Theorem rpnnen2lem12 16248

Description: Lemma for rpnnen2 16249. (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 7443	. 2 ⊢ (0[,]1) ∈ V
2		elpwi 4587	. . . . 5 ⊢ (𝑦 ∈ 𝒫 ℕ → 𝑦 ⊆ ℕ)
3		nnuz 12900	. . . . . . 7 ⊢ ℕ = (ℤ≥‘1)
4	3	sumeq1i 15718	. . . . . 6 ⊢ Σ𝑘 ∈ ℕ ((𝐹‘𝑦)‘𝑘) = Σ𝑘 ∈ (ℤ≥‘1)((𝐹‘𝑦)‘𝑘)
5		1nn 12256	. . . . . . 7 ⊢ 1 ∈ ℕ
6		rpnnen2.1	. . . . . . . 8 ⊢ 𝐹 = (𝑥 ∈ 𝒫 ℕ ↦ (𝑛 ∈ ℕ ↦ if(𝑛 ∈ 𝑥, ((1 / 3)↑𝑛), 0)))
7	6	rpnnen2lem6 16242	. . . . . . 7 ⊢ ((𝑦 ⊆ ℕ ∧ 1 ∈ ℕ) → Σ𝑘 ∈ (ℤ≥‘1)((𝐹‘𝑦)‘𝑘) ∈ ℝ)
8	5, 7	mpan2 691	. . . . . 6 ⊢ (𝑦 ⊆ ℕ → Σ𝑘 ∈ (ℤ≥‘1)((𝐹‘𝑦)‘𝑘) ∈ ℝ)
9	4, 8	eqeltrid 2839	. . . . 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 12628	. . . . 5 ⊢ (𝑦 ∈ 𝒫 ℕ → 1 ∈ ℤ)
12		eqidd 2737	. . . . 5 ⊢ ((𝑦 ∈ 𝒫 ℕ ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑦)‘𝑘) = ((𝐹‘𝑦)‘𝑘))
13	6	rpnnen2lem2 16238	. . . . . . 7 ⊢ (𝑦 ⊆ ℕ → (𝐹‘𝑦):ℕ⟶ℝ)
14	2, 13	syl 17	. . . . . 6 ⊢ (𝑦 ∈ 𝒫 ℕ → (𝐹‘𝑦):ℕ⟶ℝ)
15	14	ffvelcdmda 7079	. . . . 5 ⊢ ((𝑦 ∈ 𝒫 ℕ ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑦)‘𝑘) ∈ ℝ)
16	6	rpnnen2lem5 16241	. . . . . 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 3986	. . . . . . . 8 ⊢ ℕ ⊆ ℕ
19	6	rpnnen2lem4 16240	. . . . . . . 8 ⊢ ((𝑦 ⊆ ℕ ∧ ℕ ⊆ ℕ ∧ 𝑘 ∈ ℕ) → (0 ≤ ((𝐹‘𝑦)‘𝑘) ∧ ((𝐹‘𝑦)‘𝑘) ≤ ((𝐹‘ℕ)‘𝑘)))
20	18, 19	mp3an2 1451	. . . . . . 7 ⊢ ((𝑦 ⊆ ℕ ∧ 𝑘 ∈ ℕ) → (0 ≤ ((𝐹‘𝑦)‘𝑘) ∧ ((𝐹‘𝑦)‘𝑘) ≤ ((𝐹‘ℕ)‘𝑘)))
21	20	simpld 494	. . . . . 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 15787	. . . 4 ⊢ (𝑦 ∈ 𝒫 ℕ → 0 ≤ Σ𝑘 ∈ ℕ ((𝐹‘𝑦)‘𝑘))
24		halfre 12459	. . . . . 6 ⊢ (1 / 2) ∈ ℝ
25	24	a1i 11	. . . . 5 ⊢ (𝑦 ∈ 𝒫 ℕ → (1 / 2) ∈ ℝ)
26		1re 11240	. . . . . 6 ⊢ 1 ∈ ℝ
27	26	a1i 11	. . . . 5 ⊢ (𝑦 ∈ 𝒫 ℕ → 1 ∈ ℝ)
28	6	rpnnen2lem7 16243	. . . . . . . . 9 ⊢ ((𝑦 ⊆ ℕ ∧ ℕ ⊆ ℕ ∧ 1 ∈ ℕ) → Σ𝑘 ∈ (ℤ≥‘1)((𝐹‘𝑦)‘𝑘) ≤ Σ𝑘 ∈ (ℤ≥‘1)((𝐹‘ℕ)‘𝑘))
29	18, 5, 28	mp3an23 1455	. . . . . . . 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 12901	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ ↔ 𝑘 ∈ (ℤ≥‘1))
34	6	rpnnen2lem2 16238	. . . . . . . . . . . . 13 ⊢ (ℕ ⊆ ℕ → (𝐹‘ℕ):ℕ⟶ℝ)
35	18, 34	ax-mp 5	. . . . . . . . . . . 12 ⊢ (𝐹‘ℕ):ℕ⟶ℝ
36	35	ffvelcdmi 7078	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → ((𝐹‘ℕ)‘𝑘) ∈ ℝ)
37	36	recnd 11268	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → ((𝐹‘ℕ)‘𝑘) ∈ ℂ)
38	33, 37	sylbir 235	. . . . . . . . 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 16239	. . . . . . . . 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 15778	. . . . . . 7 ⊢ (𝑦 ∈ 𝒫 ℕ → Σ𝑘 ∈ (ℤ≥‘1)((𝐹‘ℕ)‘𝑘) = (1 / 2))
43	30, 42	breqtrd 5150	. . . . . 6 ⊢ (𝑦 ∈ 𝒫 ℕ → Σ𝑘 ∈ (ℤ≥‘1)((𝐹‘𝑦)‘𝑘) ≤ (1 / 2))
44	4, 43	eqbrtrid 5159	. . . . 5 ⊢ (𝑦 ∈ 𝒫 ℕ → Σ𝑘 ∈ ℕ ((𝐹‘𝑦)‘𝑘) ≤ (1 / 2))
45		halflt1 12463	. . . . . . 7 ⊢ (1 / 2) < 1
46	24, 26, 45	ltleii 11363	. . . . . 6 ⊢ (1 / 2) ≤ 1
47	46	a1i 11	. . . . 5 ⊢ (𝑦 ∈ 𝒫 ℕ → (1 / 2) ≤ 1)
48	10, 25, 27, 44, 47	letrd 11397	. . . 4 ⊢ (𝑦 ∈ 𝒫 ℕ → Σ𝑘 ∈ ℕ ((𝐹‘𝑦)‘𝑘) ≤ 1)
49		elicc01 13488	. . . 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 1344	. . 3 ⊢ (𝑦 ∈ 𝒫 ℕ → Σ𝑘 ∈ ℕ ((𝐹‘𝑦)‘𝑘) ∈ (0[,]1))
51		elpwi 4587	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝒫 ℕ → 𝑧 ⊆ ℕ)
52		ssdifss 4120	. . . . . . . . . . . 12 ⊢ (𝑦 ⊆ ℕ → (𝑦 ∖ 𝑧) ⊆ ℕ)
53		ssdifss 4120	. . . . . . . . . . . 12 ⊢ (𝑧 ⊆ ℕ → (𝑧 ∖ 𝑦) ⊆ ℕ)
54		unss 4170	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∖ 𝑧) ⊆ ℕ ∧ (𝑧 ∖ 𝑦) ⊆ ℕ) ↔ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦)) ⊆ ℕ)
55	54	biimpi 216	. . . . . . . . . . . 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 3979	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑧 ↔ (𝑦 ⊆ 𝑧 ∧ 𝑧 ⊆ 𝑦))
59		ssdif0 4346	. . . . . . . . . . . . . 14 ⊢ (𝑦 ⊆ 𝑧 ↔ (𝑦 ∖ 𝑧) = ∅)
60		ssdif0 4346	. . . . . . . . . . . . . 14 ⊢ (𝑧 ⊆ 𝑦 ↔ (𝑧 ∖ 𝑦) = ∅)
61	59, 60	anbi12i 628	. . . . . . . . . . . . 13 ⊢ ((𝑦 ⊆ 𝑧 ∧ 𝑧 ⊆ 𝑦) ↔ ((𝑦 ∖ 𝑧) = ∅ ∧ (𝑧 ∖ 𝑦) = ∅))
62		un00 4425	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∖ 𝑧) = ∅ ∧ (𝑧 ∖ 𝑦) = ∅) ↔ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦)) = ∅)
63	58, 61, 62	3bitri 297	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑧 ↔ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦)) = ∅)
64	63	necon3bii 2985	. . . . . . . . . . 11 ⊢ (𝑦 ≠ 𝑧 ↔ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦)) ≠ ∅)
65	64	biimpi 216	. . . . . . . . . 10 ⊢ (𝑦 ≠ 𝑧 → ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦)) ≠ ∅)
66		nnwo 12934	. . . . . . . . . 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 412	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝒫 ℕ ∧ 𝑧 ∈ 𝒫 ℕ) → (𝑦 ≠ 𝑧 → ∃𝑚 ∈ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦))∀𝑛 ∈ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦))𝑚 ≤ 𝑛))
69	57	sselda 3963	. . . . . . . . . 10 ⊢ (((𝑦 ∈ 𝒫 ℕ ∧ 𝑧 ∈ 𝒫 ℕ) ∧ 𝑚 ∈ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦))) → 𝑚 ∈ ℕ)
70		df-ral 3053	. . . . . . . . . . . 12 ⊢ (∀𝑛 ∈ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦))𝑚 ≤ 𝑛 ↔ ∀𝑛(𝑛 ∈ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦)) → 𝑚 ≤ 𝑛))
71		con34b 316	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦)) → 𝑚 ≤ 𝑛) ↔ (¬ 𝑚 ≤ 𝑛 → ¬ 𝑛 ∈ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦))))
72		eldif 3941	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (𝑦 ∖ 𝑧) ↔ (𝑛 ∈ 𝑦 ∧ ¬ 𝑛 ∈ 𝑧))
73		eldif 3941	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (𝑧 ∖ 𝑦) ↔ (𝑛 ∈ 𝑧 ∧ ¬ 𝑛 ∈ 𝑦))
74	72, 73	orbi12i 914	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ (𝑦 ∖ 𝑧) ∨ 𝑛 ∈ (𝑧 ∖ 𝑦)) ↔ ((𝑛 ∈ 𝑦 ∧ ¬ 𝑛 ∈ 𝑧) ∨ (𝑛 ∈ 𝑧 ∧ ¬ 𝑛 ∈ 𝑦)))
75		elun 4133	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦)) ↔ (𝑛 ∈ (𝑦 ∖ 𝑧) ∨ 𝑛 ∈ (𝑧 ∖ 𝑦)))
76		xor 1016	. . . . . . . . . . . . . . . . 17 ⊢ (¬ (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧) ↔ ((𝑛 ∈ 𝑦 ∧ ¬ 𝑛 ∈ 𝑧) ∨ (𝑛 ∈ 𝑧 ∧ ¬ 𝑛 ∈ 𝑦)))
77	74, 75, 76	3bitr4ri 304	. . . . . . . . . . . . . . . 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 336	. . . . . . . . . . . . . 14 ⊢ ((¬ 𝑚 ≤ 𝑛 → ¬ 𝑛 ∈ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦))) ↔ (¬ 𝑚 ≤ 𝑛 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)))
80	71, 79	bitri 275	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦)) → 𝑚 ≤ 𝑛) ↔ (¬ 𝑚 ≤ 𝑛 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)))
81	80	albii 1819	. . . . . . . . . . . 12 ⊢ (∀𝑛(𝑛 ∈ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦)) → 𝑚 ≤ 𝑛) ↔ ∀𝑛(¬ 𝑚 ≤ 𝑛 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)))
82	70, 81	bitri 275	. . . . . . . . . . 11 ⊢ (∀𝑛 ∈ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦))𝑚 ≤ 𝑛 ↔ ∀𝑛(¬ 𝑚 ≤ 𝑛 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)))
83		alral 3066	. . . . . . . . . . . 12 ⊢ (∀𝑛(¬ 𝑚 ≤ 𝑛 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)) → ∀𝑛 ∈ ℕ (¬ 𝑚 ≤ 𝑛 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)))
84		nnre 12252	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
85		nnre 12252	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℝ)
86		ltnle 11319	. . . . . . . . . . . . . . 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 3162	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ → (∀𝑛 ∈ ℕ (𝑛 < 𝑚 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)) ↔ ∀𝑛 ∈ ℕ (¬ 𝑚 ≤ 𝑛 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧))))
90	83, 89	imbitrrid 246	. . . . . . . . . . 11 ⊢ (𝑚 ∈ ℕ → (∀𝑛(¬ 𝑚 ≤ 𝑛 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)) → ∀𝑛 ∈ ℕ (𝑛 < 𝑚 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧))))
91	82, 90	biimtrid 242	. . . . . . . . . 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 3154	. . . . . . . 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 4176	. . . . . . 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	imbitrdi 251	. . . . . 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 766	. . . . . . . . . 10 ⊢ (((𝑦 ⊆ ℕ ∧ 𝑧 ⊆ ℕ) ∧ (𝑚 ∈ (𝑦 ∖ 𝑧) ∧ ∀𝑛 ∈ ℕ (𝑛 < 𝑚 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)))) → 𝑦 ⊆ ℕ)
98		simplr 768	. . . . . . . . . 10 ⊢ (((𝑦 ⊆ ℕ ∧ 𝑧 ⊆ ℕ) ∧ (𝑚 ∈ (𝑦 ∖ 𝑧) ∧ ∀𝑛 ∈ ℕ (𝑛 < 𝑚 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)))) → 𝑧 ⊆ ℕ)
99		simprl 770	. . . . . . . . . 10 ⊢ (((𝑦 ⊆ ℕ ∧ 𝑧 ⊆ ℕ) ∧ (𝑚 ∈ (𝑦 ∖ 𝑧) ∧ ∀𝑛 ∈ ℕ (𝑛 < 𝑚 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)))) → 𝑚 ∈ (𝑦 ∖ 𝑧))
100		simprr 772	. . . . . . . . . 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 261	. . . . . . . . . 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 16247	. . . . . . . . 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 3143	. . . . . . . 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 768	. . . . . . . . . 10 ⊢ (((𝑦 ⊆ ℕ ∧ 𝑧 ⊆ ℕ) ∧ (𝑚 ∈ (𝑧 ∖ 𝑦) ∧ ∀𝑛 ∈ ℕ (𝑛 < 𝑚 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)))) → 𝑧 ⊆ ℕ)
105		simpll 766	. . . . . . . . . 10 ⊢ (((𝑦 ⊆ ℕ ∧ 𝑧 ⊆ ℕ) ∧ (𝑚 ∈ (𝑧 ∖ 𝑦) ∧ ∀𝑛 ∈ ℕ (𝑛 < 𝑚 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)))) → 𝑦 ⊆ ℕ)
106		simprl 770	. . . . . . . . . 10 ⊢ (((𝑦 ⊆ ℕ ∧ 𝑧 ⊆ ℕ) ∧ (𝑚 ∈ (𝑧 ∖ 𝑦) ∧ ∀𝑛 ∈ ℕ (𝑛 < 𝑚 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)))) → 𝑚 ∈ (𝑧 ∖ 𝑦))
107		simprr 772	. . . . . . . . . . 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 222	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ 𝑧 ↔ 𝑛 ∈ 𝑦) ↔ (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧))
109	108	imbi2i 336	. . . . . . . . . . . 12 ⊢ ((𝑛 < 𝑚 → (𝑛 ∈ 𝑧 ↔ 𝑛 ∈ 𝑦)) ↔ (𝑛 < 𝑚 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)))
110	109	ralbii 3083	. . . . . . . . . . 11 ⊢ (∀𝑛 ∈ ℕ (𝑛 < 𝑚 → (𝑛 ∈ 𝑧 ↔ 𝑛 ∈ 𝑦)) ↔ ∀𝑛 ∈ ℕ (𝑛 < 𝑚 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)))
111	107, 110	sylibr 234	. . . . . . . . . 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 16247	. . . . . . . . 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 3143	. . . . . . . 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 859	. . . . . . 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 2952	. . . 4 ⊢ ((𝑦 ∈ 𝒫 ℕ ∧ 𝑧 ∈ 𝒫 ℕ) → (Σ𝑘 ∈ ℕ ((𝐹‘𝑦)‘𝑘) = Σ𝑘 ∈ ℕ ((𝐹‘𝑧)‘𝑘) → 𝑦 = 𝑧))
119		fveq2 6881	. . . . . 6 ⊢ (𝑦 = 𝑧 → (𝐹‘𝑦) = (𝐹‘𝑧))
120	119	fveq1d 6883	. . . . 5 ⊢ (𝑦 = 𝑧 → ((𝐹‘𝑦)‘𝑘) = ((𝐹‘𝑧)‘𝑘))
121	120	sumeq2sdv 15724	. . . 4 ⊢ (𝑦 = 𝑧 → Σ𝑘 ∈ ℕ ((𝐹‘𝑦)‘𝑘) = Σ𝑘 ∈ ℕ ((𝐹‘𝑧)‘𝑘))
122	118, 121	impbid1 225	. . 3 ⊢ ((𝑦 ∈ 𝒫 ℕ ∧ 𝑧 ∈ 𝒫 ℕ) → (Σ𝑘 ∈ ℕ ((𝐹‘𝑦)‘𝑘) = Σ𝑘 ∈ ℕ ((𝐹‘𝑧)‘𝑘) ↔ 𝑦 = 𝑧))
123	50, 122	dom2 9014	. 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 206   ∧ wa 395   ∨ wo 847  ∀wal 1538   = wceq 1540   ∈ wcel 2109   ≠ wne 2933  ∀wral 3052  ∃wrex 3061  Vcvv 3464   ∖ cdif 3928   ∪ cun 3929   ⊆ wss 3931  ∅c0 4313  ifcif 4505  𝒫 cpw 4580   class class class wbr 5124   ↦ cmpt 5206  dom cdm 5659  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410   ≼ cdom 8962  ℂcc 11132  ℝcr 11133  0cc0 11134  1c1 11135   + caddc 11137   < clt 11274   ≤ cle 11275   / cdiv 11899  ℕcn 12245  2c2 12300  3c3 12301  ℤ_≥cuz 12857  [,]cicc 13370  seqcseq 14024  ↑cexp 14084   ⇝ cli 15505  Σcsu 15707

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-inf2 9660  ax-cnex 11190  ax-resscn 11191  ax-1cn 11192  ax-icn 11193  ax-addcl 11194  ax-addrcl 11195  ax-mulcl 11196  ax-mulrcl 11197  ax-mulcom 11198  ax-addass 11199  ax-mulass 11200  ax-distr 11201  ax-i2m1 11202  ax-1ne0 11203  ax-1rid 11204  ax-rnegex 11205  ax-rrecex 11206  ax-cnre 11207  ax-pre-lttri 11208  ax-pre-lttrn 11209  ax-pre-ltadd 11210  ax-pre-mulgt0 11211  ax-pre-sup 11212

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-int 4928  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7867  df-1st 7993  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-er 8724  df-pm 8848  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-sup 9459  df-inf 9460  df-oi 9529  df-card 9958  df-pnf 11276  df-mnf 11277  df-xr 11278  df-ltxr 11279  df-le 11280  df-sub 11473  df-neg 11474  df-div 11900  df-nn 12246  df-2 12308  df-3 12309  df-n0 12507  df-z 12594  df-uz 12858  df-rp 13014  df-ico 13373  df-icc 13374  df-fz 13530  df-fzo 13677  df-fl 13814  df-seq 14025  df-exp 14085  df-hash 14354  df-cj 15123  df-re 15124  df-im 15125  df-sqrt 15259  df-abs 15260  df-limsup 15492  df-clim 15509  df-rlim 15510  df-sum 15708

This theorem is referenced by:  rpnnen2  16249