Theorem rpnnen2lem12 15572

Description: Lemma for rpnnen2 15573. (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 7183	. 2 ⊢ (0[,]1) ∈ V
2		elpwi 4551	. . . . 5 ⊢ (𝑦 ∈ 𝒫 ℕ → 𝑦 ⊆ ℕ)
3		nnuz 12275	. . . . . . 7 ⊢ ℕ = (ℤ≥‘1)
4	3	sumeq1i 15049	. . . . . 6 ⊢ Σ𝑘 ∈ ℕ ((𝐹‘𝑦)‘𝑘) = Σ𝑘 ∈ (ℤ≥‘1)((𝐹‘𝑦)‘𝑘)
5		1nn 11643	. . . . . . 7 ⊢ 1 ∈ ℕ
6		rpnnen2.1	. . . . . . . 8 ⊢ 𝐹 = (𝑥 ∈ 𝒫 ℕ ↦ (𝑛 ∈ ℕ ↦ if(𝑛 ∈ 𝑥, ((1 / 3)↑𝑛), 0)))
7	6	rpnnen2lem6 15566	. . . . . . 7 ⊢ ((𝑦 ⊆ ℕ ∧ 1 ∈ ℕ) → Σ𝑘 ∈ (ℤ≥‘1)((𝐹‘𝑦)‘𝑘) ∈ ℝ)
8	5, 7	mpan2 689	. . . . . 6 ⊢ (𝑦 ⊆ ℕ → Σ𝑘 ∈ (ℤ≥‘1)((𝐹‘𝑦)‘𝑘) ∈ ℝ)
9	4, 8	eqeltrid 2917	. . . . 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 12007	. . . . 5 ⊢ (𝑦 ∈ 𝒫 ℕ → 1 ∈ ℤ)
12		eqidd 2822	. . . . 5 ⊢ ((𝑦 ∈ 𝒫 ℕ ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑦)‘𝑘) = ((𝐹‘𝑦)‘𝑘))
13	6	rpnnen2lem2 15562	. . . . . . 7 ⊢ (𝑦 ⊆ ℕ → (𝐹‘𝑦):ℕ⟶ℝ)
14	2, 13	syl 17	. . . . . 6 ⊢ (𝑦 ∈ 𝒫 ℕ → (𝐹‘𝑦):ℕ⟶ℝ)
15	14	ffvelrnda 6846	. . . . 5 ⊢ ((𝑦 ∈ 𝒫 ℕ ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑦)‘𝑘) ∈ ℝ)
16	6	rpnnen2lem5 15565	. . . . . 6 ⊢ ((𝑦 ⊆ ℕ ∧ 1 ∈ ℕ) → seq1( + , (𝐹‘𝑦)) ∈ dom ⇝ )
17	2, 5, 16	sylancl 588	. . . . 5 ⊢ (𝑦 ∈ 𝒫 ℕ → seq1( + , (𝐹‘𝑦)) ∈ dom ⇝ )
18		ssid 3989	. . . . . . . 8 ⊢ ℕ ⊆ ℕ
19	6	rpnnen2lem4 15564	. . . . . . . 8 ⊢ ((𝑦 ⊆ ℕ ∧ ℕ ⊆ ℕ ∧ 𝑘 ∈ ℕ) → (0 ≤ ((𝐹‘𝑦)‘𝑘) ∧ ((𝐹‘𝑦)‘𝑘) ≤ ((𝐹‘ℕ)‘𝑘)))
20	18, 19	mp3an2 1445	. . . . . . 7 ⊢ ((𝑦 ⊆ ℕ ∧ 𝑘 ∈ ℕ) → (0 ≤ ((𝐹‘𝑦)‘𝑘) ∧ ((𝐹‘𝑦)‘𝑘) ≤ ((𝐹‘ℕ)‘𝑘)))
21	20	simpld 497	. . . . . 6 ⊢ ((𝑦 ⊆ ℕ ∧ 𝑘 ∈ ℕ) → 0 ≤ ((𝐹‘𝑦)‘𝑘))
22	2, 21	sylan 582	. . . . 5 ⊢ ((𝑦 ∈ 𝒫 ℕ ∧ 𝑘 ∈ ℕ) → 0 ≤ ((𝐹‘𝑦)‘𝑘))
23	3, 11, 12, 15, 17, 22	isumge0 15115	. . . 4 ⊢ (𝑦 ∈ 𝒫 ℕ → 0 ≤ Σ𝑘 ∈ ℕ ((𝐹‘𝑦)‘𝑘))
24		halfre 11845	. . . . . 6 ⊢ (1 / 2) ∈ ℝ
25	24	a1i 11	. . . . 5 ⊢ (𝑦 ∈ 𝒫 ℕ → (1 / 2) ∈ ℝ)
26		1re 10635	. . . . . 6 ⊢ 1 ∈ ℝ
27	26	a1i 11	. . . . 5 ⊢ (𝑦 ∈ 𝒫 ℕ → 1 ∈ ℝ)
28	6	rpnnen2lem7 15567	. . . . . . . . 9 ⊢ ((𝑦 ⊆ ℕ ∧ ℕ ⊆ ℕ ∧ 1 ∈ ℕ) → Σ𝑘 ∈ (ℤ≥‘1)((𝐹‘𝑦)‘𝑘) ≤ Σ𝑘 ∈ (ℤ≥‘1)((𝐹‘ℕ)‘𝑘))
29	18, 5, 28	mp3an23 1449	. . . . . . . 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 2821	. . . . . . . 8 ⊢ (ℤ≥‘1) = (ℤ≥‘1)
32		eqidd 2822	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝒫 ℕ ∧ 𝑘 ∈ (ℤ≥‘1)) → ((𝐹‘ℕ)‘𝑘) = ((𝐹‘ℕ)‘𝑘))
33		elnnuz 12276	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ ↔ 𝑘 ∈ (ℤ≥‘1))
34	6	rpnnen2lem2 15562	. . . . . . . . . . . . 13 ⊢ (ℕ ⊆ ℕ → (𝐹‘ℕ):ℕ⟶ℝ)
35	18, 34	ax-mp 5	. . . . . . . . . . . 12 ⊢ (𝐹‘ℕ):ℕ⟶ℝ
36	35	ffvelrni 6845	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → ((𝐹‘ℕ)‘𝑘) ∈ ℝ)
37	36	recnd 10663	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → ((𝐹‘ℕ)‘𝑘) ∈ ℂ)
38	33, 37	sylbir 237	. . . . . . . . 9 ⊢ (𝑘 ∈ (ℤ≥‘1) → ((𝐹‘ℕ)‘𝑘) ∈ ℂ)
39	38	adantl 484	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝒫 ℕ ∧ 𝑘 ∈ (ℤ≥‘1)) → ((𝐹‘ℕ)‘𝑘) ∈ ℂ)
40	6	rpnnen2lem3 15563	. . . . . . . . 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 15106	. . . . . . 7 ⊢ (𝑦 ∈ 𝒫 ℕ → Σ𝑘 ∈ (ℤ≥‘1)((𝐹‘ℕ)‘𝑘) = (1 / 2))
43	30, 42	breqtrd 5085	. . . . . 6 ⊢ (𝑦 ∈ 𝒫 ℕ → Σ𝑘 ∈ (ℤ≥‘1)((𝐹‘𝑦)‘𝑘) ≤ (1 / 2))
44	4, 43	eqbrtrid 5094	. . . . 5 ⊢ (𝑦 ∈ 𝒫 ℕ → Σ𝑘 ∈ ℕ ((𝐹‘𝑦)‘𝑘) ≤ (1 / 2))
45		halflt1 11849	. . . . . . 7 ⊢ (1 / 2) < 1
46	24, 26, 45	ltleii 10757	. . . . . 6 ⊢ (1 / 2) ≤ 1
47	46	a1i 11	. . . . 5 ⊢ (𝑦 ∈ 𝒫 ℕ → (1 / 2) ≤ 1)
48	10, 25, 27, 44, 47	letrd 10791	. . . 4 ⊢ (𝑦 ∈ 𝒫 ℕ → Σ𝑘 ∈ ℕ ((𝐹‘𝑦)‘𝑘) ≤ 1)
49		elicc01 12848	. . . 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))
50	10, 23, 48, 49	syl3anbrc 1339	. . 3 ⊢ (𝑦 ∈ 𝒫 ℕ → Σ𝑘 ∈ ℕ ((𝐹‘𝑦)‘𝑘) ∈ (0[,]1))
51		elpwi 4551	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝒫 ℕ → 𝑧 ⊆ ℕ)
52		ssdifss 4112	. . . . . . . . . . . 12 ⊢ (𝑦 ⊆ ℕ → (𝑦 ∖ 𝑧) ⊆ ℕ)
53		ssdifss 4112	. . . . . . . . . . . 12 ⊢ (𝑧 ⊆ ℕ → (𝑧 ∖ 𝑦) ⊆ ℕ)
54		unss 4160	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∖ 𝑧) ⊆ ℕ ∧ (𝑧 ∖ 𝑦) ⊆ ℕ) ↔ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦)) ⊆ ℕ)
55	54	biimpi 218	. . . . . . . . . . . 12 ⊢ (((𝑦 ∖ 𝑧) ⊆ ℕ ∧ (𝑧 ∖ 𝑦) ⊆ ℕ) → ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦)) ⊆ ℕ)
56	52, 53, 55	syl2an 597	. . . . . . . . . . 11 ⊢ ((𝑦 ⊆ ℕ ∧ 𝑧 ⊆ ℕ) → ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦)) ⊆ ℕ)
57	2, 51, 56	syl2an 597	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝒫 ℕ ∧ 𝑧 ∈ 𝒫 ℕ) → ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦)) ⊆ ℕ)
58		eqss 3982	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑧 ↔ (𝑦 ⊆ 𝑧 ∧ 𝑧 ⊆ 𝑦))
59		ssdif0 4323	. . . . . . . . . . . . . 14 ⊢ (𝑦 ⊆ 𝑧 ↔ (𝑦 ∖ 𝑧) = ∅)
60		ssdif0 4323	. . . . . . . . . . . . . 14 ⊢ (𝑧 ⊆ 𝑦 ↔ (𝑧 ∖ 𝑦) = ∅)
61	59, 60	anbi12i 628	. . . . . . . . . . . . 13 ⊢ ((𝑦 ⊆ 𝑧 ∧ 𝑧 ⊆ 𝑦) ↔ ((𝑦 ∖ 𝑧) = ∅ ∧ (𝑧 ∖ 𝑦) = ∅))
62		un00 4394	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∖ 𝑧) = ∅ ∧ (𝑧 ∖ 𝑦) = ∅) ↔ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦)) = ∅)
63	58, 61, 62	3bitri 299	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑧 ↔ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦)) = ∅)
64	63	necon3bii 3068	. . . . . . . . . . 11 ⊢ (𝑦 ≠ 𝑧 ↔ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦)) ≠ ∅)
65	64	biimpi 218	. . . . . . . . . 10 ⊢ (𝑦 ≠ 𝑧 → ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦)) ≠ ∅)
66		nnwo 12307	. . . . . . . . . 10 ⊢ ((((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦)) ⊆ ℕ ∧ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦)) ≠ ∅) → ∃𝑚 ∈ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦))∀𝑛 ∈ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦))𝑚 ≤ 𝑛)
67	57, 65, 66	syl2an 597	. . . . . . . . 9 ⊢ (((𝑦 ∈ 𝒫 ℕ ∧ 𝑧 ∈ 𝒫 ℕ) ∧ 𝑦 ≠ 𝑧) → ∃𝑚 ∈ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦))∀𝑛 ∈ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦))𝑚 ≤ 𝑛)
68	67	ex 415	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝒫 ℕ ∧ 𝑧 ∈ 𝒫 ℕ) → (𝑦 ≠ 𝑧 → ∃𝑚 ∈ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦))∀𝑛 ∈ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦))𝑚 ≤ 𝑛))
69	57	sselda 3967	. . . . . . . . . 10 ⊢ (((𝑦 ∈ 𝒫 ℕ ∧ 𝑧 ∈ 𝒫 ℕ) ∧ 𝑚 ∈ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦))) → 𝑚 ∈ ℕ)
70		df-ral 3143	. . . . . . . . . . . 12 ⊢ (∀𝑛 ∈ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦))𝑚 ≤ 𝑛 ↔ ∀𝑛(𝑛 ∈ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦)) → 𝑚 ≤ 𝑛))
71		con34b 318	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦)) → 𝑚 ≤ 𝑛) ↔ (¬ 𝑚 ≤ 𝑛 → ¬ 𝑛 ∈ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦))))
72		eldif 3946	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (𝑦 ∖ 𝑧) ↔ (𝑛 ∈ 𝑦 ∧ ¬ 𝑛 ∈ 𝑧))
73		eldif 3946	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (𝑧 ∖ 𝑦) ↔ (𝑛 ∈ 𝑧 ∧ ¬ 𝑛 ∈ 𝑦))
74	72, 73	orbi12i 911	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ (𝑦 ∖ 𝑧) ∨ 𝑛 ∈ (𝑧 ∖ 𝑦)) ↔ ((𝑛 ∈ 𝑦 ∧ ¬ 𝑛 ∈ 𝑧) ∨ (𝑛 ∈ 𝑧 ∧ ¬ 𝑛 ∈ 𝑦)))
75		elun 4125	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦)) ↔ (𝑛 ∈ (𝑦 ∖ 𝑧) ∨ 𝑛 ∈ (𝑧 ∖ 𝑦)))
76		xor 1011	. . . . . . . . . . . . . . . . 17 ⊢ (¬ (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧) ↔ ((𝑛 ∈ 𝑦 ∧ ¬ 𝑛 ∈ 𝑧) ∨ (𝑛 ∈ 𝑧 ∧ ¬ 𝑛 ∈ 𝑦)))
77	74, 75, 76	3bitr4ri 306	. . . . . . . . . . . . . . . 16 ⊢ (¬ (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧) ↔ 𝑛 ∈ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦)))
78	77	con1bii 359	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑛 ∈ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦)) ↔ (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧))
79	78	imbi2i 338	. . . . . . . . . . . . . 14 ⊢ ((¬ 𝑚 ≤ 𝑛 → ¬ 𝑛 ∈ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦))) ↔ (¬ 𝑚 ≤ 𝑛 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)))
80	71, 79	bitri 277	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦)) → 𝑚 ≤ 𝑛) ↔ (¬ 𝑚 ≤ 𝑛 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)))
81	80	albii 1816	. . . . . . . . . . . 12 ⊢ (∀𝑛(𝑛 ∈ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦)) → 𝑚 ≤ 𝑛) ↔ ∀𝑛(¬ 𝑚 ≤ 𝑛 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)))
82	70, 81	bitri 277	. . . . . . . . . . 11 ⊢ (∀𝑛 ∈ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦))𝑚 ≤ 𝑛 ↔ ∀𝑛(¬ 𝑚 ≤ 𝑛 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)))
83		alral 3154	. . . . . . . . . . . 12 ⊢ (∀𝑛(¬ 𝑚 ≤ 𝑛 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)) → ∀𝑛 ∈ ℕ (¬ 𝑚 ≤ 𝑛 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)))
84		nnre 11639	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
85		nnre 11639	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℝ)
86		ltnle 10714	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℝ ∧ 𝑚 ∈ ℝ) → (𝑛 < 𝑚 ↔ ¬ 𝑚 ≤ 𝑛))
87	84, 85, 86	syl2anr 598	. . . . . . . . . . . . . 14 ⊢ ((𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) → (𝑛 < 𝑚 ↔ ¬ 𝑚 ≤ 𝑛))
88	87	imbi1d 344	. . . . . . . . . . . . 13 ⊢ ((𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) → ((𝑛 < 𝑚 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)) ↔ (¬ 𝑚 ≤ 𝑛 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧))))
89	88	ralbidva 3196	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ → (∀𝑛 ∈ ℕ (𝑛 < 𝑚 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)) ↔ ∀𝑛 ∈ ℕ (¬ 𝑚 ≤ 𝑛 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧))))
90	83, 89	syl5ibr 248	. . . . . . . . . . 11 ⊢ (𝑚 ∈ ℕ → (∀𝑛(¬ 𝑚 ≤ 𝑛 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)) → ∀𝑛 ∈ ℕ (𝑛 < 𝑚 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧))))
91	82, 90	syl5bi 244	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ → (∀𝑛 ∈ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦))𝑚 ≤ 𝑛 → ∀𝑛 ∈ ℕ (𝑛 < 𝑚 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧))))
92	69, 91	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))))
93	92	reximdva 3274	. . . . . . . 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))))
94	68, 93	syld 47	. . . . . . 7 ⊢ ((𝑦 ∈ 𝒫 ℕ ∧ 𝑧 ∈ 𝒫 ℕ) → (𝑦 ≠ 𝑧 → ∃𝑚 ∈ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦))∀𝑛 ∈ ℕ (𝑛 < 𝑚 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧))))
95		rexun 4166	. . . . . . 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))))
96	94, 95	syl6ib 253	. . . . . 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)))))
97		simpll 765	. . . . . . . . . 10 ⊢ (((𝑦 ⊆ ℕ ∧ 𝑧 ⊆ ℕ) ∧ (𝑚 ∈ (𝑦 ∖ 𝑧) ∧ ∀𝑛 ∈ ℕ (𝑛 < 𝑚 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)))) → 𝑦 ⊆ ℕ)
98		simplr 767	. . . . . . . . . 10 ⊢ (((𝑦 ⊆ ℕ ∧ 𝑧 ⊆ ℕ) ∧ (𝑚 ∈ (𝑦 ∖ 𝑧) ∧ ∀𝑛 ∈ ℕ (𝑛 < 𝑚 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)))) → 𝑧 ⊆ ℕ)
99		simprl 769	. . . . . . . . . 10 ⊢ (((𝑦 ⊆ ℕ ∧ 𝑧 ⊆ ℕ) ∧ (𝑚 ∈ (𝑦 ∖ 𝑧) ∧ ∀𝑛 ∈ ℕ (𝑛 < 𝑚 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)))) → 𝑚 ∈ (𝑦 ∖ 𝑧))
100		simprr 771	. . . . . . . . . 10 ⊢ (((𝑦 ⊆ ℕ ∧ 𝑧 ⊆ ℕ) ∧ (𝑚 ∈ (𝑦 ∖ 𝑧) ∧ ∀𝑛 ∈ ℕ (𝑛 < 𝑚 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)))) → ∀𝑛 ∈ ℕ (𝑛 < 𝑚 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)))
101		biid 263	. . . . . . . . . 10 ⊢ (Σ𝑘 ∈ ℕ ((𝐹‘𝑦)‘𝑘) = Σ𝑘 ∈ ℕ ((𝐹‘𝑧)‘𝑘) ↔ Σ𝑘 ∈ ℕ ((𝐹‘𝑦)‘𝑘) = Σ𝑘 ∈ ℕ ((𝐹‘𝑧)‘𝑘))
102	6, 97, 98, 99, 100, 101	rpnnen2lem11 15571	. . . . . . . . 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))
103	102	rexlimdvaa 3285	. . . . . . . 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)))
104		simplr 767	. . . . . . . . . 10 ⊢ (((𝑦 ⊆ ℕ ∧ 𝑧 ⊆ ℕ) ∧ (𝑚 ∈ (𝑧 ∖ 𝑦) ∧ ∀𝑛 ∈ ℕ (𝑛 < 𝑚 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)))) → 𝑧 ⊆ ℕ)
105		simpll 765	. . . . . . . . . 10 ⊢ (((𝑦 ⊆ ℕ ∧ 𝑧 ⊆ ℕ) ∧ (𝑚 ∈ (𝑧 ∖ 𝑦) ∧ ∀𝑛 ∈ ℕ (𝑛 < 𝑚 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)))) → 𝑦 ⊆ ℕ)
106		simprl 769	. . . . . . . . . 10 ⊢ (((𝑦 ⊆ ℕ ∧ 𝑧 ⊆ ℕ) ∧ (𝑚 ∈ (𝑧 ∖ 𝑦) ∧ ∀𝑛 ∈ ℕ (𝑛 < 𝑚 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)))) → 𝑚 ∈ (𝑧 ∖ 𝑦))
107		simprr 771	. . . . . . . . . . 11 ⊢ (((𝑦 ⊆ ℕ ∧ 𝑧 ⊆ ℕ) ∧ (𝑚 ∈ (𝑧 ∖ 𝑦) ∧ ∀𝑛 ∈ ℕ (𝑛 < 𝑚 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)))) → ∀𝑛 ∈ ℕ (𝑛 < 𝑚 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)))
108		bicom 224	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ 𝑧 ↔ 𝑛 ∈ 𝑦) ↔ (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧))
109	108	imbi2i 338	. . . . . . . . . . . 12 ⊢ ((𝑛 < 𝑚 → (𝑛 ∈ 𝑧 ↔ 𝑛 ∈ 𝑦)) ↔ (𝑛 < 𝑚 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)))
110	109	ralbii 3165	. . . . . . . . . . 11 ⊢ (∀𝑛 ∈ ℕ (𝑛 < 𝑚 → (𝑛 ∈ 𝑧 ↔ 𝑛 ∈ 𝑦)) ↔ ∀𝑛 ∈ ℕ (𝑛 < 𝑚 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)))
111	107, 110	sylibr 236	. . . . . . . . . 10 ⊢ (((𝑦 ⊆ ℕ ∧ 𝑧 ⊆ ℕ) ∧ (𝑚 ∈ (𝑧 ∖ 𝑦) ∧ ∀𝑛 ∈ ℕ (𝑛 < 𝑚 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)))) → ∀𝑛 ∈ ℕ (𝑛 < 𝑚 → (𝑛 ∈ 𝑧 ↔ 𝑛 ∈ 𝑦)))
112		eqcom 2828	. . . . . . . . . 10 ⊢ (Σ𝑘 ∈ ℕ ((𝐹‘𝑦)‘𝑘) = Σ𝑘 ∈ ℕ ((𝐹‘𝑧)‘𝑘) ↔ Σ𝑘 ∈ ℕ ((𝐹‘𝑧)‘𝑘) = Σ𝑘 ∈ ℕ ((𝐹‘𝑦)‘𝑘))
113	6, 104, 105, 106, 111, 112	rpnnen2lem11 15571	. . . . . . . . 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))
114	113	rexlimdvaa 3285	. . . . . . . 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)))
115	103, 114	jaod 855	. . . . . . 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)))
116	2, 51, 115	syl2an 597	. . . . . 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)))
117	96, 116	syld 47	. . . . 5 ⊢ ((𝑦 ∈ 𝒫 ℕ ∧ 𝑧 ∈ 𝒫 ℕ) → (𝑦 ≠ 𝑧 → ¬ Σ𝑘 ∈ ℕ ((𝐹‘𝑦)‘𝑘) = Σ𝑘 ∈ ℕ ((𝐹‘𝑧)‘𝑘)))
118	117	necon4ad 3035	. . . 4 ⊢ ((𝑦 ∈ 𝒫 ℕ ∧ 𝑧 ∈ 𝒫 ℕ) → (Σ𝑘 ∈ ℕ ((𝐹‘𝑦)‘𝑘) = Σ𝑘 ∈ ℕ ((𝐹‘𝑧)‘𝑘) → 𝑦 = 𝑧))
119		fveq2 6665	. . . . . 6 ⊢ (𝑦 = 𝑧 → (𝐹‘𝑦) = (𝐹‘𝑧))
120	119	fveq1d 6667	. . . . 5 ⊢ (𝑦 = 𝑧 → ((𝐹‘𝑦)‘𝑘) = ((𝐹‘𝑧)‘𝑘))
121	120	sumeq2sdv 15055	. . . 4 ⊢ (𝑦 = 𝑧 → Σ𝑘 ∈ ℕ ((𝐹‘𝑦)‘𝑘) = Σ𝑘 ∈ ℕ ((𝐹‘𝑧)‘𝑘))
122	118, 121	impbid1 227	. . 3 ⊢ ((𝑦 ∈ 𝒫 ℕ ∧ 𝑧 ∈ 𝒫 ℕ) → (Σ𝑘 ∈ ℕ ((𝐹‘𝑦)‘𝑘) = Σ𝑘 ∈ ℕ ((𝐹‘𝑧)‘𝑘) ↔ 𝑦 = 𝑧))
123	50, 122	dom2 8546	. 2 ⊢ ((0[,]1) ∈ V → 𝒫 ℕ ≼ (0[,]1))
124	1, 123	ax-mp 5	1 ⊢ 𝒫 ℕ ≼ (0[,]1)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∨ wo 843  ∀wal 1531   = wceq 1533   ∈ wcel 2110   ≠ wne 3016  ∀wral 3138  ∃wrex 3139  Vcvv 3495   ∖ cdif 3933   ∪ cun 3934   ⊆ wss 3936  ∅c0 4291  ifcif 4467  𝒫 cpw 4539   class class class wbr 5059   ↦ cmpt 5139  dom cdm 5550  ⟶wf 6346  ‘cfv 6350  (class class class)co 7150   ≼ cdom 8501  ℂcc 10529  ℝcr 10530  0cc0 10531  1c1 10532   + caddc 10534   < clt 10669   ≤ cle 10670   / cdiv 11291  ℕcn 11632  2c2 11686  3c3 11687  ℤ_≥cuz 12237  [,]cicc 12735  seqcseq 13363  ↑cexp 13423   ⇝ cli 14835  Σcsu 15036

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455  ax-inf2 9098  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608  ax-pre-sup 10609

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-fal 1546  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4833  df-int 4870  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-tr 5166  df-id 5455  df-eprel 5460  df-po 5469  df-so 5470  df-fr 5509  df-se 5510  df-we 5511  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-pred 6143  df-ord 6189  df-on 6190  df-lim 6191  df-suc 6192  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-isom 6359  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7575  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-oadd 8100  df-er 8283  df-pm 8403  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-sup 8900  df-inf 8901  df-oi 8968  df-card 9362  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-div 11292  df-nn 11633  df-2 11694  df-3 11695  df-n0 11892  df-z 11976  df-uz 12238  df-rp 12384  df-ico 12738  df-icc 12739  df-fz 12887  df-fzo 13028  df-fl 13156  df-seq 13364  df-exp 13424  df-hash 13685  df-cj 14452  df-re 14453  df-im 14454  df-sqrt 14588  df-abs 14589  df-limsup 14822  df-clim 14839  df-rlim 14840  df-sum 15037

This theorem is referenced by:  rpnnen2  15573