Theorem rpnnen2lem12 16169

Description: Lemma for rpnnen2 16170. (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 7402	. 2 ⊢ (0[,]1) ∈ V
2		elpwi 4566	. . . . 5 ⊢ (𝑦 ∈ 𝒫 ℕ → 𝑦 ⊆ ℕ)
3		nnuz 12812	. . . . . . 7 ⊢ ℕ = (ℤ≥‘1)
4	3	sumeq1i 15639	. . . . . 6 ⊢ Σ𝑘 ∈ ℕ ((𝐹‘𝑦)‘𝑘) = Σ𝑘 ∈ (ℤ≥‘1)((𝐹‘𝑦)‘𝑘)
5		1nn 12173	. . . . . . 7 ⊢ 1 ∈ ℕ
6		rpnnen2.1	. . . . . . . 8 ⊢ 𝐹 = (𝑥 ∈ 𝒫 ℕ ↦ (𝑛 ∈ ℕ ↦ if(𝑛 ∈ 𝑥, ((1 / 3)↑𝑛), 0)))
7	6	rpnnen2lem6 16163	. . . . . . 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 2832	. . . . 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 12540	. . . . 5 ⊢ (𝑦 ∈ 𝒫 ℕ → 1 ∈ ℤ)
12		eqidd 2730	. . . . 5 ⊢ ((𝑦 ∈ 𝒫 ℕ ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑦)‘𝑘) = ((𝐹‘𝑦)‘𝑘))
13	6	rpnnen2lem2 16159	. . . . . . 7 ⊢ (𝑦 ⊆ ℕ → (𝐹‘𝑦):ℕ⟶ℝ)
14	2, 13	syl 17	. . . . . 6 ⊢ (𝑦 ∈ 𝒫 ℕ → (𝐹‘𝑦):ℕ⟶ℝ)
15	14	ffvelcdmda 7038	. . . . 5 ⊢ ((𝑦 ∈ 𝒫 ℕ ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑦)‘𝑘) ∈ ℝ)
16	6	rpnnen2lem5 16162	. . . . . 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 3966	. . . . . . . 8 ⊢ ℕ ⊆ ℕ
19	6	rpnnen2lem4 16161	. . . . . . . 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 15708	. . . 4 ⊢ (𝑦 ∈ 𝒫 ℕ → 0 ≤ Σ𝑘 ∈ ℕ ((𝐹‘𝑦)‘𝑘))
24		halfre 12371	. . . . . 6 ⊢ (1 / 2) ∈ ℝ
25	24	a1i 11	. . . . 5 ⊢ (𝑦 ∈ 𝒫 ℕ → (1 / 2) ∈ ℝ)
26		1re 11150	. . . . . 6 ⊢ 1 ∈ ℝ
27	26	a1i 11	. . . . 5 ⊢ (𝑦 ∈ 𝒫 ℕ → 1 ∈ ℝ)
28	6	rpnnen2lem7 16164	. . . . . . . . 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 2729	. . . . . . . 8 ⊢ (ℤ≥‘1) = (ℤ≥‘1)
32		eqidd 2730	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝒫 ℕ ∧ 𝑘 ∈ (ℤ≥‘1)) → ((𝐹‘ℕ)‘𝑘) = ((𝐹‘ℕ)‘𝑘))
33		elnnuz 12813	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ ↔ 𝑘 ∈ (ℤ≥‘1))
34	6	rpnnen2lem2 16159	. . . . . . . . . . . . 13 ⊢ (ℕ ⊆ ℕ → (𝐹‘ℕ):ℕ⟶ℝ)
35	18, 34	ax-mp 5	. . . . . . . . . . . 12 ⊢ (𝐹‘ℕ):ℕ⟶ℝ
36	35	ffvelcdmi 7037	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → ((𝐹‘ℕ)‘𝑘) ∈ ℝ)
37	36	recnd 11178	. . . . . . . . . 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 16160	. . . . . . . . 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 15699	. . . . . . 7 ⊢ (𝑦 ∈ 𝒫 ℕ → Σ𝑘 ∈ (ℤ≥‘1)((𝐹‘ℕ)‘𝑘) = (1 / 2))
43	30, 42	breqtrd 5128	. . . . . 6 ⊢ (𝑦 ∈ 𝒫 ℕ → Σ𝑘 ∈ (ℤ≥‘1)((𝐹‘𝑦)‘𝑘) ≤ (1 / 2))
44	4, 43	eqbrtrid 5137	. . . . 5 ⊢ (𝑦 ∈ 𝒫 ℕ → Σ𝑘 ∈ ℕ ((𝐹‘𝑦)‘𝑘) ≤ (1 / 2))
45		halflt1 12375	. . . . . . 7 ⊢ (1 / 2) < 1
46	24, 26, 45	ltleii 11273	. . . . . 6 ⊢ (1 / 2) ≤ 1
47	46	a1i 11	. . . . 5 ⊢ (𝑦 ∈ 𝒫 ℕ → (1 / 2) ≤ 1)
48	10, 25, 27, 44, 47	letrd 11307	. . . 4 ⊢ (𝑦 ∈ 𝒫 ℕ → Σ𝑘 ∈ ℕ ((𝐹‘𝑦)‘𝑘) ≤ 1)
49		elicc01 13403	. . . 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 4566	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝒫 ℕ → 𝑧 ⊆ ℕ)
52		ssdifss 4099	. . . . . . . . . . . 12 ⊢ (𝑦 ⊆ ℕ → (𝑦 ∖ 𝑧) ⊆ ℕ)
53		ssdifss 4099	. . . . . . . . . . . 12 ⊢ (𝑧 ⊆ ℕ → (𝑧 ∖ 𝑦) ⊆ ℕ)
54		unss 4149	. . . . . . . . . . . . 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 3959	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑧 ↔ (𝑦 ⊆ 𝑧 ∧ 𝑧 ⊆ 𝑦))
59		ssdif0 4325	. . . . . . . . . . . . . 14 ⊢ (𝑦 ⊆ 𝑧 ↔ (𝑦 ∖ 𝑧) = ∅)
60		ssdif0 4325	. . . . . . . . . . . . . 14 ⊢ (𝑧 ⊆ 𝑦 ↔ (𝑧 ∖ 𝑦) = ∅)
61	59, 60	anbi12i 628	. . . . . . . . . . . . 13 ⊢ ((𝑦 ⊆ 𝑧 ∧ 𝑧 ⊆ 𝑦) ↔ ((𝑦 ∖ 𝑧) = ∅ ∧ (𝑧 ∖ 𝑦) = ∅))
62		un00 4404	. . . . . . . . . . . . 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 2977	. . . . . . . . . . 11 ⊢ (𝑦 ≠ 𝑧 ↔ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦)) ≠ ∅)
65	64	biimpi 216	. . . . . . . . . 10 ⊢ (𝑦 ≠ 𝑧 → ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦)) ≠ ∅)
66		nnwo 12848	. . . . . . . . . 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 3943	. . . . . . . . . 10 ⊢ (((𝑦 ∈ 𝒫 ℕ ∧ 𝑧 ∈ 𝒫 ℕ) ∧ 𝑚 ∈ ((𝑦 ∖ 𝑧) ∪ (𝑧 ∖ 𝑦))) → 𝑚 ∈ ℕ)
70		df-ral 3045	. . . . . . . . . . . 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 3921	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (𝑦 ∖ 𝑧) ↔ (𝑛 ∈ 𝑦 ∧ ¬ 𝑛 ∈ 𝑧))
73		eldif 3921	. . . . . . . . . . . . . . . . . 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 4112	. . . . . . . . . . . . . . . . 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 3058	. . . . . . . . . . . 12 ⊢ (∀𝑛(¬ 𝑚 ≤ 𝑛 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)) → ∀𝑛 ∈ ℕ (¬ 𝑚 ≤ 𝑛 → (𝑛 ∈ 𝑦 ↔ 𝑛 ∈ 𝑧)))
84		nnre 12169	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
85		nnre 12169	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℝ)
86		ltnle 11229	. . . . . . . . . . . . . . 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 3154	. . . . . . . . . . . 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 3146	. . . . . . . 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 4155	. . . . . . 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 16168	. . . . . . . . 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 3135	. . . . . . . 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 3075	. . . . . . . . . . 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 2736	. . . . . . . . . 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 16168	. . . . . . . . 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 3135	. . . . . . . 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 2944	. . . 4 ⊢ ((𝑦 ∈ 𝒫 ℕ ∧ 𝑧 ∈ 𝒫 ℕ) → (Σ𝑘 ∈ ℕ ((𝐹‘𝑦)‘𝑘) = Σ𝑘 ∈ ℕ ((𝐹‘𝑧)‘𝑘) → 𝑦 = 𝑧))
119		fveq2 6840	. . . . . 6 ⊢ (𝑦 = 𝑧 → (𝐹‘𝑦) = (𝐹‘𝑧))
120	119	fveq1d 6842	. . . . 5 ⊢ (𝑦 = 𝑧 → ((𝐹‘𝑦)‘𝑘) = ((𝐹‘𝑧)‘𝑘))
121	120	sumeq2sdv 15645	. . . 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 8943	. 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 2925  ∀wral 3044  ∃wrex 3053  Vcvv 3444   ∖ cdif 3908   ∪ cun 3909   ⊆ wss 3911  ∅c0 4292  ifcif 4484  𝒫 cpw 4559   class class class wbr 5102   ↦ cmpt 5183  dom cdm 5631  ⟶wf 6495  ‘cfv 6499  (class class class)co 7369   ≼ cdom 8893  ℂcc 11042  ℝcr 11043  0cc0 11044  1c1 11045   + caddc 11047   < clt 11184   ≤ cle 11185   / cdiv 11811  ℕcn 12162  2c2 12217  3c3 12218  ℤ_≥cuz 12769  [,]cicc 13285  seqcseq 13942  ↑cexp 14002   ⇝ cli 15426  Σcsu 15628

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 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-inf2 9570  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121  ax-pre-sup 11122

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-isom 6508  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-1st 7947  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-1o 8411  df-er 8648  df-pm 8779  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-sup 9369  df-inf 9370  df-oi 9439  df-card 9868  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-div 11812  df-nn 12163  df-2 12225  df-3 12226  df-n0 12419  df-z 12506  df-uz 12770  df-rp 12928  df-ico 13288  df-icc 13289  df-fz 13445  df-fzo 13592  df-fl 13730  df-seq 13943  df-exp 14003  df-hash 14272  df-cj 15041  df-re 15042  df-im 15043  df-sqrt 15177  df-abs 15178  df-limsup 15413  df-clim 15430  df-rlim 15431  df-sum 15629

This theorem is referenced by:  rpnnen2  16170