Theorem rpnnen2lem12 16010

Description: Lemma for rpnnen2 16011. (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 7349	. 2 ⊢ (0[,]1) ∈ V
2		elpwi 4551	. . . . 5 ⊢ (𝑦 ∈ 𝒫 ℕ → 𝑦 ⊆ ℕ)
3		nnuz 12700	. . . . . . 7 ⊢ ℕ = (ℤ≥‘1)
4	3	sumeq1i 15486	. . . . . 6 ⊢ Σ𝑘 ∈ ℕ ((𝐹‘𝑦)‘𝑘) = Σ𝑘 ∈ (ℤ≥‘1)((𝐹‘𝑦)‘𝑘)
5		1nn 12063	. . . . . . 7 ⊢ 1 ∈ ℕ
6		rpnnen2.1	. . . . . . . 8 ⊢ 𝐹 = (𝑥 ∈ 𝒫 ℕ ↦ (𝑛 ∈ ℕ ↦ if(𝑛 ∈ 𝑥, ((1 / 3)↑𝑛), 0)))
7	6	rpnnen2lem6 16004	. . . . . . 7 ⊢ ((𝑦 ⊆ ℕ ∧ 1 ∈ ℕ) → Σ𝑘 ∈ (ℤ≥‘1)((𝐹‘𝑦)‘𝑘) ∈ ℝ)
8	5, 7	mpan2 688	. . . . . 6 ⊢ (𝑦 ⊆ ℕ → Σ𝑘 ∈ (ℤ≥‘1)((𝐹‘𝑦)‘𝑘) ∈ ℝ)
9	4, 8	eqeltrid 2841	. . . . 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 12430	. . . . 5 ⊢ (𝑦 ∈ 𝒫 ℕ → 1 ∈ ℤ)
12		eqidd 2737	. . . . 5 ⊢ ((𝑦 ∈ 𝒫 ℕ ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑦)‘𝑘) = ((𝐹‘𝑦)‘𝑘))
13	6	rpnnen2lem2 16000	. . . . . . 7 ⊢ (𝑦 ⊆ ℕ → (𝐹‘𝑦):ℕ⟶ℝ)
14	2, 13	syl 17	. . . . . 6 ⊢ (𝑦 ∈ 𝒫 ℕ → (𝐹‘𝑦):ℕ⟶ℝ)
15	14	ffvelcdmda 7000	. . . . 5 ⊢ ((𝑦 ∈ 𝒫 ℕ ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑦)‘𝑘) ∈ ℝ)
16	6	rpnnen2lem5 16003	. . . . . 6 ⊢ ((𝑦 ⊆ ℕ ∧ 1 ∈ ℕ) → seq1( + , (𝐹‘𝑦)) ∈ dom ⇝ )
17	2, 5, 16	sylancl 586	. . . . 5 ⊢ (𝑦 ∈ 𝒫 ℕ → seq1( + , (𝐹‘𝑦)) ∈ dom ⇝ )
18		ssid 3952	. . . . . . . 8 ⊢ ℕ ⊆ ℕ
19	6	rpnnen2lem4 16002	. . . . . . . 8 ⊢ ((𝑦 ⊆ ℕ ∧ ℕ ⊆ ℕ ∧ 𝑘 ∈ ℕ) → (0 ≤ ((𝐹‘𝑦)‘𝑘) ∧ ((𝐹‘𝑦)‘𝑘) ≤ ((𝐹‘ℕ)‘𝑘)))
20	18, 19	mp3an2 1448	. . . . . . 7 ⊢ ((𝑦 ⊆ ℕ ∧ 𝑘 ∈ ℕ) → (0 ≤ ((𝐹‘𝑦)‘𝑘) ∧ ((𝐹‘𝑦)‘𝑘) ≤ ((𝐹‘ℕ)‘𝑘)))
21	20	simpld 495	. . . . . 6 ⊢ ((𝑦 ⊆ ℕ ∧ 𝑘 ∈ ℕ) → 0 ≤ ((𝐹‘𝑦)‘𝑘))
22	2, 21	sylan 580	. . . . 5 ⊢ ((𝑦 ∈ 𝒫 ℕ ∧ 𝑘 ∈ ℕ) → 0 ≤ ((𝐹‘𝑦)‘𝑘))
23	3, 11, 12, 15, 17, 22	isumge0 15554	. . . 4 ⊢ (𝑦 ∈ 𝒫 ℕ → 0 ≤ Σ𝑘 ∈ ℕ ((𝐹‘𝑦)‘𝑘))
24		halfre 12266	. . . . . 6 ⊢ (1 / 2) ∈ ℝ
25	24	a1i 11	. . . . 5 ⊢ (𝑦 ∈ 𝒫 ℕ → (1 / 2) ∈ ℝ)
26		1re 11054	. . . . . 6 ⊢ 1 ∈ ℝ
27	26	a1i 11	. . . . 5 ⊢ (𝑦 ∈ 𝒫 ℕ → 1 ∈ ℝ)
28	6	rpnnen2lem7 16005	. . . . . . . . 9 ⊢ ((𝑦 ⊆ ℕ ∧ ℕ ⊆ ℕ ∧ 1 ∈ ℕ) → Σ𝑘 ∈ (ℤ≥‘1)((𝐹‘𝑦)‘𝑘) ≤ Σ𝑘 ∈ (ℤ≥‘1)((𝐹‘ℕ)‘𝑘))
29	18, 5, 28	mp3an23 1452	. . . . . . . 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 2736	. . . . . . . 8 ⊢ (ℤ≥‘1) = (ℤ≥‘1)
32		eqidd 2737	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝒫 ℕ ∧ 𝑘 ∈ (ℤ≥‘1)) → ((𝐹‘ℕ)‘𝑘) = ((𝐹‘ℕ)‘𝑘))
33		elnnuz 12701	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ ↔ 𝑘 ∈ (ℤ≥‘1))
34	6	rpnnen2lem2 16000	. . . . . . . . . . . . 13 ⊢ (ℕ ⊆ ℕ → (𝐹‘ℕ):ℕ⟶ℝ)
35	18, 34	ax-mp 5	. . . . . . . . . . . 12 ⊢ (𝐹‘ℕ):ℕ⟶ℝ
36	35	ffvelcdmi 6999	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → ((𝐹‘ℕ)‘𝑘) ∈ ℝ)
37	36	recnd 11082	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → ((𝐹‘ℕ)‘𝑘) ∈ ℂ)
38	33, 37	sylbir 234	. . . . . . . . 9 ⊢ (𝑘 ∈ (ℤ≥‘1) → ((𝐹‘ℕ)‘𝑘) ∈ ℂ)
39	38	adantl 482	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝒫 ℕ ∧ 𝑘 ∈ (ℤ≥‘1)) → ((𝐹‘ℕ)‘𝑘) ∈ ℂ)
40	6	rpnnen2lem3 16001	. . . . . . . . 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 15545	. . . . . . 7 ⊢ (𝑦 ∈ 𝒫 ℕ → Σ𝑘 ∈ (ℤ≥‘1)((𝐹‘ℕ)‘𝑘) = (1 / 2))
43	30, 42	breqtrd 5112	. . . . . 6 ⊢ (𝑦 ∈ 𝒫 ℕ → Σ𝑘 ∈ (ℤ≥‘1)((𝐹‘𝑦)‘𝑘) ≤ (1 / 2))
44	4, 43	eqbrtrid 5121	. . . . 5 ⊢ (𝑦 ∈ 𝒫 ℕ → Σ𝑘 ∈ ℕ ((𝐹‘𝑦)‘𝑘) ≤ (1 / 2))
45		halflt1 12270	. . . . . . 7 ⊢ (1 / 2) < 1
46	24, 26, 45	ltleii 11177	. . . . . 6 ⊢ (1 / 2) ≤ 1
47	46	a1i 11	. . . . 5 ⊢ (𝑦 ∈ 𝒫 ℕ → (1 / 2) ≤ 1)
48	10, 25, 27, 44, 47	letrd 11211	. . . 4 ⊢ (𝑦 ∈ 𝒫 ℕ → Σ𝑘 ∈ ℕ ((𝐹‘𝑦)‘𝑘) ≤ 1)
49		elicc01 13277	. . . 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 1342	. . 3 ⊢ (𝑦 ∈ 𝒫 ℕ → Σ𝑘 ∈ ℕ ((𝐹‘𝑦)‘𝑘) ∈ (0[,]1))
51		elpwi 4551	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝒫 ℕ → 𝑧 ⊆ ℕ)
52		ssdifss 4080	. . . . . . . . . . . 12 ⊢ (𝑦 ⊆ ℕ → (𝑦 ∖ 𝑧) ⊆ ℕ)
53		ssdifss 4080	. . . . . . . . . . . 12 ⊢ (𝑧 ⊆ ℕ → (𝑧 ∖ 𝑦) ⊆ ℕ)
54		unss 4128	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∖ 𝑧) ⊆ ℕ ∧ (𝑧 ∖ 𝑦) ⊆ ℕ) ↔ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦)) ⊆ ℕ)
55	54	biimpi 215	. . . . . . . . . . . 12 ⊢ (((𝑦 ∖ 𝑧) ⊆ ℕ ∧ (𝑧 ∖ 𝑦) ⊆ ℕ) → ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦)) ⊆ ℕ)
56	52, 53, 55	syl2an 596	. . . . . . . . . . 11 ⊢ ((𝑦 ⊆ ℕ ∧ 𝑧 ⊆ ℕ) → ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦)) ⊆ ℕ)
57	2, 51, 56	syl2an 596	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝒫 ℕ ∧ 𝑧 ∈ 𝒫 ℕ) → ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦)) ⊆ ℕ)
58		eqss 3945	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑧 ↔ (𝑦 ⊆ 𝑧 ∧ 𝑧 ⊆ 𝑦))
59		ssdif0 4307	. . . . . . . . . . . . . 14 ⊢ (𝑦 ⊆ 𝑧 ↔ (𝑦 ∖ 𝑧) = ∅)
60		ssdif0 4307	. . . . . . . . . . . . . 14 ⊢ (𝑧 ⊆ 𝑦 ↔ (𝑧 ∖ 𝑦) = ∅)
61	59, 60	anbi12i 627	. . . . . . . . . . . . 13 ⊢ ((𝑦 ⊆ 𝑧 ∧ 𝑧 ⊆ 𝑦) ↔ ((𝑦 ∖ 𝑧) = ∅ ∧ (𝑧 ∖ 𝑦) = ∅))
62		un00 4386	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∖ 𝑧) = ∅ ∧ (𝑧 ∖ 𝑦) = ∅) ↔ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦)) = ∅)
63	58, 61, 62	3bitri 296	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑧 ↔ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦)) = ∅)
64	63	necon3bii 2993	. . . . . . . . . . 11 ⊢ (𝑦 ≠ 𝑧 ↔ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦)) ≠ ∅)
65	64	biimpi 215	. . . . . . . . . 10 ⊢ (𝑦 ≠ 𝑧 → ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦)) ≠ ∅)
66		nnwo 12732	. . . . . . . . . 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 596	. . . . . . . . 9 ⊢ (((𝑦 ∈ 𝒫 ℕ ∧ 𝑧 ∈ 𝒫 ℕ) ∧ 𝑦 ≠ 𝑧) → ∃𝑚 ∈ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦))∀𝑛 ∈ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦))𝑚 ≤ 𝑛)
68	67	ex 413	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝒫 ℕ ∧ 𝑧 ∈ 𝒫 ℕ) → (𝑦 ≠ 𝑧 → ∃𝑚 ∈ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦))∀𝑛 ∈ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦))𝑚 ≤ 𝑛))
69	57	sselda 3930	. . . . . . . . . 10 ⊢ (((𝑦 ∈ 𝒫 ℕ ∧ 𝑧 ∈ 𝒫 ℕ) ∧ 𝑚 ∈ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦))) → 𝑚 ∈ ℕ)
70		df-ral 3062	. . . . . . . . . . . 12 ⊢ (∀𝑛 ∈ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦))𝑚 ≤ 𝑛 ↔ ∀𝑛(𝑛 ∈ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦)) → 𝑚 ≤ 𝑛))
71		con34b 315	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦)) → 𝑚 ≤ 𝑛) ↔ (¬ 𝑚 ≤ 𝑛 → ¬ 𝑛 ∈ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦))))
72		eldif 3906	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (𝑦 ∖ 𝑧) ↔ (𝑛 ∈ 𝑦 ∧ ¬ 𝑛 ∈ 𝑧))
73		eldif 3906	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (𝑧 ∖ 𝑦) ↔ (𝑛 ∈ 𝑧 ∧ ¬ 𝑛 ∈ 𝑦))
74	72, 73	orbi12i 912	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ (𝑦 ∖ 𝑧) ∨ 𝑛 ∈ (𝑧 ∖ 𝑦)) ↔ ((𝑛 ∈ 𝑦 ∧ ¬ 𝑛 ∈ 𝑧) ∨ (𝑛 ∈ 𝑧 ∧ ¬ 𝑛 ∈ 𝑦)))
75		elun 4093	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦)) ↔ (𝑛 ∈ (𝑦 ∖ 𝑧) ∨ 𝑛 ∈ (𝑧 ∖ 𝑦)))
76		xor 1012	. . . . . . . . . . . . . . . . 17 ⊢ (¬ (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧) ↔ ((𝑛 ∈ 𝑦 ∧ ¬ 𝑛 ∈ 𝑧) ∨ (𝑛 ∈ 𝑧 ∧ ¬ 𝑛 ∈ 𝑦)))
77	74, 75, 76	3bitr4ri 303	. . . . . . . . . . . . . . . 16 ⊢ (¬ (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧) ↔ 𝑛 ∈ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦)))
78	77	con1bii 356	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑛 ∈ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦)) ↔ (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧))
79	78	imbi2i 335	. . . . . . . . . . . . . 14 ⊢ ((¬ 𝑚 ≤ 𝑛 → ¬ 𝑛 ∈ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦))) ↔ (¬ 𝑚 ≤ 𝑛 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)))
80	71, 79	bitri 274	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦)) → 𝑚 ≤ 𝑛) ↔ (¬ 𝑚 ≤ 𝑛 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)))
81	80	albii 1820	. . . . . . . . . . . 12 ⊢ (∀𝑛(𝑛 ∈ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦)) → 𝑚 ≤ 𝑛) ↔ ∀𝑛(¬ 𝑚 ≤ 𝑛 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)))
82	70, 81	bitri 274	. . . . . . . . . . 11 ⊢ (∀𝑛 ∈ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦))𝑚 ≤ 𝑛 ↔ ∀𝑛(¬ 𝑚 ≤ 𝑛 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)))
83		alral 3075	. . . . . . . . . . . 12 ⊢ (∀𝑛(¬ 𝑚 ≤ 𝑛 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)) → ∀𝑛 ∈ ℕ (¬ 𝑚 ≤ 𝑛 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)))
84		nnre 12059	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
85		nnre 12059	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℝ)
86		ltnle 11133	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℝ ∧ 𝑚 ∈ ℝ) → (𝑛 < 𝑚 ↔ ¬ 𝑚 ≤ 𝑛))
87	84, 85, 86	syl2anr 597	. . . . . . . . . . . . . 14 ⊢ ((𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) → (𝑛 < 𝑚 ↔ ¬ 𝑚 ≤ 𝑛))
88	87	imbi1d 341	. . . . . . . . . . . . 13 ⊢ ((𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) → ((𝑛 < 𝑚 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)) ↔ (¬ 𝑚 ≤ 𝑛 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧))))
89	88	ralbidva 3168	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ → (∀𝑛 ∈ ℕ (𝑛 < 𝑚 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)) ↔ ∀𝑛 ∈ ℕ (¬ 𝑚 ≤ 𝑛 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧))))
90	83, 89	syl5ibr 245	. . . . . . . . . . 11 ⊢ (𝑚 ∈ ℕ → (∀𝑛(¬ 𝑚 ≤ 𝑛 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)) → ∀𝑛 ∈ ℕ (𝑛 < 𝑚 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧))))
91	82, 90	biimtrid 241	. . . . . . . . . 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 3161	. . . . . . . 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 4134	. . . . . . 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 250	. . . . . 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 764	. . . . . . . . . 10 ⊢ (((𝑦 ⊆ ℕ ∧ 𝑧 ⊆ ℕ) ∧ (𝑚 ∈ (𝑦 ∖ 𝑧) ∧ ∀𝑛 ∈ ℕ (𝑛 < 𝑚 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)))) → 𝑦 ⊆ ℕ)
98		simplr 766	. . . . . . . . . 10 ⊢ (((𝑦 ⊆ ℕ ∧ 𝑧 ⊆ ℕ) ∧ (𝑚 ∈ (𝑦 ∖ 𝑧) ∧ ∀𝑛 ∈ ℕ (𝑛 < 𝑚 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)))) → 𝑧 ⊆ ℕ)
99		simprl 768	. . . . . . . . . 10 ⊢ (((𝑦 ⊆ ℕ ∧ 𝑧 ⊆ ℕ) ∧ (𝑚 ∈ (𝑦 ∖ 𝑧) ∧ ∀𝑛 ∈ ℕ (𝑛 < 𝑚 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)))) → 𝑚 ∈ (𝑦 ∖ 𝑧))
100		simprr 770	. . . . . . . . . 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 260	. . . . . . . . . 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 16009	. . . . . . . . 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 3149	. . . . . . . 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 766	. . . . . . . . . 10 ⊢ (((𝑦 ⊆ ℕ ∧ 𝑧 ⊆ ℕ) ∧ (𝑚 ∈ (𝑧 ∖ 𝑦) ∧ ∀𝑛 ∈ ℕ (𝑛 < 𝑚 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)))) → 𝑧 ⊆ ℕ)
105		simpll 764	. . . . . . . . . 10 ⊢ (((𝑦 ⊆ ℕ ∧ 𝑧 ⊆ ℕ) ∧ (𝑚 ∈ (𝑧 ∖ 𝑦) ∧ ∀𝑛 ∈ ℕ (𝑛 < 𝑚 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)))) → 𝑦 ⊆ ℕ)
106		simprl 768	. . . . . . . . . 10 ⊢ (((𝑦 ⊆ ℕ ∧ 𝑧 ⊆ ℕ) ∧ (𝑚 ∈ (𝑧 ∖ 𝑦) ∧ ∀𝑛 ∈ ℕ (𝑛 < 𝑚 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)))) → 𝑚 ∈ (𝑧 ∖ 𝑦))
107		simprr 770	. . . . . . . . . . 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 221	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ 𝑧 ↔ 𝑛 ∈ 𝑦) ↔ (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧))
109	108	imbi2i 335	. . . . . . . . . . . 12 ⊢ ((𝑛 < 𝑚 → (𝑛 ∈ 𝑧 ↔ 𝑛 ∈ 𝑦)) ↔ (𝑛 < 𝑚 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)))
110	109	ralbii 3092	. . . . . . . . . . 11 ⊢ (∀𝑛 ∈ ℕ (𝑛 < 𝑚 → (𝑛 ∈ 𝑧 ↔ 𝑛 ∈ 𝑦)) ↔ ∀𝑛 ∈ ℕ (𝑛 < 𝑚 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)))
111	107, 110	sylibr 233	. . . . . . . . . 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 2743	. . . . . . . . . 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 16009	. . . . . . . . 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 3149	. . . . . . . 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 856	. . . . . . 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 596	. . . . . 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 2959	. . . 4 ⊢ ((𝑦 ∈ 𝒫 ℕ ∧ 𝑧 ∈ 𝒫 ℕ) → (Σ𝑘 ∈ ℕ ((𝐹‘𝑦)‘𝑘) = Σ𝑘 ∈ ℕ ((𝐹‘𝑧)‘𝑘) → 𝑦 = 𝑧))
119		fveq2 6811	. . . . . 6 ⊢ (𝑦 = 𝑧 → (𝐹‘𝑦) = (𝐹‘𝑧))
120	119	fveq1d 6813	. . . . 5 ⊢ (𝑦 = 𝑧 → ((𝐹‘𝑦)‘𝑘) = ((𝐹‘𝑧)‘𝑘))
121	120	sumeq2sdv 15492	. . . 4 ⊢ (𝑦 = 𝑧 → Σ𝑘 ∈ ℕ ((𝐹‘𝑦)‘𝑘) = Σ𝑘 ∈ ℕ ((𝐹‘𝑧)‘𝑘))
122	118, 121	impbid1 224	. . 3 ⊢ ((𝑦 ∈ 𝒫 ℕ ∧ 𝑧 ∈ 𝒫 ℕ) → (Σ𝑘 ∈ ℕ ((𝐹‘𝑦)‘𝑘) = Σ𝑘 ∈ ℕ ((𝐹‘𝑧)‘𝑘) ↔ 𝑦 = 𝑧))
123	50, 122	dom2 8834	. 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 205   ∧ wa 396   ∨ wo 844  ∀wal 1538   = wceq 1540   ∈ wcel 2105   ≠ wne 2940  ∀wral 3061  ∃wrex 3070  Vcvv 3440   ∖ cdif 3893   ∪ cun 3894   ⊆ wss 3896  ∅c0 4266  ifcif 4470  𝒫 cpw 4544   class class class wbr 5086   ↦ cmpt 5169  dom cdm 5607  ⟶wf 6461  ‘cfv 6465  (class class class)co 7316   ≼ cdom 8780  ℂcc 10948  ℝcr 10949  0cc0 10950  1c1 10951   + caddc 10953   < clt 11088   ≤ cle 11089   / cdiv 11711  ℕcn 12052  2c2 12107  3c3 12108  ℤ_≥cuz 12661  [,]cicc 13161  seqcseq 13800  ↑cexp 13861   ⇝ cli 15269  Σcsu 15473

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5223  ax-sep 5237  ax-nul 5244  ax-pow 5302  ax-pr 5366  ax-un 7629  ax-inf2 9476  ax-cnex 11006  ax-resscn 11007  ax-1cn 11008  ax-icn 11009  ax-addcl 11010  ax-addrcl 11011  ax-mulcl 11012  ax-mulrcl 11013  ax-mulcom 11014  ax-addass 11015  ax-mulass 11016  ax-distr 11017  ax-i2m1 11018  ax-1ne0 11019  ax-1rid 11020  ax-rnegex 11021  ax-rrecex 11022  ax-cnre 11023  ax-pre-lttri 11024  ax-pre-lttrn 11025  ax-pre-ltadd 11026  ax-pre-mulgt0 11027  ax-pre-sup 11028

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3349  df-reu 3350  df-rab 3404  df-v 3442  df-sbc 3726  df-csb 3842  df-dif 3899  df-un 3901  df-in 3903  df-ss 3913  df-pss 3915  df-nul 4267  df-if 4471  df-pw 4546  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4850  df-int 4892  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5170  df-tr 5204  df-id 5506  df-eprel 5512  df-po 5520  df-so 5521  df-fr 5562  df-se 5563  df-we 5564  df-xp 5613  df-rel 5614  df-cnv 5615  df-co 5616  df-dm 5617  df-rn 5618  df-res 5619  df-ima 5620  df-pred 6224  df-ord 6291  df-on 6292  df-lim 6293  df-suc 6294  df-iota 6417  df-fun 6467  df-fn 6468  df-f 6469  df-f1 6470  df-fo 6471  df-f1o 6472  df-fv 6473  df-isom 6474  df-riota 7273  df-ov 7319  df-oprab 7320  df-mpo 7321  df-om 7759  df-1st 7877  df-2nd 7878  df-frecs 8145  df-wrecs 8176  df-recs 8250  df-rdg 8289  df-1o 8345  df-er 8547  df-pm 8667  df-en 8783  df-dom 8784  df-sdom 8785  df-fin 8786  df-sup 9277  df-inf 9278  df-oi 9345  df-card 9774  df-pnf 11090  df-mnf 11091  df-xr 11092  df-ltxr 11093  df-le 11094  df-sub 11286  df-neg 11287  df-div 11712  df-nn 12053  df-2 12115  df-3 12116  df-n0 12313  df-z 12399  df-uz 12662  df-rp 12810  df-ico 13164  df-icc 13165  df-fz 13319  df-fzo 13462  df-fl 13591  df-seq 13801  df-exp 13862  df-hash 14124  df-cj 14886  df-re 14887  df-im 14888  df-sqrt 15022  df-abs 15023  df-limsup 15256  df-clim 15273  df-rlim 15274  df-sum 15474

This theorem is referenced by:  rpnnen2  16011