	75, 94	mpd 13	. . . . . 6 ⊢ ((𝜑 ∧ (𝑟 ∈ N ∧ (𝑋 +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩)<_P ((𝐹‘𝑟) +_P 𝑄))) → ((𝑋 +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩)<_P ((𝐿 +_P 𝑄) +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩) ∨ ((𝐿 +_P 𝑄) +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩)<_P ((𝐹‘𝑟) +_P 𝑄)))
96	74, 95	ecased 1385	. . . . 5 ⊢ ((𝜑 ∧ (𝑟 ∈ N ∧ (𝑋 +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩)<_P ((𝐹‘𝑟) +_P 𝑄))) → (𝑋 +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩)<_P ((𝐿 +_P 𝑄) +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩))
97	36	adantl 277	. . . . . 6 ⊢ (((𝜑 ∧ (𝑟 ∈ N ∧ (𝑋 +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩)<_P ((𝐹‘𝑟) +_P 𝑄))) ∧ (𝑓 ∈ P ∧ 𝑔 ∈ P ∧ ℎ ∈ P)) → (𝑓<_P 𝑔 ↔ (ℎ +_P 𝑓)<_P (ℎ +_P 𝑔)))
98	41	adantl 277	. . . . . 6 ⊢ (((𝜑 ∧ (𝑟 ∈ N ∧ (𝑋 +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩)<_P ((𝐹‘𝑟) +_P 𝑄))) ∧ (𝑓 ∈ P ∧ 𝑔 ∈ P)) → (𝑓 +_P 𝑔) = (𝑔 +_P 𝑓))
99	97, 77, 88, 79, 98	caovord2d 6191	. . . . 5 ⊢ ((𝜑 ∧ (𝑟 ∈ N ∧ (𝑋 +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩)<_P ((𝐹‘𝑟) +_P 𝑄))) → (𝑋<_P (𝐿 +_P 𝑄) ↔ (𝑋 +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩)<_P ((𝐿 +_P 𝑄) +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩)))
100	96, 99	mpbird 167	. . . 4 ⊢ ((𝜑 ∧ (𝑟 ∈ N ∧ (𝑋 +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩)<_P ((𝐹‘𝑟) +_P 𝑄))) → 𝑋<_P (𝐿 +_P 𝑄))
101	100	exp32 365	. . 3 ⊢ (𝜑 → (𝑟 ∈ N → ((𝑋 +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩)<_P ((𝐹‘𝑟) +_P 𝑄) → 𝑋<_P (𝐿 +_P 𝑄))))
102	2, 16, 101	rexlimd 2647	. 2 ⊢ (𝜑 → (∃𝑟 ∈ N (𝑋 +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞}⟩)<_P ((𝐹‘𝑟) +_P 𝑄) → 𝑋<_P (𝐿 +_P 𝑄)))
103	1, 102	mpd 13	1 ⊢ (𝜑 → 𝑋<_P (𝐿 +_P 𝑄))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 715 ∧ w3a 1004 = wceq 1397 ∈ wcel 2202 {cab 2217 ∀wral 2510 ∃wrex 2511 {crab 2514 ⟨cop 3672 class class class wbr 4088 Or wor 4392 ⟶wf 5322 ‘cfv 5326 (class class class)co 6017 1_oc1o 6574 [cec 6699 Ncnpi 7491 <_N clti 7494 ~_Q ceq 7498 Qcnq 7499 +_Q cplq 7501 *_Qcrq 7503 <_Q cltq 7504 Pcnp 7510 +_P cpp 7512 <_P cltp 7514

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 619 ax-in2 620 ax-io 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4204 ax-sep 4207 ax-nul 4215 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-setind 4635 ax-iinf 4686

This theorem depends on definitions: df-bi 117 df-dc 842 df-3or 1005 df-3an 1006 df-tru 1400 df-fal 1403 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-ral 2515 df-rex 2516 df-reu 2517 df-rab 2519 df-v 2804 df-sbc 3032 df-csb 3128 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-nul 3495 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-int 3929 df-iun 3972 df-br 4089 df-opab 4151 df-mpt 4152 df-tr 4188 df-eprel 4386 df-id 4390 df-po 4393 df-iso 4394 df-iord 4463 df-on 4465 df-suc 4468 df-iom 4689 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-rn 4736 df-res 4737 df-ima 4738 df-iota 5286 df-fun 5328 df-fn 5329 df-f 5330 df-f1 5331 df-fo 5332 df-f1o 5333 df-fv 5334 df-ov 6020 df-oprab 6021 df-mpo 6022 df-1st 6302 df-2nd 6303 df-recs 6470 df-irdg 6535 df-1o 6581 df-2o 6582 df-oadd 6585 df-omul 6586 df-er 6701 df-ec 6703 df-qs 6707 df-ni 7523 df-pli 7524 df-mi 7525 df-lti 7526 df-plpq 7563 df-mpq 7564 df-enq 7566 df-nqqs 7567 df-plqqs 7568 df-mqqs 7569 df-1nqqs 7570 df-rq 7571 df-ltnqqs 7572 df-enq0 7643 df-nq0 7644 df-0nq0 7645 df-plq0 7646 df-mq0 7647 df-inp 7685 df-iplp 7687 df-iltp 7689

This theorem is referenced by: caucvgprprlem1 7928

W3C validator