rpnnen2lem11 - Metamath Proof Explorer

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Theorem rpnnen2lem11 16192

Description: Lemma for rpnnen2 16194. (Contributed by Mario Carneiro, 13-May-2013.)

**Hypotheses**
Ref	Expression
rpnnen2.1	⊢ 𝐹 = (𝑥 ∈ 𝒫 ℕ ↦ (𝑛 ∈ ℕ ↦ if(𝑛 ∈ 𝑥, ((1 / 3)↑𝑛), 0)))
rpnnen2.2	⊢ (𝜑 → 𝐴 ⊆ ℕ)
rpnnen2.3	⊢ (𝜑 → 𝐵 ⊆ ℕ)
rpnnen2.4	⊢ (𝜑 → 𝑚 ∈ (𝐴 ∖ 𝐵))
rpnnen2.5	⊢ (𝜑 → ∀𝑛 ∈ ℕ (𝑛 < 𝑚 → (𝑛 ∈ 𝐴 ↔ 𝑛 ∈ 𝐵)))
rpnnen2.6	⊢ (𝜓 ↔ Σ𝑘 ∈ ℕ ((𝐹‘𝐴)‘𝑘) = Σ𝑘 ∈ ℕ ((𝐹‘𝐵)‘𝑘))

**Assertion**
Ref	Expression
rpnnen2lem11	⊢ (𝜑 → ¬ 𝜓)

Distinct variable groups: 𝑚,𝑛,𝑥,𝑘 𝐴,𝑘,𝑛,𝑥 𝐵,𝑘,𝑛,𝑥 𝑘,𝑚,𝐹 𝜑,𝑘 Allowed substitution hints: 𝜑(𝑥,𝑚,𝑛) 𝜓(𝑥,𝑘,𝑚,𝑛) 𝐴(𝑚) 𝐵(𝑚) 𝐹(𝑥,𝑛)

**Proof of Theorem rpnnen2lem11**
Step	Hyp	Ref	Expression
1		rpnnen2.3	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ ℕ)
2		rpnnen2.2	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ ℕ)
3		rpnnen2.4	. . . . 5 ⊢ (𝜑 → 𝑚 ∈ (𝐴 ∖ 𝐵))
4		eldifi 4122	. . . . . 6 ⊢ (𝑚 ∈ (𝐴 ∖ 𝐵) → 𝑚 ∈ 𝐴)
5		ssel2 3973	. . . . . 6 ⊢ ((𝐴 ⊆ ℕ ∧ 𝑚 ∈ 𝐴) → 𝑚 ∈ ℕ)
6	4, 5	sylan2 592	. . . . 5 ⊢ ((𝐴 ⊆ ℕ ∧ 𝑚 ∈ (𝐴 ∖ 𝐵)) → 𝑚 ∈ ℕ)
7	2, 3, 6	syl2anc 583	. . . 4 ⊢ (𝜑 → 𝑚 ∈ ℕ)
8		rpnnen2.1	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝒫 ℕ ↦ (𝑛 ∈ ℕ ↦ if(𝑛 ∈ 𝑥, ((1 / 3)↑𝑛), 0)))
9	8	rpnnen2lem6 16187	. . . 4 ⊢ ((𝐵 ⊆ ℕ ∧ 𝑚 ∈ ℕ) → Σ𝑘 ∈ (ℤ_≥‘𝑚)((𝐹‘𝐵)‘𝑘) ∈ ℝ)
10	1, 7, 9	syl2anc 583	. . 3 ⊢ (𝜑 → Σ𝑘 ∈ (ℤ_≥‘𝑚)((𝐹‘𝐵)‘𝑘) ∈ ℝ)
11		3nn 12313	. . . . . 6 ⊢ 3 ∈ ℕ
12		nnrecre 12276	. . . . . 6 ⊢ (3 ∈ ℕ → (1 / 3) ∈ ℝ)
13	11, 12	ax-mp 5	. . . . 5 ⊢ (1 / 3) ∈ ℝ
14	7	nnnn0d 12554	. . . . 5 ⊢ (𝜑 → 𝑚 ∈ ℕ₀)
15		reexpcl 14067	. . . . 5 ⊢ (((1 / 3) ∈ ℝ ∧ 𝑚 ∈ ℕ₀) → ((1 / 3)↑𝑚) ∈ ℝ)
16	13, 14, 15	sylancr 586	. . . 4 ⊢ (𝜑 → ((1 / 3)↑𝑚) ∈ ℝ)
17	8	rpnnen2lem6 16187	. . . . 5 ⊢ ((𝐴 ⊆ ℕ ∧ 𝑚 ∈ ℕ) → Σ𝑘 ∈ (ℤ_≥‘𝑚)((𝐹‘𝐴)‘𝑘) ∈ ℝ)
18	2, 7, 17	syl2anc 583	. . . 4 ⊢ (𝜑 → Σ𝑘 ∈ (ℤ_≥‘𝑚)((𝐹‘𝐴)‘𝑘) ∈ ℝ)
19		nnrp 13009	. . . . . . . . 9 ⊢ (3 ∈ ℕ → 3 ∈ ℝ⁺)
20		rpreccl 13024	. . . . . . . . 9 ⊢ (3 ∈ ℝ⁺ → (1 / 3) ∈ ℝ⁺)
21	11, 19, 20	mp2b 10	. . . . . . . 8 ⊢ (1 / 3) ∈ ℝ⁺
22	7	nnzd 12607	. . . . . . . 8 ⊢ (𝜑 → 𝑚 ∈ ℤ)
23		rpexpcl 14069	. . . . . . . 8 ⊢ (((1 / 3) ∈ ℝ⁺ ∧ 𝑚 ∈ ℤ) → ((1 / 3)↑𝑚) ∈ ℝ⁺)
24	21, 22, 23	sylancr 586	. . . . . . 7 ⊢ (𝜑 → ((1 / 3)↑𝑚) ∈ ℝ⁺)
25	24	rpred 13040	. . . . . 6 ⊢ (𝜑 → ((1 / 3)↑𝑚) ∈ ℝ)
26	25	rehalfcld 12481	. . . . 5 ⊢ (𝜑 → (((1 / 3)↑𝑚) / 2) ∈ ℝ)
27	3	snssd 4808	. . . . . . . . 9 ⊢ (𝜑 → {𝑚} ⊆ (𝐴 ∖ 𝐵))
28	2	ssdifd 4136	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 ∖ 𝐵) ⊆ (ℕ ∖ 𝐵))
29	27, 28	sstrd 3988	. . . . . . . 8 ⊢ (𝜑 → {𝑚} ⊆ (ℕ ∖ 𝐵))
30	7	snssd 4808	. . . . . . . . 9 ⊢ (𝜑 → {𝑚} ⊆ ℕ)
31		ssconb 4133	. . . . . . . . 9 ⊢ ((𝐵 ⊆ ℕ ∧ {𝑚} ⊆ ℕ) → (𝐵 ⊆ (ℕ ∖ {𝑚}) ↔ {𝑚} ⊆ (ℕ ∖ 𝐵)))
32	1, 30, 31	syl2anc 583	. . . . . . . 8 ⊢ (𝜑 → (𝐵 ⊆ (ℕ ∖ {𝑚}) ↔ {𝑚} ⊆ (ℕ ∖ 𝐵)))
33	29, 32	mpbird 257	. . . . . . 7 ⊢ (𝜑 → 𝐵 ⊆ (ℕ ∖ {𝑚}))
34		difssd 4128	. . . . . . 7 ⊢ (𝜑 → (ℕ ∖ {𝑚}) ⊆ ℕ)
35	8	rpnnen2lem7 16188	. . . . . . 7 ⊢ ((𝐵 ⊆ (ℕ ∖ {𝑚}) ∧ (ℕ ∖ {𝑚}) ⊆ ℕ ∧ 𝑚 ∈ ℕ) → Σ𝑘 ∈ (ℤ_≥‘𝑚)((𝐹‘𝐵)‘𝑘) ≤ Σ𝑘 ∈ (ℤ_≥‘𝑚)((𝐹‘(ℕ ∖ {𝑚}))‘𝑘))
36	33, 34, 7, 35	syl3anc 1369	. . . . . 6 ⊢ (𝜑 → Σ𝑘 ∈ (ℤ_≥‘𝑚)((𝐹‘𝐵)‘𝑘) ≤ Σ𝑘 ∈ (ℤ_≥‘𝑚)((𝐹‘(ℕ ∖ {𝑚}))‘𝑘))
37	8	rpnnen2lem9 16190	. . . . . . . 8 ⊢ (𝑚 ∈ ℕ → Σ𝑘 ∈ (ℤ_≥‘𝑚)((𝐹‘(ℕ ∖ {𝑚}))‘𝑘) = (0 + (((1 / 3)↑(𝑚 + 1)) / (1 − (1 / 3)))))
38	7, 37	syl 17	. . . . . . 7 ⊢ (𝜑 → Σ𝑘 ∈ (ℤ_≥‘𝑚)((𝐹‘(ℕ ∖ {𝑚}))‘𝑘) = (0 + (((1 / 3)↑(𝑚 + 1)) / (1 − (1 / 3)))))
39	13	recni 11250	. . . . . . . . . . . 12 ⊢ (1 / 3) ∈ ℂ
40		expp1 14057	. . . . . . . . . . . 12 ⊢ (((1 / 3) ∈ ℂ ∧ 𝑚 ∈ ℕ₀) → ((1 / 3)↑(𝑚 + 1)) = (((1 / 3)↑𝑚) · (1 / 3)))
41	39, 14, 40	sylancr 586	. . . . . . . . . . 11 ⊢ (𝜑 → ((1 / 3)↑(𝑚 + 1)) = (((1 / 3)↑𝑚) · (1 / 3)))
42	25	recnd 11264	. . . . . . . . . . . 12 ⊢ (𝜑 → ((1 / 3)↑𝑚) ∈ ℂ)
43		3cn 12315	. . . . . . . . . . . . 13 ⊢ 3 ∈ ℂ
44		3ne0 12340	. . . . . . . . . . . . 13 ⊢ 3 ≠ 0
45		divrec 11910	. . . . . . . . . . . . 13 ⊢ ((((1 / 3)↑𝑚) ∈ ℂ ∧ 3 ∈ ℂ ∧ 3 ≠ 0) → (((1 / 3)↑𝑚) / 3) = (((1 / 3)↑𝑚) · (1 / 3)))
46	43, 44, 45	mp3an23 1450	. . . . . . . . . . . 12 ⊢ (((1 / 3)↑𝑚) ∈ ℂ → (((1 / 3)↑𝑚) / 3) = (((1 / 3)↑𝑚) · (1 / 3)))
47	42, 46	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (((1 / 3)↑𝑚) / 3) = (((1 / 3)↑𝑚) · (1 / 3)))
48	41, 47	eqtr4d 2770	. . . . . . . . . 10 ⊢ (𝜑 → ((1 / 3)↑(𝑚 + 1)) = (((1 / 3)↑𝑚) / 3))
49	48	oveq1d 7429	. . . . . . . . 9 ⊢ (𝜑 → (((1 / 3)↑(𝑚 + 1)) / (1 − (1 / 3))) = ((((1 / 3)↑𝑚) / 3) / (1 − (1 / 3))))
50		ax-1cn 11188	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℂ
51	43, 44	pm3.2i 470	. . . . . . . . . . . . 13 ⊢ (3 ∈ ℂ ∧ 3 ≠ 0)
52		divsubdir 11930	. . . . . . . . . . . . 13 ⊢ ((3 ∈ ℂ ∧ 1 ∈ ℂ ∧ (3 ∈ ℂ ∧ 3 ≠ 0)) → ((3 − 1) / 3) = ((3 / 3) − (1 / 3)))
53	43, 50, 51, 52	mp3an 1458	. . . . . . . . . . . 12 ⊢ ((3 − 1) / 3) = ((3 / 3) − (1 / 3))
54		3m1e2 12362	. . . . . . . . . . . . 13 ⊢ (3 − 1) = 2
55	54	oveq1i 7424	. . . . . . . . . . . 12 ⊢ ((3 − 1) / 3) = (2 / 3)
56	43, 44	dividi 11969	. . . . . . . . . . . . 13 ⊢ (3 / 3) = 1
57	56	oveq1i 7424	. . . . . . . . . . . 12 ⊢ ((3 / 3) − (1 / 3)) = (1 − (1 / 3))
58	53, 55, 57	3eqtr3ri 2764	. . . . . . . . . . 11 ⊢ (1 − (1 / 3)) = (2 / 3)
59	58	oveq2i 7425	. . . . . . . . . 10 ⊢ ((((1 / 3)↑𝑚) / 3) / (1 − (1 / 3))) = ((((1 / 3)↑𝑚) / 3) / (2 / 3))
60		2cnne0 12444	. . . . . . . . . . . 12 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0)
61		divcan7 11945	. . . . . . . . . . . 12 ⊢ ((((1 / 3)↑𝑚) ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0) ∧ (3 ∈ ℂ ∧ 3 ≠ 0)) → ((((1 / 3)↑𝑚) / 3) / (2 / 3)) = (((1 / 3)↑𝑚) / 2))
62	60, 51, 61	mp3an23 1450	. . . . . . . . . . 11 ⊢ (((1 / 3)↑𝑚) ∈ ℂ → ((((1 / 3)↑𝑚) / 3) / (2 / 3)) = (((1 / 3)↑𝑚) / 2))
63	42, 62	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ((((1 / 3)↑𝑚) / 3) / (2 / 3)) = (((1 / 3)↑𝑚) / 2))
64	59, 63	eqtrid 2779	. . . . . . . . 9 ⊢ (𝜑 → ((((1 / 3)↑𝑚) / 3) / (1 − (1 / 3))) = (((1 / 3)↑𝑚) / 2))
65	49, 64	eqtrd 2767	. . . . . . . 8 ⊢ (𝜑 → (((1 / 3)↑(𝑚 + 1)) / (1 − (1 / 3))) = (((1 / 3)↑𝑚) / 2))
66	65	oveq2d 7430	. . . . . . 7 ⊢ (𝜑 → (0 + (((1 / 3)↑(𝑚 + 1)) / (1 − (1 / 3)))) = (0 + (((1 / 3)↑𝑚) / 2)))
67	26	recnd 11264	. . . . . . . 8 ⊢ (𝜑 → (((1 / 3)↑𝑚) / 2) ∈ ℂ)
68	67	addlidd 11437	. . . . . . 7 ⊢ (𝜑 → (0 + (((1 / 3)↑𝑚) / 2)) = (((1 / 3)↑𝑚) / 2))
69	38, 66, 68	3eqtrd 2771	. . . . . 6 ⊢ (𝜑 → Σ𝑘 ∈ (ℤ_≥‘𝑚)((𝐹‘(ℕ ∖ {𝑚}))‘𝑘) = (((1 / 3)↑𝑚) / 2))
70	36, 69	breqtrd 5168	. . . . 5 ⊢ (𝜑 → Σ𝑘 ∈ (ℤ_≥‘𝑚)((𝐹‘𝐵)‘𝑘) ≤ (((1 / 3)↑𝑚) / 2))
71		rphalflt 13027	. . . . . 6 ⊢ (((1 / 3)↑𝑚) ∈ ℝ⁺ → (((1 / 3)↑𝑚) / 2) < ((1 / 3)↑𝑚))
72	24, 71	syl 17	. . . . 5 ⊢ (𝜑 → (((1 / 3)↑𝑚) / 2) < ((1 / 3)↑𝑚))
73	10, 26, 25, 70, 72	lelttrd 11394	. . . 4 ⊢ (𝜑 → Σ𝑘 ∈ (ℤ_≥‘𝑚)((𝐹‘𝐵)‘𝑘) < ((1 / 3)↑𝑚))
74		eluznn 12924	. . . . . . . . 9 ⊢ ((𝑚 ∈ ℕ ∧ 𝑘 ∈ (ℤ_≥‘𝑚)) → 𝑘 ∈ ℕ)
75	7, 74	sylan 579	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑚)) → 𝑘 ∈ ℕ)
76	8	rpnnen2lem1 16182	. . . . . . . 8 ⊢ (({𝑚} ⊆ ℕ ∧ 𝑘 ∈ ℕ) → ((𝐹‘{𝑚})‘𝑘) = if(𝑘 ∈ {𝑚}, ((1 / 3)↑𝑘), 0))
77	30, 75, 76	syl2an2r 684	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑚)) → ((𝐹‘{𝑚})‘𝑘) = if(𝑘 ∈ {𝑚}, ((1 / 3)↑𝑘), 0))
78	77	sumeq2dv 15673	. . . . . 6 ⊢ (𝜑 → Σ𝑘 ∈ (ℤ_≥‘𝑚)((𝐹‘{𝑚})‘𝑘) = Σ𝑘 ∈ (ℤ_≥‘𝑚)if(𝑘 ∈ {𝑚}, ((1 / 3)↑𝑘), 0))
79		uzid 12859	. . . . . . . . 9 ⊢ (𝑚 ∈ ℤ → 𝑚 ∈ (ℤ_≥‘𝑚))
80	22, 79	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑚 ∈ (ℤ_≥‘𝑚))
81	80	snssd 4808	. . . . . . 7 ⊢ (𝜑 → {𝑚} ⊆ (ℤ_≥‘𝑚))
82		vex 3473	. . . . . . . . 9 ⊢ 𝑚 ∈ V
83		oveq2 7422	. . . . . . . . . 10 ⊢ (𝑘 = 𝑚 → ((1 / 3)↑𝑘) = ((1 / 3)↑𝑚))
84	83	eleq1d 2813	. . . . . . . . 9 ⊢ (𝑘 = 𝑚 → (((1 / 3)↑𝑘) ∈ ℂ ↔ ((1 / 3)↑𝑚) ∈ ℂ))
85	82, 84	ralsn 4681	. . . . . . . 8 ⊢ (∀𝑘 ∈ {𝑚} ((1 / 3)↑𝑘) ∈ ℂ ↔ ((1 / 3)↑𝑚) ∈ ℂ)
86	42, 85	sylibr 233	. . . . . . 7 ⊢ (𝜑 → ∀𝑘 ∈ {𝑚} ((1 / 3)↑𝑘) ∈ ℂ)
87		ssidd 4001	. . . . . . . 8 ⊢ (𝜑 → (ℤ_≥‘𝑚) ⊆ (ℤ_≥‘𝑚))
88	87	orcd 872	. . . . . . 7 ⊢ (𝜑 → ((ℤ_≥‘𝑚) ⊆ (ℤ_≥‘𝑚) ∨ (ℤ_≥‘𝑚) ∈ Fin))
89		sumss2 15696	. . . . . . 7 ⊢ ((({𝑚} ⊆ (ℤ_≥‘𝑚) ∧ ∀𝑘 ∈ {𝑚} ((1 / 3)↑𝑘) ∈ ℂ) ∧ ((ℤ_≥‘𝑚) ⊆ (ℤ_≥‘𝑚) ∨ (ℤ_≥‘𝑚) ∈ Fin)) → Σ𝑘 ∈ {𝑚} ((1 / 3)↑𝑘) = Σ𝑘 ∈ (ℤ_≥‘𝑚)if(𝑘 ∈ {𝑚}, ((1 / 3)↑𝑘), 0))
90	81, 86, 88, 89	syl21anc 837	. . . . . 6 ⊢ (𝜑 → Σ𝑘 ∈ {𝑚} ((1 / 3)↑𝑘) = Σ𝑘 ∈ (ℤ_≥‘𝑚)if(𝑘 ∈ {𝑚}, ((1 / 3)↑𝑘), 0))
91	83	sumsn 15716	. . . . . . 7 ⊢ ((𝑚 ∈ ℕ ∧ ((1 / 3)↑𝑚) ∈ ℂ) → Σ𝑘 ∈ {𝑚} ((1 / 3)↑𝑘) = ((1 / 3)↑𝑚))
92	7, 42, 91	syl2anc 583	. . . . . 6 ⊢ (𝜑 → Σ𝑘 ∈ {𝑚} ((1 / 3)↑𝑘) = ((1 / 3)↑𝑚))
93	78, 90, 92	3eqtr2d 2773	. . . . 5 ⊢ (𝜑 → Σ𝑘 ∈ (ℤ_≥‘𝑚)((𝐹‘{𝑚})‘𝑘) = ((1 / 3)↑𝑚))
94	3, 4	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑚 ∈ 𝐴)
95	94	snssd 4808	. . . . . 6 ⊢ (𝜑 → {𝑚} ⊆ 𝐴)
96	8	rpnnen2lem7 16188	. . . . . 6 ⊢ (({𝑚} ⊆ 𝐴 ∧ 𝐴 ⊆ ℕ ∧ 𝑚 ∈ ℕ) → Σ𝑘 ∈ (ℤ_≥‘𝑚)((𝐹‘{𝑚})‘𝑘) ≤ Σ𝑘 ∈ (ℤ_≥‘𝑚)((𝐹‘𝐴)‘𝑘))
97	95, 2, 7, 96	syl3anc 1369	. . . . 5 ⊢ (𝜑 → Σ𝑘 ∈ (ℤ_≥‘𝑚)((𝐹‘{𝑚})‘𝑘) ≤ Σ𝑘 ∈ (ℤ_≥‘𝑚)((𝐹‘𝐴)‘𝑘))
98	93, 97	eqbrtrrd 5166	. . . 4 ⊢ (𝜑 → ((1 / 3)↑𝑚) ≤ Σ𝑘 ∈ (ℤ_≥‘𝑚)((𝐹‘𝐴)‘𝑘))
99	10, 16, 18, 73, 98	ltletrd 11396	. . 3 ⊢ (𝜑 → Σ𝑘 ∈ (ℤ_≥‘𝑚)((𝐹‘𝐵)‘𝑘) < Σ𝑘 ∈ (ℤ_≥‘𝑚)((𝐹‘𝐴)‘𝑘))
100	10, 99	gtned 11371	. 2 ⊢ (𝜑 → Σ𝑘 ∈ (ℤ_≥‘𝑚)((𝐹‘𝐴)‘𝑘) ≠ Σ𝑘 ∈ (ℤ_≥‘𝑚)((𝐹‘𝐵)‘𝑘))
101		rpnnen2.5	. . . . 5 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (𝑛 < 𝑚 → (𝑛 ∈ 𝐴 ↔ 𝑛 ∈ 𝐵)))
102		rpnnen2.6	. . . . 5 ⊢ (𝜓 ↔ Σ𝑘 ∈ ℕ ((𝐹‘𝐴)‘𝑘) = Σ𝑘 ∈ ℕ ((𝐹‘𝐵)‘𝑘))
103	8, 2, 1, 3, 101, 102	rpnnen2lem10 16191	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → Σ𝑘 ∈ (ℤ_≥‘𝑚)((𝐹‘𝐴)‘𝑘) = Σ𝑘 ∈ (ℤ_≥‘𝑚)((𝐹‘𝐵)‘𝑘))
104	103	ex 412	. . 3 ⊢ (𝜑 → (𝜓 → Σ𝑘 ∈ (ℤ_≥‘𝑚)((𝐹‘𝐴)‘𝑘) = Σ𝑘 ∈ (ℤ_≥‘𝑚)((𝐹‘𝐵)‘𝑘)))
105	104	necon3ad 2948	. 2 ⊢ (𝜑 → (Σ𝑘 ∈ (ℤ_≥‘𝑚)((𝐹‘𝐴)‘𝑘) ≠ Σ𝑘 ∈ (ℤ_≥‘𝑚)((𝐹‘𝐵)‘𝑘) → ¬ 𝜓))
106	100, 105	mpd 15	1 ⊢ (𝜑 → ¬ 𝜓)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∨ wo 846 = wceq 1534 ∈ wcel 2099 ≠ wne 2935 ∀wral 3056 ∖ cdif 3941 ⊆ wss 3944 ifcif 4524 𝒫 cpw 4598 {csn 4624 class class class wbr 5142 ↦ cmpt 5225 ‘cfv 6542 (class class class)co 7414 Fincfn 8955 ℂcc 11128 ℝcr 11129 0cc0 11130 1c1 11131 + caddc 11133 · cmul 11135 < clt 11270 ≤ cle 11271 − cmin 11466 / cdiv 11893 ℕcn 12234 2c2 12289 3c3 12290 ℕ₀cn0 12494 ℤcz 12580 ℤ_≥cuz 12844 ℝ⁺crp 12998 ↑cexp 14050 Σcsu 15656

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-inf2 9656 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 ax-pre-sup 11208

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-pm 8839 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-sup 9457 df-inf 9458 df-oi 9525 df-card 9954 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-div 11894 df-nn 12235 df-2 12297 df-3 12298 df-n0 12495 df-z 12581 df-uz 12845 df-rp 12999 df-ico 13354 df-fz 13509 df-fzo 13652 df-fl 13781 df-seq 13991 df-exp 14051 df-hash 14314 df-cj 15070 df-re 15071 df-im 15072 df-sqrt 15206 df-abs 15207 df-limsup 15439 df-clim 15456 df-rlim 15457 df-sum 15657

This theorem is referenced by: rpnnen2lem12 16193

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