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Theorem caucvgprprlemaddq 8075

Description: Lemma for caucvgprpr 8079. 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 1581	. . 3 ⊢ Ⅎ𝑟𝜑
3		nfcv 2392	. . . 4 ⊢ Ⅎ𝑟𝑋
4		nfcv 2392	. . . 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 2593	. . . . . . . 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 2392	. . . . . . . 8 ⊢ Ⅎ𝑟Q
8	6, 7	nfrabw 2733	. . . . . . 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 2593	. . . . . . . 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 2733	. . . . . . 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 3920	. . . . . 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 2389	. . . . 5 ⊢ Ⅎ𝑟𝐿
13		nfcv 2392	. . . . 5 ⊢ Ⅎ𝑟 +_P
14		nfcv 2392	. . . . 5 ⊢ Ⅎ𝑟𝑄
15	12, 13, 14	nfov 6115	. . . 4 ⊢ Ⅎ𝑟(𝐿 +_P 𝑄)
16	3, 4, 15	nfbr 4177	. . 3 ⊢ Ⅎ𝑟 𝑋<_P (𝐿 +_P 𝑄)
17		caucvgprpr.f	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:N⟶P)
18	17	ad2antrr 492	. . . . . . . . . . 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 492	. . . . . . . . . . 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 541	. . . . . . . . . . 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 8060	. . . . . . . . . 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 5844	. . . . . . . . . . . . . 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 7915	. . . . . . . . . . . . . . 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 7904	. . . . . . . . . . . . . 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 415	. . . . . . . . . . . . 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 7915	. . . . . . . . . . . . . 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 492	. . . . . . . . . . . . 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 7946	. . . . . . . . . . . . 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 1278	. . . . . . . . . . . 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 4140	. . . . . . . . . . 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 7986	. . . . . . . . . . . . 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 7904	. . . . . . . . . . . . 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 415	. . . . . . . . . . . 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 5844	. . . . . . . . . . . 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 7945	. . . . . . . . . . . . 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 6259	. . . . . . . . . . 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 7945	. . . . . . . . . . . . . 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 415	. . . . . . . . . . . . 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 6101	. . . . . . . . . . . 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 4140	. . . . . . . . . . 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 684	. . . . . . . . 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 2643	. . . . . . . 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 4133	. . . . . . . . . . . . . 14 ⊢ (𝑝 = 𝑙 → (𝑝 <_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q ) ↔ 𝑙 <_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )))
52	51	cbvabv 2365	. . . . . . . . . . . . 13 ⊢ {𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )} = {𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑏, 1_o⟩] ~_Q )}
53		breq2 4134	. . . . . . . . . . . . . 14 ⊢ (𝑞 = 𝑢 → ((_Q‘[⟨𝑏, 1_o⟩] ~_Q ) <_Q 𝑞 ↔ (_Q‘[⟨𝑏, 1_o⟩] ~_Q ) <_Q 𝑢))
54	53	cbvabv 2365	. . . . . . . . . . . . 13 ⊢ {𝑞 ∣ (_Q‘[⟨𝑏, 1_o⟩] ~_Q ) <_Q 𝑞} = {𝑢 ∣ (_Q‘[⟨𝑏, 1_o⟩] ~_Q ) <_Q 𝑢}
55	52, 54	opeq12i 3909	. . . . . . . . . . . 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 6096	. . . . . . . . . . 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 4133	. . . . . . . . . . . . . 14 ⊢ (𝑝 = 𝑙 → (𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) ↔ 𝑙 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )))
58	57	cbvabv 2365	. . . . . . . . . . . . 13 ⊢ {𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )} = {𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑟, 1_o⟩] ~_Q )}
59		breq2 4134	. . . . . . . . . . . . . 14 ⊢ (𝑞 = 𝑢 → ((_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞 ↔ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑢))
60	59	cbvabv 2365	. . . . . . . . . . . . 13 ⊢ {𝑞 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑞} = {𝑢 ∣ (_Q‘[⟨𝑟, 1_o⟩] ~_Q ) <_Q 𝑢}
61	58, 60	opeq12i 3909	. . . . . . . . . . . 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 6096	. . . . . . . . . . 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 6097	. . . . . . . . . 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 4137	. . . . . . . . 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 2557	. . . . . . . 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 688	. . . . . . 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 535	. . . . . . . 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 8074	. . . . . . 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 673	. . . . . 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 537	. . . . . . 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 7915	. . . . . . . . . 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 7904	. . . . . . . . 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 415	. . . . . . . 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 5844	. . . . . . . . 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 7904	. . . . . . . . 9 ⊢ (((𝐹‘𝑟) ∈ P ∧ 𝑄 ∈ P) → ((𝐹‘𝑟) +_P 𝑄) ∈ P)
84	82, 71, 83	syl2anc 415	. . . . . . . 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 8071	. . . . . . . . . . 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 7904	. . . . . . . . . 10 ⊢ ((𝐿 ∈ P ∧ 𝑄 ∈ P) → (𝐿 +_P 𝑄) ∈ P)
88	86, 71, 87	syl2anc 415	. . . . . . . . 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 7904	. . . . . . . . 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 415	. . . . . . . 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 7963	. . . . . . . . 9 ⊢ <_P Or P
92		sowlin 4465	. . . . . . . . 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 428	. . . . . . . 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 1278	. . . . . . 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 1390	. . . . 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 6259	. . . . 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 2665	. 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

This proof depends on syntax axioms: → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 720 ∧ w3a 1009 = wceq 1402 ∈ wcel 2209 {cab 2224 ∀wral 2528 ∃wrex 2529 {crab 2532 ⟨cop 3712 class class class wbr 4130 Or wor 4440 ⟶wf 5373 ‘cfv 5377 (class class class)co 6085 1_oc1o 6680 [cec 6805 Ncnpi 7639 <_N clti 7642 ~_Q ceq 7646 Qcnq 7647 +_Q cplq 7649 *_Qcrq 7651 <_Q cltq 7652 Pcnp 7658 +_P cpp 7660 <_P cltp 7662

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4246 ax-sep 4249 ax-nul 4259 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-setind 4684 ax-iinf 4735

This proof depends on definitions: df-bi 117 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-ral 2533 df-rex 2534 df-reu 2535 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-br 4131 df-opab 4193 df-mpt 4194 df-tr 4230 df-eprel 4434 df-id 4438 df-po 4441 df-iso 4442 df-iord 4511 df-on 4513 df-suc 4516 df-iom 4738 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-rn 4785 df-res 4786 df-ima 4787 df-iota 5337 df-fun 5379 df-fn 5380 df-f 5381 df-f1 5382 df-fo 5383 df-f1o 5384 df-fv 5385 df-ov 6088 df-oprab 6089 df-mpo 6090 df-1st 6374 df-2nd 6375 df-recs 6576 df-irdg 6641 df-1o 6687 df-2o 6688 df-oadd 6691 df-omul 6692 df-er 6807 df-ec 6809 df-qs 6813 df-ni 7671 df-pli 7672 df-mi 7673 df-lti 7674 df-plpq 7711 df-mpq 7712 df-enq 7714 df-nqqs 7715 df-plqqs 7716 df-mqqs 7717 df-1nqqs 7718 df-rq 7719 df-ltnqqs 7720 df-enq0 7791 df-nq0 7792 df-0nq0 7793 df-plq0 7794 df-mq0 7795 df-inp 7833 df-iplp 7835 df-iltp 7837

This theorem is used by: caucvgprprlem1 8076

W3C validator