caucvgprprlemaddq - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > caucvgprprlemaddq		GIF version

Theorem caucvgprprlemaddq 6898

Description: Lemma for caucvgprpr 6902. 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 1461	. . 3 ⊢ Ⅎ𝑟𝜑
3		nfcv 2219	. . . 4 ⊢ Ⅎ𝑟𝑋
4		nfcv 2219	. . . 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 2407	. . . . . . . 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 2219	. . . . . . . 8 ⊢ Ⅎ𝑟Q
8	6, 7	nfrabxy 2534	. . . . . . 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 2407	. . . . . . . 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 2534	. . . . . . 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 3586	. . . . . 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 2216	. . . . 5 ⊢ Ⅎ𝑟𝐿
13		nfcv 2219	. . . . 5 ⊢ Ⅎ𝑟 +_P
14		nfcv 2219	. . . . 5 ⊢ Ⅎ𝑟𝑄
15	12, 13, 14	nfov 5555	. . . 4 ⊢ Ⅎ𝑟(𝐿 +_P 𝑄)
16	3, 4, 15	nfbr 3829	. . 3 ⊢ Ⅎ𝑟 𝑋<_P (𝐿 +_P 𝑄)
17		caucvgprpr.f	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:N⟶P)
18	17	ad2antrr 471	. . . . . . . . . . 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 471	. . . . . . . . . . 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 108	. . . . . . . . . . 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 501	. . . . . . . . . . 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 6883	. . . . . . . . . 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 5324	. . . . . . . . . . . . . 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 6738	. . . . . . . . . . . . . . 15 ⊢ (𝑏 ∈ N → ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑏, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑏, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩ ∈ P)
26	25	adantl 271	. . . . . . . . . . . . . 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 6727	. . . . . . . . . . . . . 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 403	. . . . . . . . . . . . 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 6738	. . . . . . . . . . . . . 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 471	. . . . . . . . . . . . 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 6769	. . . . . . . . . . . . 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 1169	. . . . . . . . . . . 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 3795	. . . . . . . . . . 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 6809	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P ∧ ℎ ∈ P) → (𝑓<_P 𝑔 ↔ (ℎ +_P 𝑓)<_P (ℎ +_P 𝑔)))
37	36	adantl 271	. . . . . . . . . . . 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 6727	. . . . . . . . . . . . 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 403	. . . . . . . . . . . 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 5324	. . . . . . . . . . . 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 6768	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P) → (𝑓 +_P 𝑔) = (𝑔 +_P 𝑓))
42	41	adantl 271	. . . . . . . . . . . 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 5690	. . . . . . . . . . 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 6768	. . . . . . . . . . . . . 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 403	. . . . . . . . . . . . 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 5548	. . . . . . . . . . . 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 3795	. . . . . . . . . . 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 219	. . . . . . . . . 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 630	. . . . . . . . 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 2454	. . . . . . . 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 3788	. . . . . . . . . . . . . 14 ⊢ (𝑝 = 𝑙 → (𝑝 <_Q (_Q‘[⟨𝑏, 1_𝑜⟩] ~_Q ) ↔ 𝑙 <_Q (_Q‘[⟨𝑏, 1_𝑜⟩] ~_Q )))
52	51	cbvabv 2202	. . . . . . . . . . . . 13 ⊢ {𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑏, 1_𝑜⟩] ~_Q )} = {𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑏, 1_𝑜⟩] ~_Q )}
53		breq2 3789	. . . . . . . . . . . . . 14 ⊢ (𝑞 = 𝑢 → ((_Q‘[⟨𝑏, 1_𝑜⟩] ~_Q ) <_Q 𝑞 ↔ (_Q‘[⟨𝑏, 1_𝑜⟩] ~_Q ) <_Q 𝑢))
54	53	cbvabv 2202	. . . . . . . . . . . . 13 ⊢ {𝑞 ∣ (_Q‘[⟨𝑏, 1_𝑜⟩] ~_Q ) <_Q 𝑞} = {𝑢 ∣ (_Q‘[⟨𝑏, 1_𝑜⟩] ~_Q ) <_Q 𝑢}
55	52, 54	opeq12i 3575	. . . . . . . . . . . 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 5543	. . . . . . . . . . 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 3788	. . . . . . . . . . . . . 14 ⊢ (𝑝 = 𝑙 → (𝑝 <_Q (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q ) ↔ 𝑙 <_Q (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q )))
58	57	cbvabv 2202	. . . . . . . . . . . . 13 ⊢ {𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q )} = {𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q )}
59		breq2 3789	. . . . . . . . . . . . . 14 ⊢ (𝑞 = 𝑢 → ((_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q ) <_Q 𝑞 ↔ (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q ) <_Q 𝑢))
60	59	cbvabv 2202	. . . . . . . . . . . . 13 ⊢ {𝑞 ∣ (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q ) <_Q 𝑞} = {𝑢 ∣ (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q ) <_Q 𝑢}
61	58, 60	opeq12i 3575	. . . . . . . . . . . 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 5543	. . . . . . . . . . 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 5544	. . . . . . . . . 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 3792	. . . . . . . . 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 2373	. . . . . . . 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 634	. . . . . . 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 270	. . . . . . . 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 270	. . . . . . . 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 270	. . . . . . . 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 270	. . . . . . . 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 497	. . . . . . . 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 6897	. . . . . . 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 621	. . . . . 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 498	. . . . . . 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 270	. . . . . . . . 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 6738	. . . . . . . . . 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 6727	. . . . . . . . 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 403	. . . . . . . 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 5324	. . . . . . . . 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 6727	. . . . . . . . 9 ⊢ (((𝐹‘𝑟) ∈ P ∧ 𝑄 ∈ P) → ((𝐹‘𝑟) +_P 𝑄) ∈ P)
84	82, 71, 83	syl2anc 403	. . . . . . . 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 6894	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐿 ∈ P)
86	85	adantr 270	. . . . . . . . . 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 6727	. . . . . . . . . 10 ⊢ ((𝐿 ∈ P ∧ 𝑄 ∈ P) → (𝐿 +_P 𝑄) ∈ P)
88	86, 71, 87	syl2anc 403	. . . . . . . . 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 6727	. . . . . . . . 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 403	. . . . . . . 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 6786	. . . . . . . . 9 ⊢ <_P Or P
92		sowlin 4075	. . . . . . . . 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 414	. . . . . . . 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 1169	. . . . . . 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 1280	. . . . 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 271	. . . . . 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 271	. . . . . 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 5690	. . . . 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 165	. . . 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 357	. . 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 2474	. 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 102 ↔ wb 103 ∨ wo 661 ∧ w3a 919 = wceq 1284 ∈ wcel 1433 {cab 2067 ∀wral 2348 ∃wrex 2349 {crab 2352 ⟨cop 3401 class class class wbr 3785 Or wor 4050 ⟶wf 4918 ‘cfv 4922 (class class class)co 5532 1_𝑜c1o 6017 [cec 6127 Ncnpi 6462 <_N clti 6465 ~_Q ceq 6469 Qcnq 6470 +_Q cplq 6472 *_Qcrq 6474 <_Q cltq 6475 Pcnp 6481 +_P cpp 6483 <_P cltp 6485

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 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329

This theorem depends on definitions: df-bi 115 df-dc 776 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-eprel 4044 df-id 4048 df-po 4051 df-iso 4052 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-recs 5943 df-irdg 5980 df-1o 6024 df-2o 6025 df-oadd 6028 df-omul 6029 df-er 6129 df-ec 6131 df-qs 6135 df-ni 6494 df-pli 6495 df-mi 6496 df-lti 6497 df-plpq 6534 df-mpq 6535 df-enq 6537 df-nqqs 6538 df-plqqs 6539 df-mqqs 6540 df-1nqqs 6541 df-rq 6542 df-ltnqqs 6543 df-enq0 6614 df-nq0 6615 df-0nq0 6616 df-plq0 6617 df-mq0 6618 df-inp 6656 df-iplp 6658 df-iltp 6660

This theorem is referenced by: caucvgprprlem1 6899

W3C validator