rpnnen2lem11 - Metamath Proof Explorer

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Theorem rpnnen2lem11 16195

Description: Lemma for rpnnen2 16197. (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 4120	. . . . . 6 ⊢ (𝑚 ∈ (𝐴 ∖ 𝐵) → 𝑚 ∈ 𝐴)
5		ssel2 3968	. . . . . 6 ⊢ ((𝐴 ⊆ ℕ ∧ 𝑚 ∈ 𝐴) → 𝑚 ∈ ℕ)
6	4, 5	sylan2 591	. . . . 5 ⊢ ((𝐴 ⊆ ℕ ∧ 𝑚 ∈ (𝐴 ∖ 𝐵)) → 𝑚 ∈ ℕ)
7	2, 3, 6	syl2anc 582	. . . 4 ⊢ (𝜑 → 𝑚 ∈ ℕ)
8		rpnnen2.1	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝒫 ℕ ↦ (𝑛 ∈ ℕ ↦ if(𝑛 ∈ 𝑥, ((1 / 3)↑𝑛), 0)))
9	8	rpnnen2lem6 16190	. . . 4 ⊢ ((𝐵 ⊆ ℕ ∧ 𝑚 ∈ ℕ) → Σ𝑘 ∈ (ℤ_≥‘𝑚)((𝐹‘𝐵)‘𝑘) ∈ ℝ)
10	1, 7, 9	syl2anc 582	. . 3 ⊢ (𝜑 → Σ𝑘 ∈ (ℤ_≥‘𝑚)((𝐹‘𝐵)‘𝑘) ∈ ℝ)
11		3nn 12316	. . . . . 6 ⊢ 3 ∈ ℕ
12		nnrecre 12279	. . . . . 6 ⊢ (3 ∈ ℕ → (1 / 3) ∈ ℝ)
13	11, 12	ax-mp 5	. . . . 5 ⊢ (1 / 3) ∈ ℝ
14	7	nnnn0d 12557	. . . . 5 ⊢ (𝜑 → 𝑚 ∈ ℕ₀)
15		reexpcl 14070	. . . . 5 ⊢ (((1 / 3) ∈ ℝ ∧ 𝑚 ∈ ℕ₀) → ((1 / 3)↑𝑚) ∈ ℝ)
16	13, 14, 15	sylancr 585	. . . 4 ⊢ (𝜑 → ((1 / 3)↑𝑚) ∈ ℝ)
17	8	rpnnen2lem6 16190	. . . . 5 ⊢ ((𝐴 ⊆ ℕ ∧ 𝑚 ∈ ℕ) → Σ𝑘 ∈ (ℤ_≥‘𝑚)((𝐹‘𝐴)‘𝑘) ∈ ℝ)
18	2, 7, 17	syl2anc 582	. . . 4 ⊢ (𝜑 → Σ𝑘 ∈ (ℤ_≥‘𝑚)((𝐹‘𝐴)‘𝑘) ∈ ℝ)
19		nnrp 13012	. . . . . . . . 9 ⊢ (3 ∈ ℕ → 3 ∈ ℝ⁺)
20		rpreccl 13027	. . . . . . . . 9 ⊢ (3 ∈ ℝ⁺ → (1 / 3) ∈ ℝ⁺)
21	11, 19, 20	mp2b 10	. . . . . . . 8 ⊢ (1 / 3) ∈ ℝ⁺
22	7	nnzd 12610	. . . . . . . 8 ⊢ (𝜑 → 𝑚 ∈ ℤ)
23		rpexpcl 14072	. . . . . . . 8 ⊢ (((1 / 3) ∈ ℝ⁺ ∧ 𝑚 ∈ ℤ) → ((1 / 3)↑𝑚) ∈ ℝ⁺)
24	21, 22, 23	sylancr 585	. . . . . . 7 ⊢ (𝜑 → ((1 / 3)↑𝑚) ∈ ℝ⁺)
25	24	rpred 13043	. . . . . 6 ⊢ (𝜑 → ((1 / 3)↑𝑚) ∈ ℝ)
26	25	rehalfcld 12484	. . . . 5 ⊢ (𝜑 → (((1 / 3)↑𝑚) / 2) ∈ ℝ)
27	3	snssd 4809	. . . . . . . . 9 ⊢ (𝜑 → {𝑚} ⊆ (𝐴 ∖ 𝐵))
28	2	ssdifd 4134	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 ∖ 𝐵) ⊆ (ℕ ∖ 𝐵))
29	27, 28	sstrd 3984	. . . . . . . 8 ⊢ (𝜑 → {𝑚} ⊆ (ℕ ∖ 𝐵))
30	7	snssd 4809	. . . . . . . . 9 ⊢ (𝜑 → {𝑚} ⊆ ℕ)
31		ssconb 4131	. . . . . . . . 9 ⊢ ((𝐵 ⊆ ℕ ∧ {𝑚} ⊆ ℕ) → (𝐵 ⊆ (ℕ ∖ {𝑚}) ↔ {𝑚} ⊆ (ℕ ∖ 𝐵)))
32	1, 30, 31	syl2anc 582	. . . . . . . 8 ⊢ (𝜑 → (𝐵 ⊆ (ℕ ∖ {𝑚}) ↔ {𝑚} ⊆ (ℕ ∖ 𝐵)))
33	29, 32	mpbird 256	. . . . . . 7 ⊢ (𝜑 → 𝐵 ⊆ (ℕ ∖ {𝑚}))
34		difssd 4126	. . . . . . 7 ⊢ (𝜑 → (ℕ ∖ {𝑚}) ⊆ ℕ)
35	8	rpnnen2lem7 16191	. . . . . . 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 1368	. . . . . 6 ⊢ (𝜑 → Σ𝑘 ∈ (ℤ_≥‘𝑚)((𝐹‘𝐵)‘𝑘) ≤ Σ𝑘 ∈ (ℤ_≥‘𝑚)((𝐹‘(ℕ ∖ {𝑚}))‘𝑘))
37	8	rpnnen2lem9 16193	. . . . . . . 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 11253	. . . . . . . . . . . 12 ⊢ (1 / 3) ∈ ℂ
40		expp1 14060	. . . . . . . . . . . 12 ⊢ (((1 / 3) ∈ ℂ ∧ 𝑚 ∈ ℕ₀) → ((1 / 3)↑(𝑚 + 1)) = (((1 / 3)↑𝑚) · (1 / 3)))
41	39, 14, 40	sylancr 585	. . . . . . . . . . 11 ⊢ (𝜑 → ((1 / 3)↑(𝑚 + 1)) = (((1 / 3)↑𝑚) · (1 / 3)))
42	25	recnd 11267	. . . . . . . . . . . 12 ⊢ (𝜑 → ((1 / 3)↑𝑚) ∈ ℂ)
43		3cn 12318	. . . . . . . . . . . . 13 ⊢ 3 ∈ ℂ
44		3ne0 12343	. . . . . . . . . . . . 13 ⊢ 3 ≠ 0
45		divrec 11913	. . . . . . . . . . . . 13 ⊢ ((((1 / 3)↑𝑚) ∈ ℂ ∧ 3 ∈ ℂ ∧ 3 ≠ 0) → (((1 / 3)↑𝑚) / 3) = (((1 / 3)↑𝑚) · (1 / 3)))
46	43, 44, 45	mp3an23 1449	. . . . . . . . . . . 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 2768	. . . . . . . . . 10 ⊢ (𝜑 → ((1 / 3)↑(𝑚 + 1)) = (((1 / 3)↑𝑚) / 3))
49	48	oveq1d 7428	. . . . . . . . 9 ⊢ (𝜑 → (((1 / 3)↑(𝑚 + 1)) / (1 − (1 / 3))) = ((((1 / 3)↑𝑚) / 3) / (1 − (1 / 3))))
50		ax-1cn 11191	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℂ
51	43, 44	pm3.2i 469	. . . . . . . . . . . . 13 ⊢ (3 ∈ ℂ ∧ 3 ≠ 0)
52		divsubdir 11933	. . . . . . . . . . . . 13 ⊢ ((3 ∈ ℂ ∧ 1 ∈ ℂ ∧ (3 ∈ ℂ ∧ 3 ≠ 0)) → ((3 − 1) / 3) = ((3 / 3) − (1 / 3)))
53	43, 50, 51, 52	mp3an 1457	. . . . . . . . . . . 12 ⊢ ((3 − 1) / 3) = ((3 / 3) − (1 / 3))
54		3m1e2 12365	. . . . . . . . . . . . 13 ⊢ (3 − 1) = 2
55	54	oveq1i 7423	. . . . . . . . . . . 12 ⊢ ((3 − 1) / 3) = (2 / 3)
56	43, 44	dividi 11972	. . . . . . . . . . . . 13 ⊢ (3 / 3) = 1
57	56	oveq1i 7423	. . . . . . . . . . . 12 ⊢ ((3 / 3) − (1 / 3)) = (1 − (1 / 3))
58	53, 55, 57	3eqtr3ri 2762	. . . . . . . . . . 11 ⊢ (1 − (1 / 3)) = (2 / 3)
59	58	oveq2i 7424	. . . . . . . . . 10 ⊢ ((((1 / 3)↑𝑚) / 3) / (1 − (1 / 3))) = ((((1 / 3)↑𝑚) / 3) / (2 / 3))
60		2cnne0 12447	. . . . . . . . . . . 12 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0)
61		divcan7 11948	. . . . . . . . . . . 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 1449	. . . . . . . . . . 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 2777	. . . . . . . . 9 ⊢ (𝜑 → ((((1 / 3)↑𝑚) / 3) / (1 − (1 / 3))) = (((1 / 3)↑𝑚) / 2))
65	49, 64	eqtrd 2765	. . . . . . . 8 ⊢ (𝜑 → (((1 / 3)↑(𝑚 + 1)) / (1 − (1 / 3))) = (((1 / 3)↑𝑚) / 2))
66	65	oveq2d 7429	. . . . . . 7 ⊢ (𝜑 → (0 + (((1 / 3)↑(𝑚 + 1)) / (1 − (1 / 3)))) = (0 + (((1 / 3)↑𝑚) / 2)))
67	26	recnd 11267	. . . . . . . 8 ⊢ (𝜑 → (((1 / 3)↑𝑚) / 2) ∈ ℂ)
68	67	addlidd 11440	. . . . . . 7 ⊢ (𝜑 → (0 + (((1 / 3)↑𝑚) / 2)) = (((1 / 3)↑𝑚) / 2))
69	38, 66, 68	3eqtrd 2769	. . . . . 6 ⊢ (𝜑 → Σ𝑘 ∈ (ℤ_≥‘𝑚)((𝐹‘(ℕ ∖ {𝑚}))‘𝑘) = (((1 / 3)↑𝑚) / 2))
70	36, 69	breqtrd 5170	. . . . 5 ⊢ (𝜑 → Σ𝑘 ∈ (ℤ_≥‘𝑚)((𝐹‘𝐵)‘𝑘) ≤ (((1 / 3)↑𝑚) / 2))
71		rphalflt 13030	. . . . . 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 11397	. . . 4 ⊢ (𝜑 → Σ𝑘 ∈ (ℤ_≥‘𝑚)((𝐹‘𝐵)‘𝑘) < ((1 / 3)↑𝑚))
74		eluznn 12927	. . . . . . . . 9 ⊢ ((𝑚 ∈ ℕ ∧ 𝑘 ∈ (ℤ_≥‘𝑚)) → 𝑘 ∈ ℕ)
75	7, 74	sylan 578	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑚)) → 𝑘 ∈ ℕ)
76	8	rpnnen2lem1 16185	. . . . . . . 8 ⊢ (({𝑚} ⊆ ℕ ∧ 𝑘 ∈ ℕ) → ((𝐹‘{𝑚})‘𝑘) = if(𝑘 ∈ {𝑚}, ((1 / 3)↑𝑘), 0))
77	30, 75, 76	syl2an2r 683	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑚)) → ((𝐹‘{𝑚})‘𝑘) = if(𝑘 ∈ {𝑚}, ((1 / 3)↑𝑘), 0))
78	77	sumeq2dv 15676	. . . . . 6 ⊢ (𝜑 → Σ𝑘 ∈ (ℤ_≥‘𝑚)((𝐹‘{𝑚})‘𝑘) = Σ𝑘 ∈ (ℤ_≥‘𝑚)if(𝑘 ∈ {𝑚}, ((1 / 3)↑𝑘), 0))
79		uzid 12862	. . . . . . . . 9 ⊢ (𝑚 ∈ ℤ → 𝑚 ∈ (ℤ_≥‘𝑚))
80	22, 79	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑚 ∈ (ℤ_≥‘𝑚))
81	80	snssd 4809	. . . . . . 7 ⊢ (𝜑 → {𝑚} ⊆ (ℤ_≥‘𝑚))
82		vex 3467	. . . . . . . . 9 ⊢ 𝑚 ∈ V
83		oveq2 7421	. . . . . . . . . 10 ⊢ (𝑘 = 𝑚 → ((1 / 3)↑𝑘) = ((1 / 3)↑𝑚))
84	83	eleq1d 2810	. . . . . . . . 9 ⊢ (𝑘 = 𝑚 → (((1 / 3)↑𝑘) ∈ ℂ ↔ ((1 / 3)↑𝑚) ∈ ℂ))
85	82, 84	ralsn 4682	. . . . . . . 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 3997	. . . . . . . 8 ⊢ (𝜑 → (ℤ_≥‘𝑚) ⊆ (ℤ_≥‘𝑚))
88	87	orcd 871	. . . . . . 7 ⊢ (𝜑 → ((ℤ_≥‘𝑚) ⊆ (ℤ_≥‘𝑚) ∨ (ℤ_≥‘𝑚) ∈ Fin))
89		sumss2 15699	. . . . . . 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 836	. . . . . 6 ⊢ (𝜑 → Σ𝑘 ∈ {𝑚} ((1 / 3)↑𝑘) = Σ𝑘 ∈ (ℤ_≥‘𝑚)if(𝑘 ∈ {𝑚}, ((1 / 3)↑𝑘), 0))
91	83	sumsn 15719	. . . . . . 7 ⊢ ((𝑚 ∈ ℕ ∧ ((1 / 3)↑𝑚) ∈ ℂ) → Σ𝑘 ∈ {𝑚} ((1 / 3)↑𝑘) = ((1 / 3)↑𝑚))
92	7, 42, 91	syl2anc 582	. . . . . 6 ⊢ (𝜑 → Σ𝑘 ∈ {𝑚} ((1 / 3)↑𝑘) = ((1 / 3)↑𝑚))
93	78, 90, 92	3eqtr2d 2771	. . . . 5 ⊢ (𝜑 → Σ𝑘 ∈ (ℤ_≥‘𝑚)((𝐹‘{𝑚})‘𝑘) = ((1 / 3)↑𝑚))
94	3, 4	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑚 ∈ 𝐴)
95	94	snssd 4809	. . . . . 6 ⊢ (𝜑 → {𝑚} ⊆ 𝐴)
96	8	rpnnen2lem7 16191	. . . . . 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 1368	. . . . 5 ⊢ (𝜑 → Σ𝑘 ∈ (ℤ_≥‘𝑚)((𝐹‘{𝑚})‘𝑘) ≤ Σ𝑘 ∈ (ℤ_≥‘𝑚)((𝐹‘𝐴)‘𝑘))
98	93, 97	eqbrtrrd 5168	. . . 4 ⊢ (𝜑 → ((1 / 3)↑𝑚) ≤ Σ𝑘 ∈ (ℤ_≥‘𝑚)((𝐹‘𝐴)‘𝑘))
99	10, 16, 18, 73, 98	ltletrd 11399	. . 3 ⊢ (𝜑 → Σ𝑘 ∈ (ℤ_≥‘𝑚)((𝐹‘𝐵)‘𝑘) < Σ𝑘 ∈ (ℤ_≥‘𝑚)((𝐹‘𝐴)‘𝑘))
100	10, 99	gtned 11374	. 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 16194	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → Σ𝑘 ∈ (ℤ_≥‘𝑚)((𝐹‘𝐴)‘𝑘) = Σ𝑘 ∈ (ℤ_≥‘𝑚)((𝐹‘𝐵)‘𝑘))
104	103	ex 411	. . 3 ⊢ (𝜑 → (𝜓 → Σ𝑘 ∈ (ℤ_≥‘𝑚)((𝐹‘𝐴)‘𝑘) = Σ𝑘 ∈ (ℤ_≥‘𝑚)((𝐹‘𝐵)‘𝑘)))
105	104	necon3ad 2943	. 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 394 ∨ wo 845 = wceq 1533 ∈ wcel 2098 ≠ wne 2930 ∀wral 3051 ∖ cdif 3938 ⊆ wss 3941 ifcif 4525 𝒫 cpw 4599 {csn 4625 class class class wbr 5144 ↦ cmpt 5227 ‘cfv 6543 (class class class)co 7413 Fincfn 8957 ℂcc 11131 ℝcr 11132 0cc0 11133 1c1 11134 + caddc 11136 · cmul 11138 < clt 11273 ≤ cle 11274 − cmin 11469 / cdiv 11896 ℕcn 12237 2c2 12292 3c3 12293 ℕ₀cn0 12497 ℤcz 12583 ℤ_≥cuz 12847 ℝ⁺crp 13001 ↑cexp 14053 Σcsu 15659

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-inf2 9659 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 ax-pre-sup 11211

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-int 4946 df-iun 4994 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-se 5629 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 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 8841 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-sup 9460 df-inf 9461 df-oi 9528 df-card 9957 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-div 11897 df-nn 12238 df-2 12300 df-3 12301 df-n0 12498 df-z 12584 df-uz 12848 df-rp 13002 df-ico 13357 df-fz 13512 df-fzo 13655 df-fl 13784 df-seq 13994 df-exp 14054 df-hash 14317 df-cj 15073 df-re 15074 df-im 15075 df-sqrt 15209 df-abs 15210 df-limsup 15442 df-clim 15459 df-rlim 15460 df-sum 15660

This theorem is referenced by: rpnnen2lem12 16196

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