Theorem rpnnen2lem12 14793

Description: Lemma for rpnnen2 14794. (Contributed by Mario Carneiro, 13-May-2013.)

Hypothesis

Ref	Expression
rpnnen2.1	⊢ 𝐹 = (𝑥 ∈ 𝒫 ℕ ↦ (𝑛 ∈ ℕ ↦ if(𝑛 ∈ 𝑥, ((1 / 3)↑𝑛), 0)))

Assertion

Ref	Expression
rpnnen2lem12	⊢ 𝒫 ℕ ≼ (0[,]1)

Distinct variable group:   𝑥,𝑛

Allowed substitution hints:   𝐹(𝑥,𝑛)

Proof of Theorem rpnnen2lem12

Dummy variables 𝑚 𝑦 𝑧 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovex 6577	. 2 ⊢ (0[,]1) ∈ V
2		elpwi 4117	. . . . 5 ⊢ (𝑦 ∈ 𝒫 ℕ → 𝑦 ⊆ ℕ)
3		nnuz 11599	. . . . . . 7 ⊢ ℕ = (ℤ≥‘1)
4	3	sumeq1i 14276	. . . . . 6 ⊢ Σ𝑘 ∈ ℕ ((𝐹‘𝑦)‘𝑘) = Σ𝑘 ∈ (ℤ≥‘1)((𝐹‘𝑦)‘𝑘)
5		1nn 10908	. . . . . . 7 ⊢ 1 ∈ ℕ
6		rpnnen2.1	. . . . . . . 8 ⊢ 𝐹 = (𝑥 ∈ 𝒫 ℕ ↦ (𝑛 ∈ ℕ ↦ if(𝑛 ∈ 𝑥, ((1 / 3)↑𝑛), 0)))
7	6	rpnnen2lem6 14787	. . . . . . 7 ⊢ ((𝑦 ⊆ ℕ ∧ 1 ∈ ℕ) → Σ𝑘 ∈ (ℤ≥‘1)((𝐹‘𝑦)‘𝑘) ∈ ℝ)
8	5, 7	mpan2 703	. . . . . 6 ⊢ (𝑦 ⊆ ℕ → Σ𝑘 ∈ (ℤ≥‘1)((𝐹‘𝑦)‘𝑘) ∈ ℝ)
9	4, 8	syl5eqel 2692	. . . . 5 ⊢ (𝑦 ⊆ ℕ → Σ𝑘 ∈ ℕ ((𝐹‘𝑦)‘𝑘) ∈ ℝ)
10	2, 9	syl 17	. . . 4 ⊢ (𝑦 ∈ 𝒫 ℕ → Σ𝑘 ∈ ℕ ((𝐹‘𝑦)‘𝑘) ∈ ℝ)
11		1zzd 11285	. . . . 5 ⊢ (𝑦 ∈ 𝒫 ℕ → 1 ∈ ℤ)
12		eqidd 2611	. . . . 5 ⊢ ((𝑦 ∈ 𝒫 ℕ ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑦)‘𝑘) = ((𝐹‘𝑦)‘𝑘))
13	6	rpnnen2lem2 14783	. . . . . . 7 ⊢ (𝑦 ⊆ ℕ → (𝐹‘𝑦):ℕ⟶ℝ)
14	2, 13	syl 17	. . . . . 6 ⊢ (𝑦 ∈ 𝒫 ℕ → (𝐹‘𝑦):ℕ⟶ℝ)
15	14	ffvelrnda 6267	. . . . 5 ⊢ ((𝑦 ∈ 𝒫 ℕ ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑦)‘𝑘) ∈ ℝ)
16	6	rpnnen2lem5 14786	. . . . . 6 ⊢ ((𝑦 ⊆ ℕ ∧ 1 ∈ ℕ) → seq1( + , (𝐹‘𝑦)) ∈ dom ⇝ )
17	2, 5, 16	sylancl 693	. . . . 5 ⊢ (𝑦 ∈ 𝒫 ℕ → seq1( + , (𝐹‘𝑦)) ∈ dom ⇝ )
18		ssid 3587	. . . . . . . 8 ⊢ ℕ ⊆ ℕ
19	6	rpnnen2lem4 14785	. . . . . . . 8 ⊢ ((𝑦 ⊆ ℕ ∧ ℕ ⊆ ℕ ∧ 𝑘 ∈ ℕ) → (0 ≤ ((𝐹‘𝑦)‘𝑘) ∧ ((𝐹‘𝑦)‘𝑘) ≤ ((𝐹‘ℕ)‘𝑘)))
20	18, 19	mp3an2 1404	. . . . . . 7 ⊢ ((𝑦 ⊆ ℕ ∧ 𝑘 ∈ ℕ) → (0 ≤ ((𝐹‘𝑦)‘𝑘) ∧ ((𝐹‘𝑦)‘𝑘) ≤ ((𝐹‘ℕ)‘𝑘)))
21	20	simpld 474	. . . . . 6 ⊢ ((𝑦 ⊆ ℕ ∧ 𝑘 ∈ ℕ) → 0 ≤ ((𝐹‘𝑦)‘𝑘))
22	2, 21	sylan 487	. . . . 5 ⊢ ((𝑦 ∈ 𝒫 ℕ ∧ 𝑘 ∈ ℕ) → 0 ≤ ((𝐹‘𝑦)‘𝑘))
23	3, 11, 12, 15, 17, 22	isumge0 14339	. . . 4 ⊢ (𝑦 ∈ 𝒫 ℕ → 0 ≤ Σ𝑘 ∈ ℕ ((𝐹‘𝑦)‘𝑘))
24		halfre 11123	. . . . . 6 ⊢ (1 / 2) ∈ ℝ
25	24	a1i 11	. . . . 5 ⊢ (𝑦 ∈ 𝒫 ℕ → (1 / 2) ∈ ℝ)
26		1re 9918	. . . . . 6 ⊢ 1 ∈ ℝ
27	26	a1i 11	. . . . 5 ⊢ (𝑦 ∈ 𝒫 ℕ → 1 ∈ ℝ)
28	6	rpnnen2lem7 14788	. . . . . . . . 9 ⊢ ((𝑦 ⊆ ℕ ∧ ℕ ⊆ ℕ ∧ 1 ∈ ℕ) → Σ𝑘 ∈ (ℤ≥‘1)((𝐹‘𝑦)‘𝑘) ≤ Σ𝑘 ∈ (ℤ≥‘1)((𝐹‘ℕ)‘𝑘))
29	18, 5, 28	mp3an23 1408	. . . . . . . 8 ⊢ (𝑦 ⊆ ℕ → Σ𝑘 ∈ (ℤ≥‘1)((𝐹‘𝑦)‘𝑘) ≤ Σ𝑘 ∈ (ℤ≥‘1)((𝐹‘ℕ)‘𝑘))
30	2, 29	syl 17	. . . . . . 7 ⊢ (𝑦 ∈ 𝒫 ℕ → Σ𝑘 ∈ (ℤ≥‘1)((𝐹‘𝑦)‘𝑘) ≤ Σ𝑘 ∈ (ℤ≥‘1)((𝐹‘ℕ)‘𝑘))
31		eqid 2610	. . . . . . . 8 ⊢ (ℤ≥‘1) = (ℤ≥‘1)
32		eqidd 2611	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝒫 ℕ ∧ 𝑘 ∈ (ℤ≥‘1)) → ((𝐹‘ℕ)‘𝑘) = ((𝐹‘ℕ)‘𝑘))
33		elnnuz 11600	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ ↔ 𝑘 ∈ (ℤ≥‘1))
34	6	rpnnen2lem2 14783	. . . . . . . . . . . . 13 ⊢ (ℕ ⊆ ℕ → (𝐹‘ℕ):ℕ⟶ℝ)
35	18, 34	ax-mp 5	. . . . . . . . . . . 12 ⊢ (𝐹‘ℕ):ℕ⟶ℝ
36	35	ffvelrni 6266	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → ((𝐹‘ℕ)‘𝑘) ∈ ℝ)
37	36	recnd 9947	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → ((𝐹‘ℕ)‘𝑘) ∈ ℂ)
38	33, 37	sylbir 224	. . . . . . . . 9 ⊢ (𝑘 ∈ (ℤ≥‘1) → ((𝐹‘ℕ)‘𝑘) ∈ ℂ)
39	38	adantl 481	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝒫 ℕ ∧ 𝑘 ∈ (ℤ≥‘1)) → ((𝐹‘ℕ)‘𝑘) ∈ ℂ)
40	6	rpnnen2lem3 14784	. . . . . . . . 9 ⊢ seq1( + , (𝐹‘ℕ)) ⇝ (1 / 2)
41	40	a1i 11	. . . . . . . 8 ⊢ (𝑦 ∈ 𝒫 ℕ → seq1( + , (𝐹‘ℕ)) ⇝ (1 / 2))
42	31, 11, 32, 39, 41	isumclim 14330	. . . . . . 7 ⊢ (𝑦 ∈ 𝒫 ℕ → Σ𝑘 ∈ (ℤ≥‘1)((𝐹‘ℕ)‘𝑘) = (1 / 2))
43	30, 42	breqtrd 4609	. . . . . 6 ⊢ (𝑦 ∈ 𝒫 ℕ → Σ𝑘 ∈ (ℤ≥‘1)((𝐹‘𝑦)‘𝑘) ≤ (1 / 2))
44	4, 43	syl5eqbr 4618	. . . . 5 ⊢ (𝑦 ∈ 𝒫 ℕ → Σ𝑘 ∈ ℕ ((𝐹‘𝑦)‘𝑘) ≤ (1 / 2))
45		halflt1 11127	. . . . . . 7 ⊢ (1 / 2) < 1
46	24, 26, 45	ltleii 10039	. . . . . 6 ⊢ (1 / 2) ≤ 1
47	46	a1i 11	. . . . 5 ⊢ (𝑦 ∈ 𝒫 ℕ → (1 / 2) ≤ 1)
48	10, 25, 27, 44, 47	letrd 10073	. . . 4 ⊢ (𝑦 ∈ 𝒫 ℕ → Σ𝑘 ∈ ℕ ((𝐹‘𝑦)‘𝑘) ≤ 1)
49		0re 9919	. . . . 5 ⊢ 0 ∈ ℝ
50	49, 26	elicc2i 12110	. . . 4 ⊢ (Σ𝑘 ∈ ℕ ((𝐹‘𝑦)‘𝑘) ∈ (0[,]1) ↔ (Σ𝑘 ∈ ℕ ((𝐹‘𝑦)‘𝑘) ∈ ℝ ∧ 0 ≤ Σ𝑘 ∈ ℕ ((𝐹‘𝑦)‘𝑘) ∧ Σ𝑘 ∈ ℕ ((𝐹‘𝑦)‘𝑘) ≤ 1))
51	10, 23, 48, 50	syl3anbrc 1239	. . 3 ⊢ (𝑦 ∈ 𝒫 ℕ → Σ𝑘 ∈ ℕ ((𝐹‘𝑦)‘𝑘) ∈ (0[,]1))
52		elpwi 4117	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝒫 ℕ → 𝑧 ⊆ ℕ)
53		ssdifss 3703	. . . . . . . . . . . 12 ⊢ (𝑦 ⊆ ℕ → (𝑦 ∖ 𝑧) ⊆ ℕ)
54		ssdifss 3703	. . . . . . . . . . . 12 ⊢ (𝑧 ⊆ ℕ → (𝑧 ∖ 𝑦) ⊆ ℕ)
55		unss 3749	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∖ 𝑧) ⊆ ℕ ∧ (𝑧 ∖ 𝑦) ⊆ ℕ) ↔ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦)) ⊆ ℕ)
56	55	biimpi 205	. . . . . . . . . . . 12 ⊢ (((𝑦 ∖ 𝑧) ⊆ ℕ ∧ (𝑧 ∖ 𝑦) ⊆ ℕ) → ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦)) ⊆ ℕ)
57	53, 54, 56	syl2an 493	. . . . . . . . . . 11 ⊢ ((𝑦 ⊆ ℕ ∧ 𝑧 ⊆ ℕ) → ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦)) ⊆ ℕ)
58	2, 52, 57	syl2an 493	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝒫 ℕ ∧ 𝑧 ∈ 𝒫 ℕ) → ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦)) ⊆ ℕ)
59		eqss 3583	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑧 ↔ (𝑦 ⊆ 𝑧 ∧ 𝑧 ⊆ 𝑦))
60		ssdif0 3896	. . . . . . . . . . . . . 14 ⊢ (𝑦 ⊆ 𝑧 ↔ (𝑦 ∖ 𝑧) = ∅)
61		ssdif0 3896	. . . . . . . . . . . . . 14 ⊢ (𝑧 ⊆ 𝑦 ↔ (𝑧 ∖ 𝑦) = ∅)
62	60, 61	anbi12i 729	. . . . . . . . . . . . 13 ⊢ ((𝑦 ⊆ 𝑧 ∧ 𝑧 ⊆ 𝑦) ↔ ((𝑦 ∖ 𝑧) = ∅ ∧ (𝑧 ∖ 𝑦) = ∅))
63		un00 3963	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∖ 𝑧) = ∅ ∧ (𝑧 ∖ 𝑦) = ∅) ↔ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦)) = ∅)
64	59, 62, 63	3bitri 285	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑧 ↔ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦)) = ∅)
65	64	necon3bii 2834	. . . . . . . . . . 11 ⊢ (𝑦 ≠ 𝑧 ↔ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦)) ≠ ∅)
66	65	biimpi 205	. . . . . . . . . 10 ⊢ (𝑦 ≠ 𝑧 → ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦)) ≠ ∅)
67		nnwo 11629	. . . . . . . . . 10 ⊢ ((((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦)) ⊆ ℕ ∧ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦)) ≠ ∅) → ∃𝑚 ∈ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦))∀𝑛 ∈ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦))𝑚 ≤ 𝑛)
68	58, 66, 67	syl2an 493	. . . . . . . . 9 ⊢ (((𝑦 ∈ 𝒫 ℕ ∧ 𝑧 ∈ 𝒫 ℕ) ∧ 𝑦 ≠ 𝑧) → ∃𝑚 ∈ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦))∀𝑛 ∈ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦))𝑚 ≤ 𝑛)
69	68	ex 449	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝒫 ℕ ∧ 𝑧 ∈ 𝒫 ℕ) → (𝑦 ≠ 𝑧 → ∃𝑚 ∈ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦))∀𝑛 ∈ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦))𝑚 ≤ 𝑛))
70	58	sselda 3568	. . . . . . . . . 10 ⊢ (((𝑦 ∈ 𝒫 ℕ ∧ 𝑧 ∈ 𝒫 ℕ) ∧ 𝑚 ∈ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦))) → 𝑚 ∈ ℕ)
71		df-ral 2901	. . . . . . . . . . . 12 ⊢ (∀𝑛 ∈ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦))𝑚 ≤ 𝑛 ↔ ∀𝑛(𝑛 ∈ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦)) → 𝑚 ≤ 𝑛))
72		con34b 305	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦)) → 𝑚 ≤ 𝑛) ↔ (¬ 𝑚 ≤ 𝑛 → ¬ 𝑛 ∈ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦))))
73		eldif 3550	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (𝑦 ∖ 𝑧) ↔ (𝑛 ∈ 𝑦 ∧ ¬ 𝑛 ∈ 𝑧))
74		eldif 3550	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (𝑧 ∖ 𝑦) ↔ (𝑛 ∈ 𝑧 ∧ ¬ 𝑛 ∈ 𝑦))
75	73, 74	orbi12i 542	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ (𝑦 ∖ 𝑧) ∨ 𝑛 ∈ (𝑧 ∖ 𝑦)) ↔ ((𝑛 ∈ 𝑦 ∧ ¬ 𝑛 ∈ 𝑧) ∨ (𝑛 ∈ 𝑧 ∧ ¬ 𝑛 ∈ 𝑦)))
76		elun 3715	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦)) ↔ (𝑛 ∈ (𝑦 ∖ 𝑧) ∨ 𝑛 ∈ (𝑧 ∖ 𝑦)))
77		xor 931	. . . . . . . . . . . . . . . . 17 ⊢ (¬ (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧) ↔ ((𝑛 ∈ 𝑦 ∧ ¬ 𝑛 ∈ 𝑧) ∨ (𝑛 ∈ 𝑧 ∧ ¬ 𝑛 ∈ 𝑦)))
78	75, 76, 77	3bitr4ri 292	. . . . . . . . . . . . . . . 16 ⊢ (¬ (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧) ↔ 𝑛 ∈ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦)))
79	78	con1bii 345	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑛 ∈ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦)) ↔ (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧))
80	79	imbi2i 325	. . . . . . . . . . . . . 14 ⊢ ((¬ 𝑚 ≤ 𝑛 → ¬ 𝑛 ∈ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦))) ↔ (¬ 𝑚 ≤ 𝑛 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)))
81	72, 80	bitri 263	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦)) → 𝑚 ≤ 𝑛) ↔ (¬ 𝑚 ≤ 𝑛 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)))
82	81	albii 1737	. . . . . . . . . . . 12 ⊢ (∀𝑛(𝑛 ∈ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦)) → 𝑚 ≤ 𝑛) ↔ ∀𝑛(¬ 𝑚 ≤ 𝑛 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)))
83	71, 82	bitri 263	. . . . . . . . . . 11 ⊢ (∀𝑛 ∈ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦))𝑚 ≤ 𝑛 ↔ ∀𝑛(¬ 𝑚 ≤ 𝑛 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)))
84		alral 2912	. . . . . . . . . . . 12 ⊢ (∀𝑛(¬ 𝑚 ≤ 𝑛 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)) → ∀𝑛 ∈ ℕ (¬ 𝑚 ≤ 𝑛 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)))
85		nnre 10904	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
86		nnre 10904	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℝ)
87		ltnle 9996	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℝ ∧ 𝑚 ∈ ℝ) → (𝑛 < 𝑚 ↔ ¬ 𝑚 ≤ 𝑛))
88	85, 86, 87	syl2anr 494	. . . . . . . . . . . . . 14 ⊢ ((𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) → (𝑛 < 𝑚 ↔ ¬ 𝑚 ≤ 𝑛))
89	88	imbi1d 330	. . . . . . . . . . . . 13 ⊢ ((𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) → ((𝑛 < 𝑚 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)) ↔ (¬ 𝑚 ≤ 𝑛 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧))))
90	89	ralbidva 2968	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ → (∀𝑛 ∈ ℕ (𝑛 < 𝑚 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)) ↔ ∀𝑛 ∈ ℕ (¬ 𝑚 ≤ 𝑛 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧))))
91	84, 90	syl5ibr 235	. . . . . . . . . . 11 ⊢ (𝑚 ∈ ℕ → (∀𝑛(¬ 𝑚 ≤ 𝑛 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)) → ∀𝑛 ∈ ℕ (𝑛 < 𝑚 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧))))
92	83, 91	syl5bi 231	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ → (∀𝑛 ∈ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦))𝑚 ≤ 𝑛 → ∀𝑛 ∈ ℕ (𝑛 < 𝑚 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧))))
93	70, 92	syl 17	. . . . . . . . 9 ⊢ (((𝑦 ∈ 𝒫 ℕ ∧ 𝑧 ∈ 𝒫 ℕ) ∧ 𝑚 ∈ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦))) → (∀𝑛 ∈ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦))𝑚 ≤ 𝑛 → ∀𝑛 ∈ ℕ (𝑛 < 𝑚 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧))))
94	93	reximdva 3000	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝒫 ℕ ∧ 𝑧 ∈ 𝒫 ℕ) → (∃𝑚 ∈ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦))∀𝑛 ∈ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦))𝑚 ≤ 𝑛 → ∃𝑚 ∈ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦))∀𝑛 ∈ ℕ (𝑛 < 𝑚 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧))))
95	69, 94	syld 46	. . . . . . 7 ⊢ ((𝑦 ∈ 𝒫 ℕ ∧ 𝑧 ∈ 𝒫 ℕ) → (𝑦 ≠ 𝑧 → ∃𝑚 ∈ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦))∀𝑛 ∈ ℕ (𝑛 < 𝑚 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧))))
96		rexun 3755	. . . . . . 7 ⊢ (∃𝑚 ∈ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦))∀𝑛 ∈ ℕ (𝑛 < 𝑚 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)) ↔ (∃𝑚 ∈ (𝑦 ∖ 𝑧)∀𝑛 ∈ ℕ (𝑛 < 𝑚 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)) ∨ ∃𝑚 ∈ (𝑧 ∖ 𝑦)∀𝑛 ∈ ℕ (𝑛 < 𝑚 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧))))
97	95, 96	syl6ib 240	. . . . . 6 ⊢ ((𝑦 ∈ 𝒫 ℕ ∧ 𝑧 ∈ 𝒫 ℕ) → (𝑦 ≠ 𝑧 → (∃𝑚 ∈ (𝑦 ∖ 𝑧)∀𝑛 ∈ ℕ (𝑛 < 𝑚 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)) ∨ ∃𝑚 ∈ (𝑧 ∖ 𝑦)∀𝑛 ∈ ℕ (𝑛 < 𝑚 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)))))
98		simpll 786	. . . . . . . . . 10 ⊢ (((𝑦 ⊆ ℕ ∧ 𝑧 ⊆ ℕ) ∧ (𝑚 ∈ (𝑦 ∖ 𝑧) ∧ ∀𝑛 ∈ ℕ (𝑛 < 𝑚 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)))) → 𝑦 ⊆ ℕ)
99		simplr 788	. . . . . . . . . 10 ⊢ (((𝑦 ⊆ ℕ ∧ 𝑧 ⊆ ℕ) ∧ (𝑚 ∈ (𝑦 ∖ 𝑧) ∧ ∀𝑛 ∈ ℕ (𝑛 < 𝑚 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)))) → 𝑧 ⊆ ℕ)
100		simprl 790	. . . . . . . . . 10 ⊢ (((𝑦 ⊆ ℕ ∧ 𝑧 ⊆ ℕ) ∧ (𝑚 ∈ (𝑦 ∖ 𝑧) ∧ ∀𝑛 ∈ ℕ (𝑛 < 𝑚 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)))) → 𝑚 ∈ (𝑦 ∖ 𝑧))
101		simprr 792	. . . . . . . . . 10 ⊢ (((𝑦 ⊆ ℕ ∧ 𝑧 ⊆ ℕ) ∧ (𝑚 ∈ (𝑦 ∖ 𝑧) ∧ ∀𝑛 ∈ ℕ (𝑛 < 𝑚 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)))) → ∀𝑛 ∈ ℕ (𝑛 < 𝑚 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)))
102		biid 250	. . . . . . . . . 10 ⊢ (Σ𝑘 ∈ ℕ ((𝐹‘𝑦)‘𝑘) = Σ𝑘 ∈ ℕ ((𝐹‘𝑧)‘𝑘) ↔ Σ𝑘 ∈ ℕ ((𝐹‘𝑦)‘𝑘) = Σ𝑘 ∈ ℕ ((𝐹‘𝑧)‘𝑘))
103	6, 98, 99, 100, 101, 102	rpnnen2lem11 14792	. . . . . . . . 9 ⊢ (((𝑦 ⊆ ℕ ∧ 𝑧 ⊆ ℕ) ∧ (𝑚 ∈ (𝑦 ∖ 𝑧) ∧ ∀𝑛 ∈ ℕ (𝑛 < 𝑚 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)))) → ¬ Σ𝑘 ∈ ℕ ((𝐹‘𝑦)‘𝑘) = Σ𝑘 ∈ ℕ ((𝐹‘𝑧)‘𝑘))
104	103	rexlimdvaa 3014	. . . . . . . 8 ⊢ ((𝑦 ⊆ ℕ ∧ 𝑧 ⊆ ℕ) → (∃𝑚 ∈ (𝑦 ∖ 𝑧)∀𝑛 ∈ ℕ (𝑛 < 𝑚 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)) → ¬ Σ𝑘 ∈ ℕ ((𝐹‘𝑦)‘𝑘) = Σ𝑘 ∈ ℕ ((𝐹‘𝑧)‘𝑘)))
105		simplr 788	. . . . . . . . . 10 ⊢ (((𝑦 ⊆ ℕ ∧ 𝑧 ⊆ ℕ) ∧ (𝑚 ∈ (𝑧 ∖ 𝑦) ∧ ∀𝑛 ∈ ℕ (𝑛 < 𝑚 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)))) → 𝑧 ⊆ ℕ)
106		simpll 786	. . . . . . . . . 10 ⊢ (((𝑦 ⊆ ℕ ∧ 𝑧 ⊆ ℕ) ∧ (𝑚 ∈ (𝑧 ∖ 𝑦) ∧ ∀𝑛 ∈ ℕ (𝑛 < 𝑚 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)))) → 𝑦 ⊆ ℕ)
107		simprl 790	. . . . . . . . . 10 ⊢ (((𝑦 ⊆ ℕ ∧ 𝑧 ⊆ ℕ) ∧ (𝑚 ∈ (𝑧 ∖ 𝑦) ∧ ∀𝑛 ∈ ℕ (𝑛 < 𝑚 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)))) → 𝑚 ∈ (𝑧 ∖ 𝑦))
108		simprr 792	. . . . . . . . . . 11 ⊢ (((𝑦 ⊆ ℕ ∧ 𝑧 ⊆ ℕ) ∧ (𝑚 ∈ (𝑧 ∖ 𝑦) ∧ ∀𝑛 ∈ ℕ (𝑛 < 𝑚 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)))) → ∀𝑛 ∈ ℕ (𝑛 < 𝑚 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)))
109		bicom 211	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ 𝑧 ↔ 𝑛 ∈ 𝑦) ↔ (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧))
110	109	imbi2i 325	. . . . . . . . . . . 12 ⊢ ((𝑛 < 𝑚 → (𝑛 ∈ 𝑧 ↔ 𝑛 ∈ 𝑦)) ↔ (𝑛 < 𝑚 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)))
111	110	ralbii 2963	. . . . . . . . . . 11 ⊢ (∀𝑛 ∈ ℕ (𝑛 < 𝑚 → (𝑛 ∈ 𝑧 ↔ 𝑛 ∈ 𝑦)) ↔ ∀𝑛 ∈ ℕ (𝑛 < 𝑚 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)))
112	108, 111	sylibr 223	. . . . . . . . . 10 ⊢ (((𝑦 ⊆ ℕ ∧ 𝑧 ⊆ ℕ) ∧ (𝑚 ∈ (𝑧 ∖ 𝑦) ∧ ∀𝑛 ∈ ℕ (𝑛 < 𝑚 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)))) → ∀𝑛 ∈ ℕ (𝑛 < 𝑚 → (𝑛 ∈ 𝑧 ↔ 𝑛 ∈ 𝑦)))
113		eqcom 2617	. . . . . . . . . 10 ⊢ (Σ𝑘 ∈ ℕ ((𝐹‘𝑦)‘𝑘) = Σ𝑘 ∈ ℕ ((𝐹‘𝑧)‘𝑘) ↔ Σ𝑘 ∈ ℕ ((𝐹‘𝑧)‘𝑘) = Σ𝑘 ∈ ℕ ((𝐹‘𝑦)‘𝑘))
114	6, 105, 106, 107, 112, 113	rpnnen2lem11 14792	. . . . . . . . 9 ⊢ (((𝑦 ⊆ ℕ ∧ 𝑧 ⊆ ℕ) ∧ (𝑚 ∈ (𝑧 ∖ 𝑦) ∧ ∀𝑛 ∈ ℕ (𝑛 < 𝑚 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)))) → ¬ Σ𝑘 ∈ ℕ ((𝐹‘𝑦)‘𝑘) = Σ𝑘 ∈ ℕ ((𝐹‘𝑧)‘𝑘))
115	114	rexlimdvaa 3014	. . . . . . . 8 ⊢ ((𝑦 ⊆ ℕ ∧ 𝑧 ⊆ ℕ) → (∃𝑚 ∈ (𝑧 ∖ 𝑦)∀𝑛 ∈ ℕ (𝑛 < 𝑚 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)) → ¬ Σ𝑘 ∈ ℕ ((𝐹‘𝑦)‘𝑘) = Σ𝑘 ∈ ℕ ((𝐹‘𝑧)‘𝑘)))
116	104, 115	jaod 394	. . . . . . 7 ⊢ ((𝑦 ⊆ ℕ ∧ 𝑧 ⊆ ℕ) → ((∃𝑚 ∈ (𝑦 ∖ 𝑧)∀𝑛 ∈ ℕ (𝑛 < 𝑚 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)) ∨ ∃𝑚 ∈ (𝑧 ∖ 𝑦)∀𝑛 ∈ ℕ (𝑛 < 𝑚 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧))) → ¬ Σ𝑘 ∈ ℕ ((𝐹‘𝑦)‘𝑘) = Σ𝑘 ∈ ℕ ((𝐹‘𝑧)‘𝑘)))
117	2, 52, 116	syl2an 493	. . . . . 6 ⊢ ((𝑦 ∈ 𝒫 ℕ ∧ 𝑧 ∈ 𝒫 ℕ) → ((∃𝑚 ∈ (𝑦 ∖ 𝑧)∀𝑛 ∈ ℕ (𝑛 < 𝑚 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)) ∨ ∃𝑚 ∈ (𝑧 ∖ 𝑦)∀𝑛 ∈ ℕ (𝑛 < 𝑚 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧))) → ¬ Σ𝑘 ∈ ℕ ((𝐹‘𝑦)‘𝑘) = Σ𝑘 ∈ ℕ ((𝐹‘𝑧)‘𝑘)))
118	97, 117	syld 46	. . . . 5 ⊢ ((𝑦 ∈ 𝒫 ℕ ∧ 𝑧 ∈ 𝒫 ℕ) → (𝑦 ≠ 𝑧 → ¬ Σ𝑘 ∈ ℕ ((𝐹‘𝑦)‘𝑘) = Σ𝑘 ∈ ℕ ((𝐹‘𝑧)‘𝑘)))
119	118	necon4ad 2801	. . . 4 ⊢ ((𝑦 ∈ 𝒫 ℕ ∧ 𝑧 ∈ 𝒫 ℕ) → (Σ𝑘 ∈ ℕ ((𝐹‘𝑦)‘𝑘) = Σ𝑘 ∈ ℕ ((𝐹‘𝑧)‘𝑘) → 𝑦 = 𝑧))
120		fveq2 6103	. . . . . 6 ⊢ (𝑦 = 𝑧 → (𝐹‘𝑦) = (𝐹‘𝑧))
121	120	fveq1d 6105	. . . . 5 ⊢ (𝑦 = 𝑧 → ((𝐹‘𝑦)‘𝑘) = ((𝐹‘𝑧)‘𝑘))
122	121	sumeq2sdv 14282	. . . 4 ⊢ (𝑦 = 𝑧 → Σ𝑘 ∈ ℕ ((𝐹‘𝑦)‘𝑘) = Σ𝑘 ∈ ℕ ((𝐹‘𝑧)‘𝑘))
123	119, 122	impbid1 214	. . 3 ⊢ ((𝑦 ∈ 𝒫 ℕ ∧ 𝑧 ∈ 𝒫 ℕ) → (Σ𝑘 ∈ ℕ ((𝐹‘𝑦)‘𝑘) = Σ𝑘 ∈ ℕ ((𝐹‘𝑧)‘𝑘) ↔ 𝑦 = 𝑧))
124	51, 123	dom2 7884	. 2 ⊢ ((0[,]1) ∈ V → 𝒫 ℕ ≼ (0[,]1))
125	1, 124	ax-mp 5	1 ⊢ 𝒫 ℕ ≼ (0[,]1)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383  ∀wal 1473   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ∖ cdif 3537   ∪ cun 3538   ⊆ wss 3540  ∅c0 3874  ifcif 4036  𝒫 cpw 4108   class class class wbr 4583   ↦ cmpt 4643  dom cdm 5038  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ≼ cdom 7839  ℂcc 9813  ℝcr 9814  0cc0 9815  1c1 9816   + caddc 9818   < clt 9953   ≤ cle 9954   / cdiv 10563  ℕcn 10897  2c2 10947  3c3 10948  ℤ_≥cuz 11563  [,]cicc 12049  seqcseq 12663  ↑cexp 12722   ⇝ cli 14063  Σcsu 14264

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-ico 12052  df-icc 12053  df-fz 12198  df-fzo 12335  df-fl 12455  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-limsup 14050  df-clim 14067  df-rlim 14068  df-sum 14265

This theorem is referenced by:  rpnnen2  14794