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Theorem caucvgprprlemaddq 6806

Description: Lemma for caucvgprpr 6810. 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 1421	. . 3 ⊢ Ⅎ𝑟𝜑
3		nfcv 2178	. . . 4 ⊢ Ⅎ𝑟𝑋
4		nfcv 2178	. . . 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 2365	. . . . . . . 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 2178	. . . . . . . 8 ⊢ Ⅎ𝑟Q
8	6, 7	nfrabxy 2490	. . . . . . 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 2365	. . . . . . . 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	nfrabxy 2490	. . . . . . 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 3565	. . . . . 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 2175	. . . . 5 ⊢ Ⅎ𝑟𝐿
13		nfcv 2178	. . . . 5 ⊢ Ⅎ𝑟 +_P
14		nfcv 2178	. . . . 5 ⊢ Ⅎ𝑟𝑄
15	12, 13, 14	nfov 5535	. . . 4 ⊢ Ⅎ𝑟(𝐿 +_P 𝑄)
16	3, 4, 15	nfbr 3808	. . 3 ⊢ Ⅎ𝑟 𝑋<_P (𝐿 +_P 𝑄)
17		caucvgprpr.f	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:N⟶P)
18	17	ad2antrr 457	. . . . . . . . . . 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 457	. . . . . . . . . . 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 103	. . . . . . . . . . 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 487	. . . . . . . . . . 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 6791	. . . . . . . . . 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	ffvelrnd 5303	. . . . . . . . . . . . . 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 6646	. . . . . . . . . . . . . . 15 ⊢ (𝑏 ∈ N → ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑏, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑏, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩ ∈ P)
26	25	adantl 262	. . . . . . . . . . . . . 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 6635	. . . . . . . . . . . . . 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 391	. . . . . . . . . . . . 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 6646	. . . . . . . . . . . . . 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 457	. . . . . . . . . . . . 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 6677	. . . . . . . . . . . . 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 1135	. . . . . . . . . . . 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 3774	. . . . . . . . . . 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 6717	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P ∧ ℎ ∈ P) → (𝑓<_P 𝑔 ↔ (ℎ +_P 𝑓)<_P (ℎ +_P 𝑔)))
37	36	adantl 262	. . . . . . . . . . . 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 6635	. . . . . . . . . . . . 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 391	. . . . . . . . . . . 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	ffvelrnd 5303	. . . . . . . . . . . 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 6676	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P) → (𝑓 +_P 𝑔) = (𝑔 +_P 𝑓))
42	41	adantl 262	. . . . . . . . . . . 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 5670	. . . . . . . . . . 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 6676	. . . . . . . . . . . . . 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 391	. . . . . . . . . . . . 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 5528	. . . . . . . . . . . 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 3774	. . . . . . . . . . 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 210	. . . . . . . . . 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 598	. . . . . . . . 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 2412	. . . . . . . 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 3767	. . . . . . . . . . . . . 14 ⊢ (𝑝 = 𝑙 → (𝑝 <_Q (_Q‘[⟨𝑏, 1_𝑜⟩] ~_Q ) ↔ 𝑙 <_Q (_Q‘[⟨𝑏, 1_𝑜⟩] ~_Q )))
52	51	cbvabv 2161	. . . . . . . . . . . . 13 ⊢ {𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑏, 1_𝑜⟩] ~_Q )} = {𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑏, 1_𝑜⟩] ~_Q )}
53		breq2 3768	. . . . . . . . . . . . . 14 ⊢ (𝑞 = 𝑢 → ((_Q‘[⟨𝑏, 1_𝑜⟩] ~_Q ) <_Q 𝑞 ↔ (_Q‘[⟨𝑏, 1_𝑜⟩] ~_Q ) <_Q 𝑢))
54	53	cbvabv 2161	. . . . . . . . . . . . 13 ⊢ {𝑞 ∣ (_Q‘[⟨𝑏, 1_𝑜⟩] ~_Q ) <_Q 𝑞} = {𝑢 ∣ (_Q‘[⟨𝑏, 1_𝑜⟩] ~_Q ) <_Q 𝑢}
55	52, 54	opeq12i 3554	. . . . . . . . . . . 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 5523	. . . . . . . . . . 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 3767	. . . . . . . . . . . . . 14 ⊢ (𝑝 = 𝑙 → (𝑝 <_Q (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q ) ↔ 𝑙 <_Q (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q )))
58	57	cbvabv 2161	. . . . . . . . . . . . 13 ⊢ {𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q )} = {𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q )}
59		breq2 3768	. . . . . . . . . . . . . 14 ⊢ (𝑞 = 𝑢 → ((_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q ) <_Q 𝑞 ↔ (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q ) <_Q 𝑢))
60	59	cbvabv 2161	. . . . . . . . . . . . 13 ⊢ {𝑞 ∣ (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q ) <_Q 𝑞} = {𝑢 ∣ (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q ) <_Q 𝑢}
61	58, 60	opeq12i 3554	. . . . . . . . . . . 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 5523	. . . . . . . . . . 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 5524	. . . . . . . . . 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 3771	. . . . . . . . 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 2331	. . . . . . . 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 602	. . . . . . 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 261	. . . . . . . 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 261	. . . . . . . 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 261	. . . . . . . 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 261	. . . . . . . 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 483	. . . . . . . 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 6805	. . . . . . 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 589	. . . . . 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 484	. . . . . . 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 261	. . . . . . . . 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 6646	. . . . . . . . . 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 6635	. . . . . . . . 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 391	. . . . . . . 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	ffvelrnd 5303	. . . . . . . . 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 6635	. . . . . . . . 9 ⊢ (((𝐹‘𝑟) ∈ P ∧ 𝑄 ∈ P) → ((𝐹‘𝑟) +_P 𝑄) ∈ P)
84	82, 71, 83	syl2anc 391	. . . . . . . 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 6802	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐿 ∈ P)
86	85	adantr 261	. . . . . . . . . 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 6635	. . . . . . . . . 10 ⊢ ((𝐿 ∈ P ∧ 𝑄 ∈ P) → (𝐿 +_P 𝑄) ∈ P)
88	86, 71, 87	syl2anc 391	. . . . . . . . 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 6635	. . . . . . . . 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 391	. . . . . . . 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 6694	. . . . . . . . 9 ⊢ <_P Or P
92		sowlin 4057	. . . . . . . . 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 400	. . . . . . . 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 1135	. . . . . . 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 1239	. . . . 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 262	. . . . . 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 262	. . . . . 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 5670	. . . . 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 156	. . . 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 347	. . 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 2430	. 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 97 ↔ wb 98 ∨ wo 629 ∧ w3a 885 = wceq 1243 ∈ wcel 1393 {cab 2026 ∀wral 2306 ∃wrex 2307 {crab 2310 ⟨cop 3378 class class class wbr 3764 Or wor 4032 ⟶wf 4898 ‘cfv 4902 (class class class)co 5512 1_𝑜c1o 5994 [cec 6104 Ncnpi 6370 <_N clti 6373 ~_Q ceq 6377 Qcnq 6378 +_Q cplq 6380 *_Qcrq 6382 <_Q cltq 6383 Pcnp 6389 +_P cpp 6391 <_P cltp 6393

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311

This theorem depends on definitions: df-bi 110 df-dc 743 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-eprel 4026 df-id 4030 df-po 4033 df-iso 4034 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-recs 5920 df-irdg 5957 df-1o 6001 df-2o 6002 df-oadd 6005 df-omul 6006 df-er 6106 df-ec 6108 df-qs 6112 df-ni 6402 df-pli 6403 df-mi 6404 df-lti 6405 df-plpq 6442 df-mpq 6443 df-enq 6445 df-nqqs 6446 df-plqqs 6447 df-mqqs 6448 df-1nqqs 6449 df-rq 6450 df-ltnqqs 6451 df-enq0 6522 df-nq0 6523 df-0nq0 6524 df-plq0 6525 df-mq0 6526 df-inp 6564 df-iplp 6566 df-iltp 6568

This theorem is referenced by: caucvgprprlem1 6807

W3C validator