nnsum4primeseven - Mathbox for Alexander van der Vekens

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Theorem nnsum4primeseven 46458

Description: If the (weak) ternary Goldbach conjecture is valid, then every even integer greater than 8 is the sum of 4 primes. (Contributed by AV, 25-Jul-2020.)

**Assertion**
Ref	Expression
nnsum4primeseven	⊢ (∀𝑚 ∈ Odd (5 < 𝑚 → 𝑚 ∈ GoldbachOddW ) → ((𝑁 ∈ (ℤ_≥‘9) ∧ 𝑁 ∈ Even ) → ∃𝑓 ∈ (ℙ ↑_m (1...4))𝑁 = Σ𝑘 ∈ (1...4)(𝑓‘𝑘)))

Distinct variable group: 𝑓,𝑁,𝑘,𝑚

Proof of Theorem nnsum4primeseven Dummy variables 𝑜 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		evengpop3 46456	. . . 4 ⊢ (∀𝑚 ∈ Odd (5 < 𝑚 → 𝑚 ∈ GoldbachOddW ) → ((𝑁 ∈ (ℤ_≥‘9) ∧ 𝑁 ∈ Even ) → ∃𝑜 ∈ GoldbachOddW 𝑁 = (𝑜 + 3)))
2	1	imp 407	. . 3 ⊢ ((∀𝑚 ∈ Odd (5 < 𝑚 → 𝑚 ∈ GoldbachOddW ) ∧ (𝑁 ∈ (ℤ_≥‘9) ∧ 𝑁 ∈ Even )) → ∃𝑜 ∈ GoldbachOddW 𝑁 = (𝑜 + 3))
3		simplll 773	. . . . . 6 ⊢ ((((∀𝑚 ∈ Odd (5 < 𝑚 → 𝑚 ∈ GoldbachOddW ) ∧ (𝑁 ∈ (ℤ_≥‘9) ∧ 𝑁 ∈ Even )) ∧ 𝑜 ∈ GoldbachOddW ) ∧ 𝑁 = (𝑜 + 3)) → ∀𝑚 ∈ Odd (5 < 𝑚 → 𝑚 ∈ GoldbachOddW ))
4		6nn 12300	. . . . . . . . . . 11 ⊢ 6 ∈ ℕ
5	4	nnzi 12585	. . . . . . . . . 10 ⊢ 6 ∈ ℤ
6	5	a1i 11	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ_≥‘9) → 6 ∈ ℤ)
7		3z 12594	. . . . . . . . . 10 ⊢ 3 ∈ ℤ
8	7	a1i 11	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ_≥‘9) → 3 ∈ ℤ)
9		6p3e9 12371	. . . . . . . . . . . . 13 ⊢ (6 + 3) = 9
10	9	eqcomi 2741	. . . . . . . . . . . 12 ⊢ 9 = (6 + 3)
11	10	fveq2i 6894	. . . . . . . . . . 11 ⊢ (ℤ_≥‘9) = (ℤ_≥‘(6 + 3))
12	11	eleq2i 2825	. . . . . . . . . 10 ⊢ (𝑁 ∈ (ℤ_≥‘9) ↔ 𝑁 ∈ (ℤ_≥‘(6 + 3)))
13	12	biimpi 215	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ_≥‘9) → 𝑁 ∈ (ℤ_≥‘(6 + 3)))
14		eluzsub 12851	. . . . . . . . 9 ⊢ ((6 ∈ ℤ ∧ 3 ∈ ℤ ∧ 𝑁 ∈ (ℤ_≥‘(6 + 3))) → (𝑁 − 3) ∈ (ℤ_≥‘6))
15	6, 8, 13, 14	syl3anc 1371	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘9) → (𝑁 − 3) ∈ (ℤ_≥‘6))
16	15	adantr 481	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ_≥‘9) ∧ 𝑁 ∈ Even ) → (𝑁 − 3) ∈ (ℤ_≥‘6))
17	16	ad3antlr 729	. . . . . 6 ⊢ ((((∀𝑚 ∈ Odd (5 < 𝑚 → 𝑚 ∈ GoldbachOddW ) ∧ (𝑁 ∈ (ℤ_≥‘9) ∧ 𝑁 ∈ Even )) ∧ 𝑜 ∈ GoldbachOddW ) ∧ 𝑁 = (𝑜 + 3)) → (𝑁 − 3) ∈ (ℤ_≥‘6))
18		3odd 46366	. . . . . . . . . . . . 13 ⊢ 3 ∈ Odd
19	18	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (ℤ_≥‘9) → 3 ∈ Odd )
20	19	anim1i 615	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ (ℤ_≥‘9) ∧ 𝑁 ∈ Even ) → (3 ∈ Odd ∧ 𝑁 ∈ Even ))
21	20	adantl 482	. . . . . . . . . 10 ⊢ ((∀𝑚 ∈ Odd (5 < 𝑚 → 𝑚 ∈ GoldbachOddW ) ∧ (𝑁 ∈ (ℤ_≥‘9) ∧ 𝑁 ∈ Even )) → (3 ∈ Odd ∧ 𝑁 ∈ Even ))
22	21	ancomd 462	. . . . . . . . 9 ⊢ ((∀𝑚 ∈ Odd (5 < 𝑚 → 𝑚 ∈ GoldbachOddW ) ∧ (𝑁 ∈ (ℤ_≥‘9) ∧ 𝑁 ∈ Even )) → (𝑁 ∈ Even ∧ 3 ∈ Odd ))
23	22	adantr 481	. . . . . . . 8 ⊢ (((∀𝑚 ∈ Odd (5 < 𝑚 → 𝑚 ∈ GoldbachOddW ) ∧ (𝑁 ∈ (ℤ_≥‘9) ∧ 𝑁 ∈ Even )) ∧ 𝑜 ∈ GoldbachOddW ) → (𝑁 ∈ Even ∧ 3 ∈ Odd ))
24	23	adantr 481	. . . . . . 7 ⊢ ((((∀𝑚 ∈ Odd (5 < 𝑚 → 𝑚 ∈ GoldbachOddW ) ∧ (𝑁 ∈ (ℤ_≥‘9) ∧ 𝑁 ∈ Even )) ∧ 𝑜 ∈ GoldbachOddW ) ∧ 𝑁 = (𝑜 + 3)) → (𝑁 ∈ Even ∧ 3 ∈ Odd ))
25		emoo 46362	. . . . . . 7 ⊢ ((𝑁 ∈ Even ∧ 3 ∈ Odd ) → (𝑁 − 3) ∈ Odd )
26	24, 25	syl 17	. . . . . 6 ⊢ ((((∀𝑚 ∈ Odd (5 < 𝑚 → 𝑚 ∈ GoldbachOddW ) ∧ (𝑁 ∈ (ℤ_≥‘9) ∧ 𝑁 ∈ Even )) ∧ 𝑜 ∈ GoldbachOddW ) ∧ 𝑁 = (𝑜 + 3)) → (𝑁 − 3) ∈ Odd )
27		nnsum4primesodd 46454	. . . . . . 7 ⊢ (∀𝑚 ∈ Odd (5 < 𝑚 → 𝑚 ∈ GoldbachOddW ) → (((𝑁 − 3) ∈ (ℤ_≥‘6) ∧ (𝑁 − 3) ∈ Odd ) → ∃𝑔 ∈ (ℙ ↑_m (1...3))(𝑁 − 3) = Σ𝑘 ∈ (1...3)(𝑔‘𝑘)))
28	27	imp 407	. . . . . 6 ⊢ ((∀𝑚 ∈ Odd (5 < 𝑚 → 𝑚 ∈ GoldbachOddW ) ∧ ((𝑁 − 3) ∈ (ℤ_≥‘6) ∧ (𝑁 − 3) ∈ Odd )) → ∃𝑔 ∈ (ℙ ↑_m (1...3))(𝑁 − 3) = Σ𝑘 ∈ (1...3)(𝑔‘𝑘))
29	3, 17, 26, 28	syl12anc 835	. . . . 5 ⊢ ((((∀𝑚 ∈ Odd (5 < 𝑚 → 𝑚 ∈ GoldbachOddW ) ∧ (𝑁 ∈ (ℤ_≥‘9) ∧ 𝑁 ∈ Even )) ∧ 𝑜 ∈ GoldbachOddW ) ∧ 𝑁 = (𝑜 + 3)) → ∃𝑔 ∈ (ℙ ↑_m (1...3))(𝑁 − 3) = Σ𝑘 ∈ (1...3)(𝑔‘𝑘))
30		simpr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ (ℤ_≥‘9) ∧ 𝑔:(1...3)⟶ℙ) → 𝑔:(1...3)⟶ℙ)
31		4z 12595	. . . . . . . . . . . . . . . . . 18 ⊢ 4 ∈ ℤ
32		fzonel 13645	. . . . . . . . . . . . . . . . . . 19 ⊢ ¬ 4 ∈ (1..^4)
33		fzoval 13632	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (4 ∈ ℤ → (1..^4) = (1...(4 − 1)))
34	31, 33	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (1..^4) = (1...(4 − 1))
35		4cn 12296	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 4 ∈ ℂ
36		ax-1cn 11167	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 1 ∈ ℂ
37		3cn 12292	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 3 ∈ ℂ
38	35, 36, 37	3pm3.2i 1339	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (4 ∈ ℂ ∧ 1 ∈ ℂ ∧ 3 ∈ ℂ)
39		3p1e4 12356	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (3 + 1) = 4
40		subadd2 11463	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((4 ∈ ℂ ∧ 1 ∈ ℂ ∧ 3 ∈ ℂ) → ((4 − 1) = 3 ↔ (3 + 1) = 4))
41	39, 40	mpbiri 257	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((4 ∈ ℂ ∧ 1 ∈ ℂ ∧ 3 ∈ ℂ) → (4 − 1) = 3)
42	38, 41	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (4 − 1) = 3
43	42	oveq2i 7419	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (1...(4 − 1)) = (1...3)
44	34, 43	eqtri 2760	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (1..^4) = (1...3)
45	44	eqcomi 2741	. . . . . . . . . . . . . . . . . . . 20 ⊢ (1...3) = (1..^4)
46	45	eleq2i 2825	. . . . . . . . . . . . . . . . . . 19 ⊢ (4 ∈ (1...3) ↔ 4 ∈ (1..^4))
47	32, 46	mtbir 322	. . . . . . . . . . . . . . . . . 18 ⊢ ¬ 4 ∈ (1...3)
48	31, 47	pm3.2i 471	. . . . . . . . . . . . . . . . 17 ⊢ (4 ∈ ℤ ∧ ¬ 4 ∈ (1...3))
49	48	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ (ℤ_≥‘9) ∧ 𝑔:(1...3)⟶ℙ) → (4 ∈ ℤ ∧ ¬ 4 ∈ (1...3)))
50		3prm 16630	. . . . . . . . . . . . . . . . 17 ⊢ 3 ∈ ℙ
51	50	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ (ℤ_≥‘9) ∧ 𝑔:(1...3)⟶ℙ) → 3 ∈ ℙ)
52		fsnunf 7182	. . . . . . . . . . . . . . . 16 ⊢ ((𝑔:(1...3)⟶ℙ ∧ (4 ∈ ℤ ∧ ¬ 4 ∈ (1...3)) ∧ 3 ∈ ℙ) → (𝑔 ∪ {⟨4, 3⟩}):((1...3) ∪ {4})⟶ℙ)
53	30, 49, 51, 52	syl3anc 1371	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ (ℤ_≥‘9) ∧ 𝑔:(1...3)⟶ℙ) → (𝑔 ∪ {⟨4, 3⟩}):((1...3) ∪ {4})⟶ℙ)
54		fzval3 13700	. . . . . . . . . . . . . . . . . 18 ⊢ (4 ∈ ℤ → (1...4) = (1..^(4 + 1)))
55	31, 54	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ (1...4) = (1..^(4 + 1))
56		1z 12591	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ ℤ
57		1re 11213	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1 ∈ ℝ
58		4re 12295	. . . . . . . . . . . . . . . . . . . 20 ⊢ 4 ∈ ℝ
59		1lt4 12387	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1 < 4
60	57, 58, 59	ltleii 11336	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ≤ 4
61		eluz2 12827	. . . . . . . . . . . . . . . . . . 19 ⊢ (4 ∈ (ℤ_≥‘1) ↔ (1 ∈ ℤ ∧ 4 ∈ ℤ ∧ 1 ≤ 4))
62	56, 31, 60, 61	mpbir3an 1341	. . . . . . . . . . . . . . . . . 18 ⊢ 4 ∈ (ℤ_≥‘1)
63		fzosplitsn 13739	. . . . . . . . . . . . . . . . . 18 ⊢ (4 ∈ (ℤ_≥‘1) → (1..^(4 + 1)) = ((1..^4) ∪ {4}))
64	62, 63	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ (1..^(4 + 1)) = ((1..^4) ∪ {4})
65	44	uneq1i 4159	. . . . . . . . . . . . . . . . 17 ⊢ ((1..^4) ∪ {4}) = ((1...3) ∪ {4})
66	55, 64, 65	3eqtri 2764	. . . . . . . . . . . . . . . 16 ⊢ (1...4) = ((1...3) ∪ {4})
67	66	feq2i 6709	. . . . . . . . . . . . . . 15 ⊢ ((𝑔 ∪ {⟨4, 3⟩}):(1...4)⟶ℙ ↔ (𝑔 ∪ {⟨4, 3⟩}):((1...3) ∪ {4})⟶ℙ)
68	53, 67	sylibr 233	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ (ℤ_≥‘9) ∧ 𝑔:(1...3)⟶ℙ) → (𝑔 ∪ {⟨4, 3⟩}):(1...4)⟶ℙ)
69		prmex 16613	. . . . . . . . . . . . . . . 16 ⊢ ℙ ∈ V
70		ovex 7441	. . . . . . . . . . . . . . . 16 ⊢ (1...4) ∈ V
71	69, 70	pm3.2i 471	. . . . . . . . . . . . . . 15 ⊢ (ℙ ∈ V ∧ (1...4) ∈ V)
72		elmapg 8832	. . . . . . . . . . . . . . 15 ⊢ ((ℙ ∈ V ∧ (1...4) ∈ V) → ((𝑔 ∪ {⟨4, 3⟩}) ∈ (ℙ ↑_m (1...4)) ↔ (𝑔 ∪ {⟨4, 3⟩}):(1...4)⟶ℙ))
73	71, 72	mp1i 13	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ (ℤ_≥‘9) ∧ 𝑔:(1...3)⟶ℙ) → ((𝑔 ∪ {⟨4, 3⟩}) ∈ (ℙ ↑_m (1...4)) ↔ (𝑔 ∪ {⟨4, 3⟩}):(1...4)⟶ℙ))
74	68, 73	mpbird 256	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ (ℤ_≥‘9) ∧ 𝑔:(1...3)⟶ℙ) → (𝑔 ∪ {⟨4, 3⟩}) ∈ (ℙ ↑_m (1...4)))
75	74	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ (ℤ_≥‘9) ∧ 𝑔:(1...3)⟶ℙ) ∧ (𝑁 − 3) = Σ𝑘 ∈ (1...3)(𝑔‘𝑘)) → (𝑔 ∪ {⟨4, 3⟩}) ∈ (ℙ ↑_m (1...4)))
76		fveq1 6890	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = (𝑔 ∪ {⟨4, 3⟩}) → (𝑓‘𝑘) = ((𝑔 ∪ {⟨4, 3⟩})‘𝑘))
77	76	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 = (𝑔 ∪ {⟨4, 3⟩}) ∧ 𝑘 ∈ (1...4)) → (𝑓‘𝑘) = ((𝑔 ∪ {⟨4, 3⟩})‘𝑘))
78	77	sumeq2dv 15648	. . . . . . . . . . . . . 14 ⊢ (𝑓 = (𝑔 ∪ {⟨4, 3⟩}) → Σ𝑘 ∈ (1...4)(𝑓‘𝑘) = Σ𝑘 ∈ (1...4)((𝑔 ∪ {⟨4, 3⟩})‘𝑘))
79	78	eqeq2d 2743	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑔 ∪ {⟨4, 3⟩}) → (𝑁 = Σ𝑘 ∈ (1...4)(𝑓‘𝑘) ↔ 𝑁 = Σ𝑘 ∈ (1...4)((𝑔 ∪ {⟨4, 3⟩})‘𝑘)))
80	79	adantl 482	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ (ℤ_≥‘9) ∧ 𝑔:(1...3)⟶ℙ) ∧ (𝑁 − 3) = Σ𝑘 ∈ (1...3)(𝑔‘𝑘)) ∧ 𝑓 = (𝑔 ∪ {⟨4, 3⟩})) → (𝑁 = Σ𝑘 ∈ (1...4)(𝑓‘𝑘) ↔ 𝑁 = Σ𝑘 ∈ (1...4)((𝑔 ∪ {⟨4, 3⟩})‘𝑘)))
81	62	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ (ℤ_≥‘9) ∧ 𝑔:(1...3)⟶ℙ) → 4 ∈ (ℤ_≥‘1))
82	66	eleq2i 2825	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ (1...4) ↔ 𝑘 ∈ ((1...3) ∪ {4}))
83		elun 4148	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ ((1...3) ∪ {4}) ↔ (𝑘 ∈ (1...3) ∨ 𝑘 ∈ {4}))
84		velsn 4644	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ {4} ↔ 𝑘 = 4)
85	84	orbi2i 911	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑘 ∈ (1...3) ∨ 𝑘 ∈ {4}) ↔ (𝑘 ∈ (1...3) ∨ 𝑘 = 4))
86	82, 83, 85	3bitri 296	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (1...4) ↔ (𝑘 ∈ (1...3) ∨ 𝑘 = 4))
87		elfz2 13490	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑘 ∈ (1...3) ↔ ((1 ∈ ℤ ∧ 3 ∈ ℤ ∧ 𝑘 ∈ ℤ) ∧ (1 ≤ 𝑘 ∧ 𝑘 ≤ 3)))
88		3re 12291	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ 3 ∈ ℝ
89	88, 58	pm3.2i 471	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (3 ∈ ℝ ∧ 4 ∈ ℝ)
90		3lt4 12385	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ 3 < 4
91		ltnle 11292	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((3 ∈ ℝ ∧ 4 ∈ ℝ) → (3 < 4 ↔ ¬ 4 ≤ 3))
92	90, 91	mpbii 232	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((3 ∈ ℝ ∧ 4 ∈ ℝ) → ¬ 4 ≤ 3)
93	89, 92	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ¬ 4 ≤ 3
94		breq1 5151	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑘 = 4 → (𝑘 ≤ 3 ↔ 4 ≤ 3))
95	94	eqcoms 2740	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (4 = 𝑘 → (𝑘 ≤ 3 ↔ 4 ≤ 3))
96	93, 95	mtbiri 326	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (4 = 𝑘 → ¬ 𝑘 ≤ 3)
97	96	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑘 ∈ ℤ → (4 = 𝑘 → ¬ 𝑘 ≤ 3))
98	97	necon2ad 2955	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑘 ∈ ℤ → (𝑘 ≤ 3 → 4 ≠ 𝑘))
99	98	adantld 491	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑘 ∈ ℤ → ((1 ≤ 𝑘 ∧ 𝑘 ≤ 3) → 4 ≠ 𝑘))
100	99	3ad2ant3 1135	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((1 ∈ ℤ ∧ 3 ∈ ℤ ∧ 𝑘 ∈ ℤ) → ((1 ≤ 𝑘 ∧ 𝑘 ≤ 3) → 4 ≠ 𝑘))
101	100	imp 407	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((1 ∈ ℤ ∧ 3 ∈ ℤ ∧ 𝑘 ∈ ℤ) ∧ (1 ≤ 𝑘 ∧ 𝑘 ≤ 3)) → 4 ≠ 𝑘)
102	87, 101	sylbi 216	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘 ∈ (1...3) → 4 ≠ 𝑘)
103	102	adantr 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑘 ∈ (1...3) ∧ 𝑔:(1...3)⟶ℙ) → 4 ≠ 𝑘)
104		fvunsn 7176	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (4 ≠ 𝑘 → ((𝑔 ∪ {⟨4, 3⟩})‘𝑘) = (𝑔‘𝑘))
105	103, 104	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑘 ∈ (1...3) ∧ 𝑔:(1...3)⟶ℙ) → ((𝑔 ∪ {⟨4, 3⟩})‘𝑘) = (𝑔‘𝑘))
106		ffvelcdm 7083	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑔:(1...3)⟶ℙ ∧ 𝑘 ∈ (1...3)) → (𝑔‘𝑘) ∈ ℙ)
107	106	ancoms 459	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑘 ∈ (1...3) ∧ 𝑔:(1...3)⟶ℙ) → (𝑔‘𝑘) ∈ ℙ)
108		prmz 16611	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑔‘𝑘) ∈ ℙ → (𝑔‘𝑘) ∈ ℤ)
109	107, 108	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑘 ∈ (1...3) ∧ 𝑔:(1...3)⟶ℙ) → (𝑔‘𝑘) ∈ ℤ)
110	109	zcnd 12666	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑘 ∈ (1...3) ∧ 𝑔:(1...3)⟶ℙ) → (𝑔‘𝑘) ∈ ℂ)
111	105, 110	eqeltrd 2833	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑘 ∈ (1...3) ∧ 𝑔:(1...3)⟶ℙ) → ((𝑔 ∪ {⟨4, 3⟩})‘𝑘) ∈ ℂ)
112	111	ex 413	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ (1...3) → (𝑔:(1...3)⟶ℙ → ((𝑔 ∪ {⟨4, 3⟩})‘𝑘) ∈ ℂ))
113	112	adantld 491	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ (1...3) → ((𝑁 ∈ (ℤ_≥‘9) ∧ 𝑔:(1...3)⟶ℙ) → ((𝑔 ∪ {⟨4, 3⟩})‘𝑘) ∈ ℂ))
114		fveq2 6891	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 = 4 → ((𝑔 ∪ {⟨4, 3⟩})‘𝑘) = ((𝑔 ∪ {⟨4, 3⟩})‘4))
115	31	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑔:(1...3)⟶ℙ → 4 ∈ ℤ)
116	7	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑔:(1...3)⟶ℙ → 3 ∈ ℤ)
117		fdm 6726	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑔:(1...3)⟶ℙ → dom 𝑔 = (1...3))
118		eleq2 2822	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (dom 𝑔 = (1...3) → (4 ∈ dom 𝑔 ↔ 4 ∈ (1...3)))
119	47, 118	mtbiri 326	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (dom 𝑔 = (1...3) → ¬ 4 ∈ dom 𝑔)
120	117, 119	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑔:(1...3)⟶ℙ → ¬ 4 ∈ dom 𝑔)
121		fsnunfv 7184	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((4 ∈ ℤ ∧ 3 ∈ ℤ ∧ ¬ 4 ∈ dom 𝑔) → ((𝑔 ∪ {⟨4, 3⟩})‘4) = 3)
122	115, 116, 120, 121	syl3anc 1371	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑔:(1...3)⟶ℙ → ((𝑔 ∪ {⟨4, 3⟩})‘4) = 3)
123	122	adantl 482	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑁 ∈ (ℤ_≥‘9) ∧ 𝑔:(1...3)⟶ℙ) → ((𝑔 ∪ {⟨4, 3⟩})‘4) = 3)
124	114, 123	sylan9eq 2792	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑘 = 4 ∧ (𝑁 ∈ (ℤ_≥‘9) ∧ 𝑔:(1...3)⟶ℙ)) → ((𝑔 ∪ {⟨4, 3⟩})‘𝑘) = 3)
125	124, 37	eqeltrdi 2841	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑘 = 4 ∧ (𝑁 ∈ (ℤ_≥‘9) ∧ 𝑔:(1...3)⟶ℙ)) → ((𝑔 ∪ {⟨4, 3⟩})‘𝑘) ∈ ℂ)
126	125	ex 413	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 4 → ((𝑁 ∈ (ℤ_≥‘9) ∧ 𝑔:(1...3)⟶ℙ) → ((𝑔 ∪ {⟨4, 3⟩})‘𝑘) ∈ ℂ))
127	113, 126	jaoi 855	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑘 ∈ (1...3) ∨ 𝑘 = 4) → ((𝑁 ∈ (ℤ_≥‘9) ∧ 𝑔:(1...3)⟶ℙ) → ((𝑔 ∪ {⟨4, 3⟩})‘𝑘) ∈ ℂ))
128	127	com12 32	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ (ℤ_≥‘9) ∧ 𝑔:(1...3)⟶ℙ) → ((𝑘 ∈ (1...3) ∨ 𝑘 = 4) → ((𝑔 ∪ {⟨4, 3⟩})‘𝑘) ∈ ℂ))
129	86, 128	biimtrid 241	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ (ℤ_≥‘9) ∧ 𝑔:(1...3)⟶ℙ) → (𝑘 ∈ (1...4) → ((𝑔 ∪ {⟨4, 3⟩})‘𝑘) ∈ ℂ))
130	129	imp 407	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ (ℤ_≥‘9) ∧ 𝑔:(1...3)⟶ℙ) ∧ 𝑘 ∈ (1...4)) → ((𝑔 ∪ {⟨4, 3⟩})‘𝑘) ∈ ℂ)
131	81, 130, 114	fsumm1 15696	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ (ℤ_≥‘9) ∧ 𝑔:(1...3)⟶ℙ) → Σ𝑘 ∈ (1...4)((𝑔 ∪ {⟨4, 3⟩})‘𝑘) = (Σ𝑘 ∈ (1...(4 − 1))((𝑔 ∪ {⟨4, 3⟩})‘𝑘) + ((𝑔 ∪ {⟨4, 3⟩})‘4)))
132	131	adantr 481	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ (ℤ_≥‘9) ∧ 𝑔:(1...3)⟶ℙ) ∧ (𝑁 − 3) = Σ𝑘 ∈ (1...3)(𝑔‘𝑘)) → Σ𝑘 ∈ (1...4)((𝑔 ∪ {⟨4, 3⟩})‘𝑘) = (Σ𝑘 ∈ (1...(4 − 1))((𝑔 ∪ {⟨4, 3⟩})‘𝑘) + ((𝑔 ∪ {⟨4, 3⟩})‘4)))
133	42	eqcomi 2741	. . . . . . . . . . . . . . . . . . . 20 ⊢ 3 = (4 − 1)
134	133	oveq2i 7419	. . . . . . . . . . . . . . . . . . 19 ⊢ (1...3) = (1...(4 − 1))
135	134	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ (ℤ_≥‘9) ∧ 𝑔:(1...3)⟶ℙ) → (1...3) = (1...(4 − 1)))
136	102	adantl 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑁 ∈ (ℤ_≥‘9) ∧ 𝑔:(1...3)⟶ℙ) ∧ 𝑘 ∈ (1...3)) → 4 ≠ 𝑘)
137	136, 104	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 ∈ (ℤ_≥‘9) ∧ 𝑔:(1...3)⟶ℙ) ∧ 𝑘 ∈ (1...3)) → ((𝑔 ∪ {⟨4, 3⟩})‘𝑘) = (𝑔‘𝑘))
138	137	eqcomd 2738	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ (ℤ_≥‘9) ∧ 𝑔:(1...3)⟶ℙ) ∧ 𝑘 ∈ (1...3)) → (𝑔‘𝑘) = ((𝑔 ∪ {⟨4, 3⟩})‘𝑘))
139	135, 138	sumeq12dv 15651	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ (ℤ_≥‘9) ∧ 𝑔:(1...3)⟶ℙ) → Σ𝑘 ∈ (1...3)(𝑔‘𝑘) = Σ𝑘 ∈ (1...(4 − 1))((𝑔 ∪ {⟨4, 3⟩})‘𝑘))
140	139	eqeq2d 2743	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ (ℤ_≥‘9) ∧ 𝑔:(1...3)⟶ℙ) → ((𝑁 − 3) = Σ𝑘 ∈ (1...3)(𝑔‘𝑘) ↔ (𝑁 − 3) = Σ𝑘 ∈ (1...(4 − 1))((𝑔 ∪ {⟨4, 3⟩})‘𝑘)))
141	140	biimpa 477	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ (ℤ_≥‘9) ∧ 𝑔:(1...3)⟶ℙ) ∧ (𝑁 − 3) = Σ𝑘 ∈ (1...3)(𝑔‘𝑘)) → (𝑁 − 3) = Σ𝑘 ∈ (1...(4 − 1))((𝑔 ∪ {⟨4, 3⟩})‘𝑘))
142	141	eqcomd 2738	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ (ℤ_≥‘9) ∧ 𝑔:(1...3)⟶ℙ) ∧ (𝑁 − 3) = Σ𝑘 ∈ (1...3)(𝑔‘𝑘)) → Σ𝑘 ∈ (1...(4 − 1))((𝑔 ∪ {⟨4, 3⟩})‘𝑘) = (𝑁 − 3))
143	142	oveq1d 7423	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ (ℤ_≥‘9) ∧ 𝑔:(1...3)⟶ℙ) ∧ (𝑁 − 3) = Σ𝑘 ∈ (1...3)(𝑔‘𝑘)) → (Σ𝑘 ∈ (1...(4 − 1))((𝑔 ∪ {⟨4, 3⟩})‘𝑘) + ((𝑔 ∪ {⟨4, 3⟩})‘4)) = ((𝑁 − 3) + ((𝑔 ∪ {⟨4, 3⟩})‘4)))
144	31	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ (ℤ_≥‘9) ∧ 𝑔:(1...3)⟶ℙ) → 4 ∈ ℤ)
145	7	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ (ℤ_≥‘9) ∧ 𝑔:(1...3)⟶ℙ) → 3 ∈ ℤ)
146	120	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ (ℤ_≥‘9) ∧ 𝑔:(1...3)⟶ℙ) → ¬ 4 ∈ dom 𝑔)
147	144, 145, 146, 121	syl3anc 1371	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ (ℤ_≥‘9) ∧ 𝑔:(1...3)⟶ℙ) → ((𝑔 ∪ {⟨4, 3⟩})‘4) = 3)
148	147	oveq2d 7424	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ (ℤ_≥‘9) ∧ 𝑔:(1...3)⟶ℙ) → ((𝑁 − 3) + ((𝑔 ∪ {⟨4, 3⟩})‘4)) = ((𝑁 − 3) + 3))
149		eluzelcn 12833	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ (ℤ_≥‘9) → 𝑁 ∈ ℂ)
150	37	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ (ℤ_≥‘9) → 3 ∈ ℂ)
151	149, 150	npcand 11574	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ (ℤ_≥‘9) → ((𝑁 − 3) + 3) = 𝑁)
152	151	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ (ℤ_≥‘9) ∧ 𝑔:(1...3)⟶ℙ) → ((𝑁 − 3) + 3) = 𝑁)
153	148, 152	eqtrd 2772	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ (ℤ_≥‘9) ∧ 𝑔:(1...3)⟶ℙ) → ((𝑁 − 3) + ((𝑔 ∪ {⟨4, 3⟩})‘4)) = 𝑁)
154	153	adantr 481	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ (ℤ_≥‘9) ∧ 𝑔:(1...3)⟶ℙ) ∧ (𝑁 − 3) = Σ𝑘 ∈ (1...3)(𝑔‘𝑘)) → ((𝑁 − 3) + ((𝑔 ∪ {⟨4, 3⟩})‘4)) = 𝑁)
155	132, 143, 154	3eqtrrd 2777	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ (ℤ_≥‘9) ∧ 𝑔:(1...3)⟶ℙ) ∧ (𝑁 − 3) = Σ𝑘 ∈ (1...3)(𝑔‘𝑘)) → 𝑁 = Σ𝑘 ∈ (1...4)((𝑔 ∪ {⟨4, 3⟩})‘𝑘))
156	75, 80, 155	rspcedvd 3614	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ (ℤ_≥‘9) ∧ 𝑔:(1...3)⟶ℙ) ∧ (𝑁 − 3) = Σ𝑘 ∈ (1...3)(𝑔‘𝑘)) → ∃𝑓 ∈ (ℙ ↑_m (1...4))𝑁 = Σ𝑘 ∈ (1...4)(𝑓‘𝑘))
157	156	ex 413	. . . . . . . . . 10 ⊢ ((𝑁 ∈ (ℤ_≥‘9) ∧ 𝑔:(1...3)⟶ℙ) → ((𝑁 − 3) = Σ𝑘 ∈ (1...3)(𝑔‘𝑘) → ∃𝑓 ∈ (ℙ ↑_m (1...4))𝑁 = Σ𝑘 ∈ (1...4)(𝑓‘𝑘)))
158	157	expcom 414	. . . . . . . . 9 ⊢ (𝑔:(1...3)⟶ℙ → (𝑁 ∈ (ℤ_≥‘9) → ((𝑁 − 3) = Σ𝑘 ∈ (1...3)(𝑔‘𝑘) → ∃𝑓 ∈ (ℙ ↑_m (1...4))𝑁 = Σ𝑘 ∈ (1...4)(𝑓‘𝑘))))
159		elmapi 8842	. . . . . . . . 9 ⊢ (𝑔 ∈ (ℙ ↑_m (1...3)) → 𝑔:(1...3)⟶ℙ)
160	158, 159	syl11 33	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘9) → (𝑔 ∈ (ℙ ↑_m (1...3)) → ((𝑁 − 3) = Σ𝑘 ∈ (1...3)(𝑔‘𝑘) → ∃𝑓 ∈ (ℙ ↑_m (1...4))𝑁 = Σ𝑘 ∈ (1...4)(𝑓‘𝑘))))
161	160	rexlimdv 3153	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘9) → (∃𝑔 ∈ (ℙ ↑_m (1...3))(𝑁 − 3) = Σ𝑘 ∈ (1...3)(𝑔‘𝑘) → ∃𝑓 ∈ (ℙ ↑_m (1...4))𝑁 = Σ𝑘 ∈ (1...4)(𝑓‘𝑘)))
162	161	adantr 481	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ_≥‘9) ∧ 𝑁 ∈ Even ) → (∃𝑔 ∈ (ℙ ↑_m (1...3))(𝑁 − 3) = Σ𝑘 ∈ (1...3)(𝑔‘𝑘) → ∃𝑓 ∈ (ℙ ↑_m (1...4))𝑁 = Σ𝑘 ∈ (1...4)(𝑓‘𝑘)))
163	162	ad3antlr 729	. . . . 5 ⊢ ((((∀𝑚 ∈ Odd (5 < 𝑚 → 𝑚 ∈ GoldbachOddW ) ∧ (𝑁 ∈ (ℤ_≥‘9) ∧ 𝑁 ∈ Even )) ∧ 𝑜 ∈ GoldbachOddW ) ∧ 𝑁 = (𝑜 + 3)) → (∃𝑔 ∈ (ℙ ↑_m (1...3))(𝑁 − 3) = Σ𝑘 ∈ (1...3)(𝑔‘𝑘) → ∃𝑓 ∈ (ℙ ↑_m (1...4))𝑁 = Σ𝑘 ∈ (1...4)(𝑓‘𝑘)))
164	29, 163	mpd 15	. . . 4 ⊢ ((((∀𝑚 ∈ Odd (5 < 𝑚 → 𝑚 ∈ GoldbachOddW ) ∧ (𝑁 ∈ (ℤ_≥‘9) ∧ 𝑁 ∈ Even )) ∧ 𝑜 ∈ GoldbachOddW ) ∧ 𝑁 = (𝑜 + 3)) → ∃𝑓 ∈ (ℙ ↑_m (1...4))𝑁 = Σ𝑘 ∈ (1...4)(𝑓‘𝑘))
165	164	rexlimdva2 3157	. . 3 ⊢ ((∀𝑚 ∈ Odd (5 < 𝑚 → 𝑚 ∈ GoldbachOddW ) ∧ (𝑁 ∈ (ℤ_≥‘9) ∧ 𝑁 ∈ Even )) → (∃𝑜 ∈ GoldbachOddW 𝑁 = (𝑜 + 3) → ∃𝑓 ∈ (ℙ ↑_m (1...4))𝑁 = Σ𝑘 ∈ (1...4)(𝑓‘𝑘)))
166	2, 165	mpd 15	. 2 ⊢ ((∀𝑚 ∈ Odd (5 < 𝑚 → 𝑚 ∈ GoldbachOddW ) ∧ (𝑁 ∈ (ℤ_≥‘9) ∧ 𝑁 ∈ Even )) → ∃𝑓 ∈ (ℙ ↑_m (1...4))𝑁 = Σ𝑘 ∈ (1...4)(𝑓‘𝑘))
167	166	ex 413	1 ⊢ (∀𝑚 ∈ Odd (5 < 𝑚 → 𝑚 ∈ GoldbachOddW ) → ((𝑁 ∈ (ℤ_≥‘9) ∧ 𝑁 ∈ Even ) → ∃𝑓 ∈ (ℙ ↑_m (1...4))𝑁 = Σ𝑘 ∈ (1...4)(𝑓‘𝑘)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 845 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 Vcvv 3474 ∪ cun 3946 {csn 4628 ⟨cop 4634 class class class wbr 5148 dom cdm 5676 ⟶wf 6539 ‘cfv 6543 (class class class)co 7408 ↑_m cmap 8819 ℂcc 11107 ℝcr 11108 1c1 11110 + caddc 11112 < clt 11247 ≤ cle 11248 − cmin 11443 3c3 12267 4c4 12268 5c5 12269 6c6 12270 9c9 12273 ℤcz 12557 ℤ_≥cuz 12821 ...cfz 13483 ..^cfzo 13626 Σcsu 15631 ℙcprime 16607 Even ceven 46282 Odd codd 46283 GoldbachOddW cgbow 46404

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-inf2 9635 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 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 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-1st 7974 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-2o 8466 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-sup 9436 df-oi 9504 df-card 9933 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-z 12558 df-uz 12822 df-rp 12974 df-fz 13484 df-fzo 13627 df-seq 13966 df-exp 14027 df-hash 14290 df-cj 15045 df-re 15046 df-im 15047 df-sqrt 15181 df-abs 15182 df-clim 15431 df-sum 15632 df-dvds 16197 df-prm 16608 df-even 46284 df-odd 46285 df-gbow 46407

This theorem is referenced by: wtgoldbnnsum4prm 46460

W3C validator