hgt750lemb - Mathbox for Thierry Arnoux

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Theorem hgt750lemb 32636

Description: An upper bound on the contribution of the non-prime terms in the Statement 7.50 of [Helfgott] p. 69. (Contributed by Thierry Arnoux, 28-Dec-2021.)

**Hypotheses**
Ref	Expression
hgt750leme.o	⊢ 𝑂 = {𝑧 ∈ ℤ ∣ ¬ 2 ∥ 𝑧}
hgt750leme.n	⊢ (𝜑 → 𝑁 ∈ ℕ)
hgt750lemb.2	⊢ (𝜑 → 2 ≤ 𝑁)
hgt750lemb.a	⊢ 𝐴 = {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)}

**Assertion**
Ref	Expression
hgt750lemb	⊢ (𝜑 → Σ𝑛 ∈ 𝐴 ((Λ‘(𝑛‘0)) · ((Λ‘(𝑛‘1)) · (Λ‘(𝑛‘2)))) ≤ ((log‘𝑁) · (Σ𝑖 ∈ (((1...𝑁) ∖ ℙ) ∪ {2})(Λ‘𝑖) · Σ𝑗 ∈ (1...𝑁)(Λ‘𝑗))))

Distinct variable groups: 𝑧,𝑂 𝐴,𝑐,𝑖,𝑗,𝑛 𝑁,𝑐,𝑖,𝑗,𝑛 𝜑,𝑐,𝑖,𝑗,𝑛 Allowed substitution hints: 𝜑(𝑧) 𝐴(𝑧) 𝑁(𝑧) 𝑂(𝑖,𝑗,𝑛,𝑐)

Proof of Theorem hgt750lemb Dummy variables 𝑑 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		hgt750leme.n	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ)
2	1	nnnn0d 12293	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
3		3nn0 12251	. . . . . 6 ⊢ 3 ∈ ℕ₀
4	3	a1i 11	. . . . 5 ⊢ (𝜑 → 3 ∈ ℕ₀)
5		ssidd 3944	. . . . 5 ⊢ (𝜑 → ℕ ⊆ ℕ)
6	2, 4, 5	reprfi2 32603	. . . 4 ⊢ (𝜑 → (ℕ(repr‘3)𝑁) ∈ Fin)
7		hgt750lemb.a	. . . . 5 ⊢ 𝐴 = {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)}
8	7	ssrab3 4015	. . . 4 ⊢ 𝐴 ⊆ (ℕ(repr‘3)𝑁)
9		ssfi 8956	. . . 4 ⊢ (((ℕ(repr‘3)𝑁) ∈ Fin ∧ 𝐴 ⊆ (ℕ(repr‘3)𝑁)) → 𝐴 ∈ Fin)
10	6, 8, 9	sylancl 586	. . 3 ⊢ (𝜑 → 𝐴 ∈ Fin)
11		vmaf 26268	. . . . . 6 ⊢ Λ:ℕ⟶ℝ
12	11	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → Λ:ℕ⟶ℝ)
13		ssidd 3944	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → ℕ ⊆ ℕ)
14	1	nnzd 12425	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℤ)
15	14	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → 𝑁 ∈ ℤ)
16	3	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → 3 ∈ ℕ₀)
17		simpr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → 𝑛 ∈ 𝐴)
18	8, 17	sselid 3919	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → 𝑛 ∈ (ℕ(repr‘3)𝑁))
19	13, 15, 16, 18	reprf 32592	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → 𝑛:(0..^3)⟶ℕ)
20		c0ex 10969	. . . . . . . . 9 ⊢ 0 ∈ V
21	20	tpid1 4704	. . . . . . . 8 ⊢ 0 ∈ {0, 1, 2}
22		fzo0to3tp 13473	. . . . . . . 8 ⊢ (0..^3) = {0, 1, 2}
23	21, 22	eleqtrri 2838	. . . . . . 7 ⊢ 0 ∈ (0..^3)
24	23	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → 0 ∈ (0..^3))
25	19, 24	ffvelrnd 6962	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → (𝑛‘0) ∈ ℕ)
26	12, 25	ffvelrnd 6962	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → (Λ‘(𝑛‘0)) ∈ ℝ)
27		1ex 10971	. . . . . . . . . 10 ⊢ 1 ∈ V
28	27	tpid2 4706	. . . . . . . . 9 ⊢ 1 ∈ {0, 1, 2}
29	28, 22	eleqtrri 2838	. . . . . . . 8 ⊢ 1 ∈ (0..^3)
30	29	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → 1 ∈ (0..^3))
31	19, 30	ffvelrnd 6962	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → (𝑛‘1) ∈ ℕ)
32	12, 31	ffvelrnd 6962	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → (Λ‘(𝑛‘1)) ∈ ℝ)
33		2ex 12050	. . . . . . . . . 10 ⊢ 2 ∈ V
34	33	tpid3 4709	. . . . . . . . 9 ⊢ 2 ∈ {0, 1, 2}
35	34, 22	eleqtrri 2838	. . . . . . . 8 ⊢ 2 ∈ (0..^3)
36	35	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → 2 ∈ (0..^3))
37	19, 36	ffvelrnd 6962	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → (𝑛‘2) ∈ ℕ)
38	12, 37	ffvelrnd 6962	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → (Λ‘(𝑛‘2)) ∈ ℝ)
39	32, 38	remulcld 11005	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → ((Λ‘(𝑛‘1)) · (Λ‘(𝑛‘2))) ∈ ℝ)
40	26, 39	remulcld 11005	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → ((Λ‘(𝑛‘0)) · ((Λ‘(𝑛‘1)) · (Λ‘(𝑛‘2)))) ∈ ℝ)
41	10, 40	fsumrecl 15446	. 2 ⊢ (𝜑 → Σ𝑛 ∈ 𝐴 ((Λ‘(𝑛‘0)) · ((Λ‘(𝑛‘1)) · (Λ‘(𝑛‘2)))) ∈ ℝ)
42	1	nnrpd 12770	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℝ⁺)
43	42	relogcld 25778	. . 3 ⊢ (𝜑 → (log‘𝑁) ∈ ℝ)
44	26, 32	remulcld 11005	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → ((Λ‘(𝑛‘0)) · (Λ‘(𝑛‘1))) ∈ ℝ)
45	10, 44	fsumrecl 15446	. . 3 ⊢ (𝜑 → Σ𝑛 ∈ 𝐴 ((Λ‘(𝑛‘0)) · (Λ‘(𝑛‘1))) ∈ ℝ)
46	43, 45	remulcld 11005	. 2 ⊢ (𝜑 → ((log‘𝑁) · Σ𝑛 ∈ 𝐴 ((Λ‘(𝑛‘0)) · (Λ‘(𝑛‘1)))) ∈ ℝ)
47		fzfi 13692	. . . . . . . 8 ⊢ (1...𝑁) ∈ Fin
48		diffi 8962	. . . . . . . 8 ⊢ ((1...𝑁) ∈ Fin → ((1...𝑁) ∖ ℙ) ∈ Fin)
49	47, 48	ax-mp 5	. . . . . . 7 ⊢ ((1...𝑁) ∖ ℙ) ∈ Fin
50		snfi 8834	. . . . . . 7 ⊢ {2} ∈ Fin
51		unfi 8955	. . . . . . 7 ⊢ ((((1...𝑁) ∖ ℙ) ∈ Fin ∧ {2} ∈ Fin) → (((1...𝑁) ∖ ℙ) ∪ {2}) ∈ Fin)
52	49, 50, 51	mp2an 689	. . . . . 6 ⊢ (((1...𝑁) ∖ ℙ) ∪ {2}) ∈ Fin
53	52	a1i 11	. . . . 5 ⊢ (𝜑 → (((1...𝑁) ∖ ℙ) ∪ {2}) ∈ Fin)
54	11	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (((1...𝑁) ∖ ℙ) ∪ {2})) → Λ:ℕ⟶ℝ)
55		difss 4066	. . . . . . . . . 10 ⊢ ((1...𝑁) ∖ ℙ) ⊆ (1...𝑁)
56	55	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ((1...𝑁) ∖ ℙ) ⊆ (1...𝑁))
57		2nn 12046	. . . . . . . . . . . 12 ⊢ 2 ∈ ℕ
58	57	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 2 ∈ ℕ)
59		hgt750lemb.2	. . . . . . . . . . 11 ⊢ (𝜑 → 2 ≤ 𝑁)
60		elfz1b 13325	. . . . . . . . . . . 12 ⊢ (2 ∈ (1...𝑁) ↔ (2 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 2 ≤ 𝑁))
61	60	biimpri 227	. . . . . . . . . . 11 ⊢ ((2 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 2 ≤ 𝑁) → 2 ∈ (1...𝑁))
62	58, 1, 59, 61	syl3anc 1370	. . . . . . . . . 10 ⊢ (𝜑 → 2 ∈ (1...𝑁))
63	62	snssd 4742	. . . . . . . . 9 ⊢ (𝜑 → {2} ⊆ (1...𝑁))
64	56, 63	unssd 4120	. . . . . . . 8 ⊢ (𝜑 → (((1...𝑁) ∖ ℙ) ∪ {2}) ⊆ (1...𝑁))
65		fz1ssnn 13287	. . . . . . . . 9 ⊢ (1...𝑁) ⊆ ℕ
66	65	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (1...𝑁) ⊆ ℕ)
67	64, 66	sstrd 3931	. . . . . . 7 ⊢ (𝜑 → (((1...𝑁) ∖ ℙ) ∪ {2}) ⊆ ℕ)
68	67	sselda 3921	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (((1...𝑁) ∖ ℙ) ∪ {2})) → 𝑖 ∈ ℕ)
69	54, 68	ffvelrnd 6962	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (((1...𝑁) ∖ ℙ) ∪ {2})) → (Λ‘𝑖) ∈ ℝ)
70	53, 69	fsumrecl 15446	. . . 4 ⊢ (𝜑 → Σ𝑖 ∈ (((1...𝑁) ∖ ℙ) ∪ {2})(Λ‘𝑖) ∈ ℝ)
71		fzfid 13693	. . . . 5 ⊢ (𝜑 → (1...𝑁) ∈ Fin)
72	11	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → Λ:ℕ⟶ℝ)
73	66	sselda 3921	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → 𝑗 ∈ ℕ)
74	72, 73	ffvelrnd 6962	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → (Λ‘𝑗) ∈ ℝ)
75	71, 74	fsumrecl 15446	. . . 4 ⊢ (𝜑 → Σ𝑗 ∈ (1...𝑁)(Λ‘𝑗) ∈ ℝ)
76	70, 75	remulcld 11005	. . 3 ⊢ (𝜑 → (Σ𝑖 ∈ (((1...𝑁) ∖ ℙ) ∪ {2})(Λ‘𝑖) · Σ𝑗 ∈ (1...𝑁)(Λ‘𝑗)) ∈ ℝ)
77	43, 76	remulcld 11005	. 2 ⊢ (𝜑 → ((log‘𝑁) · (Σ𝑖 ∈ (((1...𝑁) ∖ ℙ) ∪ {2})(Λ‘𝑖) · Σ𝑗 ∈ (1...𝑁)(Λ‘𝑗))) ∈ ℝ)
78	1	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → 𝑁 ∈ ℕ)
79	78	nnrpd 12770	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → 𝑁 ∈ ℝ⁺)
80		relogcl 25731	. . . . . . 7 ⊢ (𝑁 ∈ ℝ⁺ → (log‘𝑁) ∈ ℝ)
81	79, 80	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → (log‘𝑁) ∈ ℝ)
82	32, 81	remulcld 11005	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → ((Λ‘(𝑛‘1)) · (log‘𝑁)) ∈ ℝ)
83	26, 82	remulcld 11005	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → ((Λ‘(𝑛‘0)) · ((Λ‘(𝑛‘1)) · (log‘𝑁))) ∈ ℝ)
84		vmage0 26270	. . . . . 6 ⊢ ((𝑛‘0) ∈ ℕ → 0 ≤ (Λ‘(𝑛‘0)))
85	25, 84	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → 0 ≤ (Λ‘(𝑛‘0)))
86		vmage0 26270	. . . . . . 7 ⊢ ((𝑛‘1) ∈ ℕ → 0 ≤ (Λ‘(𝑛‘1)))
87	31, 86	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → 0 ≤ (Λ‘(𝑛‘1)))
88	37	nnrpd 12770	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → (𝑛‘2) ∈ ℝ⁺)
89	88	relogcld 25778	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → (log‘(𝑛‘2)) ∈ ℝ)
90		vmalelog 26353	. . . . . . . 8 ⊢ ((𝑛‘2) ∈ ℕ → (Λ‘(𝑛‘2)) ≤ (log‘(𝑛‘2)))
91	37, 90	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → (Λ‘(𝑛‘2)) ≤ (log‘(𝑛‘2)))
92	13, 15, 16, 18, 36	reprle 32594	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → (𝑛‘2) ≤ 𝑁)
93		logleb 25758	. . . . . . . . 9 ⊢ (((𝑛‘2) ∈ ℝ⁺ ∧ 𝑁 ∈ ℝ⁺) → ((𝑛‘2) ≤ 𝑁 ↔ (log‘(𝑛‘2)) ≤ (log‘𝑁)))
94	93	biimpa 477	. . . . . . . 8 ⊢ ((((𝑛‘2) ∈ ℝ⁺ ∧ 𝑁 ∈ ℝ⁺) ∧ (𝑛‘2) ≤ 𝑁) → (log‘(𝑛‘2)) ≤ (log‘𝑁))
95	88, 79, 92, 94	syl21anc 835	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → (log‘(𝑛‘2)) ≤ (log‘𝑁))
96	38, 89, 81, 91, 95	letrd 11132	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → (Λ‘(𝑛‘2)) ≤ (log‘𝑁))
97	38, 81, 32, 87, 96	lemul2ad 11915	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → ((Λ‘(𝑛‘1)) · (Λ‘(𝑛‘2))) ≤ ((Λ‘(𝑛‘1)) · (log‘𝑁)))
98	39, 82, 26, 85, 97	lemul2ad 11915	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → ((Λ‘(𝑛‘0)) · ((Λ‘(𝑛‘1)) · (Λ‘(𝑛‘2)))) ≤ ((Λ‘(𝑛‘0)) · ((Λ‘(𝑛‘1)) · (log‘𝑁))))
99	10, 40, 83, 98	fsumle 15511	. . 3 ⊢ (𝜑 → Σ𝑛 ∈ 𝐴 ((Λ‘(𝑛‘0)) · ((Λ‘(𝑛‘1)) · (Λ‘(𝑛‘2)))) ≤ Σ𝑛 ∈ 𝐴 ((Λ‘(𝑛‘0)) · ((Λ‘(𝑛‘1)) · (log‘𝑁))))
100	1	nncnd 11989	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℂ)
101	1	nnne0d 12023	. . . . . 6 ⊢ (𝜑 → 𝑁 ≠ 0)
102	100, 101	logcld 25726	. . . . 5 ⊢ (𝜑 → (log‘𝑁) ∈ ℂ)
103	44	recnd 11003	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → ((Λ‘(𝑛‘0)) · (Λ‘(𝑛‘1))) ∈ ℂ)
104	10, 102, 103	fsummulc2 15496	. . . 4 ⊢ (𝜑 → ((log‘𝑁) · Σ𝑛 ∈ 𝐴 ((Λ‘(𝑛‘0)) · (Λ‘(𝑛‘1)))) = Σ𝑛 ∈ 𝐴 ((log‘𝑁) · ((Λ‘(𝑛‘0)) · (Λ‘(𝑛‘1)))))
105	102	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → (log‘𝑁) ∈ ℂ)
106	105, 103	mulcomd 10996	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → ((log‘𝑁) · ((Λ‘(𝑛‘0)) · (Λ‘(𝑛‘1)))) = (((Λ‘(𝑛‘0)) · (Λ‘(𝑛‘1))) · (log‘𝑁)))
107	26	recnd 11003	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → (Λ‘(𝑛‘0)) ∈ ℂ)
108	32	recnd 11003	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → (Λ‘(𝑛‘1)) ∈ ℂ)
109	107, 108, 105	mulassd 10998	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → (((Λ‘(𝑛‘0)) · (Λ‘(𝑛‘1))) · (log‘𝑁)) = ((Λ‘(𝑛‘0)) · ((Λ‘(𝑛‘1)) · (log‘𝑁))))
110	106, 109	eqtrd 2778	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → ((log‘𝑁) · ((Λ‘(𝑛‘0)) · (Λ‘(𝑛‘1)))) = ((Λ‘(𝑛‘0)) · ((Λ‘(𝑛‘1)) · (log‘𝑁))))
111	110	sumeq2dv 15415	. . . 4 ⊢ (𝜑 → Σ𝑛 ∈ 𝐴 ((log‘𝑁) · ((Λ‘(𝑛‘0)) · (Λ‘(𝑛‘1)))) = Σ𝑛 ∈ 𝐴 ((Λ‘(𝑛‘0)) · ((Λ‘(𝑛‘1)) · (log‘𝑁))))
112	104, 111	eqtr2d 2779	. . 3 ⊢ (𝜑 → Σ𝑛 ∈ 𝐴 ((Λ‘(𝑛‘0)) · ((Λ‘(𝑛‘1)) · (log‘𝑁))) = ((log‘𝑁) · Σ𝑛 ∈ 𝐴 ((Λ‘(𝑛‘0)) · (Λ‘(𝑛‘1)))))
113	99, 112	breqtrd 5100	. 2 ⊢ (𝜑 → Σ𝑛 ∈ 𝐴 ((Λ‘(𝑛‘0)) · ((Λ‘(𝑛‘1)) · (Λ‘(𝑛‘2)))) ≤ ((log‘𝑁) · Σ𝑛 ∈ 𝐴 ((Λ‘(𝑛‘0)) · (Λ‘(𝑛‘1)))))
114	1	nnred 11988	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℝ)
115	1	nnge1d 12021	. . . 4 ⊢ (𝜑 → 1 ≤ 𝑁)
116	114, 115	logge0d 25785	. . 3 ⊢ (𝜑 → 0 ≤ (log‘𝑁))
117		xpfi 9085	. . . . . 6 ⊢ (((((1...𝑁) ∖ ℙ) ∪ {2}) ∈ Fin ∧ (1...𝑁) ∈ Fin) → ((((1...𝑁) ∖ ℙ) ∪ {2}) × (1...𝑁)) ∈ Fin)
118	53, 71, 117	syl2anc 584	. . . . 5 ⊢ (𝜑 → ((((1...𝑁) ∖ ℙ) ∪ {2}) × (1...𝑁)) ∈ Fin)
119	11	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ ((((1...𝑁) ∖ ℙ) ∪ {2}) × (1...𝑁))) → Λ:ℕ⟶ℝ)
120	67	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ ((((1...𝑁) ∖ ℙ) ∪ {2}) × (1...𝑁))) → (((1...𝑁) ∖ ℙ) ∪ {2}) ⊆ ℕ)
121		xp1st 7863	. . . . . . . . 9 ⊢ (𝑢 ∈ ((((1...𝑁) ∖ ℙ) ∪ {2}) × (1...𝑁)) → (1^st ‘𝑢) ∈ (((1...𝑁) ∖ ℙ) ∪ {2}))
122	121	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ ((((1...𝑁) ∖ ℙ) ∪ {2}) × (1...𝑁))) → (1^st ‘𝑢) ∈ (((1...𝑁) ∖ ℙ) ∪ {2}))
123	120, 122	sseldd 3922	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ ((((1...𝑁) ∖ ℙ) ∪ {2}) × (1...𝑁))) → (1^st ‘𝑢) ∈ ℕ)
124	119, 123	ffvelrnd 6962	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ ((((1...𝑁) ∖ ℙ) ∪ {2}) × (1...𝑁))) → (Λ‘(1^st ‘𝑢)) ∈ ℝ)
125		xp2nd 7864	. . . . . . . . 9 ⊢ (𝑢 ∈ ((((1...𝑁) ∖ ℙ) ∪ {2}) × (1...𝑁)) → (2^nd ‘𝑢) ∈ (1...𝑁))
126	125	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ ((((1...𝑁) ∖ ℙ) ∪ {2}) × (1...𝑁))) → (2^nd ‘𝑢) ∈ (1...𝑁))
127	65, 126	sselid 3919	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ ((((1...𝑁) ∖ ℙ) ∪ {2}) × (1...𝑁))) → (2^nd ‘𝑢) ∈ ℕ)
128	119, 127	ffvelrnd 6962	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ ((((1...𝑁) ∖ ℙ) ∪ {2}) × (1...𝑁))) → (Λ‘(2^nd ‘𝑢)) ∈ ℝ)
129	124, 128	remulcld 11005	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 ∈ ((((1...𝑁) ∖ ℙ) ∪ {2}) × (1...𝑁))) → ((Λ‘(1^st ‘𝑢)) · (Λ‘(2^nd ‘𝑢))) ∈ ℝ)
130		vmage0 26270	. . . . . . 7 ⊢ ((1^st ‘𝑢) ∈ ℕ → 0 ≤ (Λ‘(1^st ‘𝑢)))
131	123, 130	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ ((((1...𝑁) ∖ ℙ) ∪ {2}) × (1...𝑁))) → 0 ≤ (Λ‘(1^st ‘𝑢)))
132		vmage0 26270	. . . . . . 7 ⊢ ((2^nd ‘𝑢) ∈ ℕ → 0 ≤ (Λ‘(2^nd ‘𝑢)))
133	127, 132	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ ((((1...𝑁) ∖ ℙ) ∪ {2}) × (1...𝑁))) → 0 ≤ (Λ‘(2^nd ‘𝑢)))
134	124, 128, 131, 133	mulge0d 11552	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 ∈ ((((1...𝑁) ∖ ℙ) ∪ {2}) × (1...𝑁))) → 0 ≤ ((Λ‘(1^st ‘𝑢)) · (Λ‘(2^nd ‘𝑢))))
135		ssidd 3944	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → ℕ ⊆ ℕ)
136	14	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → 𝑁 ∈ ℤ)
137	3	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → 3 ∈ ℕ₀)
138		simpr 485	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → 𝑐 ∈ 𝐴)
139	8, 138	sselid 3919	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → 𝑐 ∈ (ℕ(repr‘3)𝑁))
140	135, 136, 137, 139	reprf 32592	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → 𝑐:(0..^3)⟶ℕ)
141	23	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → 0 ∈ (0..^3))
142	140, 141	ffvelrnd 6962	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (𝑐‘0) ∈ ℕ)
143	1	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → 𝑁 ∈ ℕ)
144	135, 136, 137, 139, 141	reprle 32594	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (𝑐‘0) ≤ 𝑁)
145		elfz1b 13325	. . . . . . . . . . . . 13 ⊢ ((𝑐‘0) ∈ (1...𝑁) ↔ ((𝑐‘0) ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑐‘0) ≤ 𝑁))
146	145	biimpri 227	. . . . . . . . . . . 12 ⊢ (((𝑐‘0) ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑐‘0) ≤ 𝑁) → (𝑐‘0) ∈ (1...𝑁))
147	142, 143, 144, 146	syl3anc 1370	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (𝑐‘0) ∈ (1...𝑁))
148	7	rabeq2i 3422	. . . . . . . . . . . . . 14 ⊢ (𝑐 ∈ 𝐴 ↔ (𝑐 ∈ (ℕ(repr‘3)𝑁) ∧ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)))
149	148	simprbi 497	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ 𝐴 → ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ))
150		hgt750leme.o	. . . . . . . . . . . . . . 15 ⊢ 𝑂 = {𝑧 ∈ ℤ ∣ ¬ 2 ∥ 𝑧}
151	150	oddprm2 32635	. . . . . . . . . . . . . 14 ⊢ (ℙ ∖ {2}) = (𝑂 ∩ ℙ)
152	151	eleq2i 2830	. . . . . . . . . . . . 13 ⊢ ((𝑐‘0) ∈ (ℙ ∖ {2}) ↔ (𝑐‘0) ∈ (𝑂 ∩ ℙ))
153	149, 152	sylnibr 329	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ 𝐴 → ¬ (𝑐‘0) ∈ (ℙ ∖ {2}))
154	138, 153	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → ¬ (𝑐‘0) ∈ (ℙ ∖ {2}))
155	147, 154	jca 512	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → ((𝑐‘0) ∈ (1...𝑁) ∧ ¬ (𝑐‘0) ∈ (ℙ ∖ {2})))
156		eldif 3897	. . . . . . . . . 10 ⊢ ((𝑐‘0) ∈ ((1...𝑁) ∖ (ℙ ∖ {2})) ↔ ((𝑐‘0) ∈ (1...𝑁) ∧ ¬ (𝑐‘0) ∈ (ℙ ∖ {2})))
157	155, 156	sylibr 233	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (𝑐‘0) ∈ ((1...𝑁) ∖ (ℙ ∖ {2})))
158		uncom 4087	. . . . . . . . . . . . 13 ⊢ (((1...𝑁) ∖ ℙ) ∪ {2}) = ({2} ∪ ((1...𝑁) ∖ ℙ))
159		undif3 4224	. . . . . . . . . . . . 13 ⊢ ({2} ∪ ((1...𝑁) ∖ ℙ)) = (({2} ∪ (1...𝑁)) ∖ (ℙ ∖ {2}))
160	158, 159	eqtri 2766	. . . . . . . . . . . 12 ⊢ (((1...𝑁) ∖ ℙ) ∪ {2}) = (({2} ∪ (1...𝑁)) ∖ (ℙ ∖ {2}))
161		ssequn1 4114	. . . . . . . . . . . . . 14 ⊢ ({2} ⊆ (1...𝑁) ↔ ({2} ∪ (1...𝑁)) = (1...𝑁))
162	63, 161	sylib 217	. . . . . . . . . . . . 13 ⊢ (𝜑 → ({2} ∪ (1...𝑁)) = (1...𝑁))
163	162	difeq1d 4056	. . . . . . . . . . . 12 ⊢ (𝜑 → (({2} ∪ (1...𝑁)) ∖ (ℙ ∖ {2})) = ((1...𝑁) ∖ (ℙ ∖ {2})))
164	160, 163	eqtrid 2790	. . . . . . . . . . 11 ⊢ (𝜑 → (((1...𝑁) ∖ ℙ) ∪ {2}) = ((1...𝑁) ∖ (ℙ ∖ {2})))
165	164	eleq2d 2824	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑐‘0) ∈ (((1...𝑁) ∖ ℙ) ∪ {2}) ↔ (𝑐‘0) ∈ ((1...𝑁) ∖ (ℙ ∖ {2}))))
166	165	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → ((𝑐‘0) ∈ (((1...𝑁) ∖ ℙ) ∪ {2}) ↔ (𝑐‘0) ∈ ((1...𝑁) ∖ (ℙ ∖ {2}))))
167	157, 166	mpbird 256	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (𝑐‘0) ∈ (((1...𝑁) ∖ ℙ) ∪ {2}))
168	29	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → 1 ∈ (0..^3))
169	140, 168	ffvelrnd 6962	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (𝑐‘1) ∈ ℕ)
170	135, 136, 137, 139, 168	reprle 32594	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (𝑐‘1) ≤ 𝑁)
171		elfz1b 13325	. . . . . . . . . 10 ⊢ ((𝑐‘1) ∈ (1...𝑁) ↔ ((𝑐‘1) ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑐‘1) ≤ 𝑁))
172	171	biimpri 227	. . . . . . . . 9 ⊢ (((𝑐‘1) ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑐‘1) ≤ 𝑁) → (𝑐‘1) ∈ (1...𝑁))
173	169, 143, 170, 172	syl3anc 1370	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (𝑐‘1) ∈ (1...𝑁))
174	167, 173	opelxpd 5627	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → ⟨(𝑐‘0), (𝑐‘1)⟩ ∈ ((((1...𝑁) ∖ ℙ) ∪ {2}) × (1...𝑁)))
175	174	ralrimiva 3103	. . . . . 6 ⊢ (𝜑 → ∀𝑐 ∈ 𝐴 ⟨(𝑐‘0), (𝑐‘1)⟩ ∈ ((((1...𝑁) ∖ ℙ) ∪ {2}) × (1...𝑁)))
176		fveq1 6773	. . . . . . . . 9 ⊢ (𝑑 = 𝑐 → (𝑑‘0) = (𝑐‘0))
177		fveq1 6773	. . . . . . . . 9 ⊢ (𝑑 = 𝑐 → (𝑑‘1) = (𝑐‘1))
178	176, 177	opeq12d 4812	. . . . . . . 8 ⊢ (𝑑 = 𝑐 → ⟨(𝑑‘0), (𝑑‘1)⟩ = ⟨(𝑐‘0), (𝑐‘1)⟩)
179	178	cbvmptv 5187	. . . . . . 7 ⊢ (𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩) = (𝑐 ∈ 𝐴 ↦ ⟨(𝑐‘0), (𝑐‘1)⟩)
180	179	rnmptss 6996	. . . . . 6 ⊢ (∀𝑐 ∈ 𝐴 ⟨(𝑐‘0), (𝑐‘1)⟩ ∈ ((((1...𝑁) ∖ ℙ) ∪ {2}) × (1...𝑁)) → ran (𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩) ⊆ ((((1...𝑁) ∖ ℙ) ∪ {2}) × (1...𝑁)))
181	175, 180	syl 17	. . . . 5 ⊢ (𝜑 → ran (𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩) ⊆ ((((1...𝑁) ∖ ℙ) ∪ {2}) × (1...𝑁)))
182	118, 129, 134, 181	fsumless 15508	. . . 4 ⊢ (𝜑 → Σ𝑢 ∈ ran (𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)((Λ‘(1^st ‘𝑢)) · (Λ‘(2^nd ‘𝑢))) ≤ Σ𝑢 ∈ ((((1...𝑁) ∖ ℙ) ∪ {2}) × (1...𝑁))((Λ‘(1^st ‘𝑢)) · (Λ‘(2^nd ‘𝑢))))
183		fvex 6787	. . . . . . . 8 ⊢ (𝑛‘0) ∈ V
184		fvex 6787	. . . . . . . 8 ⊢ (𝑛‘1) ∈ V
185	183, 184	op1std 7841	. . . . . . 7 ⊢ (𝑢 = ⟨(𝑛‘0), (𝑛‘1)⟩ → (1^st ‘𝑢) = (𝑛‘0))
186	185	fveq2d 6778	. . . . . 6 ⊢ (𝑢 = ⟨(𝑛‘0), (𝑛‘1)⟩ → (Λ‘(1^st ‘𝑢)) = (Λ‘(𝑛‘0)))
187	183, 184	op2ndd 7842	. . . . . . 7 ⊢ (𝑢 = ⟨(𝑛‘0), (𝑛‘1)⟩ → (2^nd ‘𝑢) = (𝑛‘1))
188	187	fveq2d 6778	. . . . . 6 ⊢ (𝑢 = ⟨(𝑛‘0), (𝑛‘1)⟩ → (Λ‘(2^nd ‘𝑢)) = (Λ‘(𝑛‘1)))
189	186, 188	oveq12d 7293	. . . . 5 ⊢ (𝑢 = ⟨(𝑛‘0), (𝑛‘1)⟩ → ((Λ‘(1^st ‘𝑢)) · (Λ‘(2^nd ‘𝑢))) = ((Λ‘(𝑛‘0)) · (Λ‘(𝑛‘1))))
190		opex 5379	. . . . . . . 8 ⊢ ⟨(𝑐‘0), (𝑐‘1)⟩ ∈ V
191	190	rgenw 3076	. . . . . . 7 ⊢ ∀𝑐 ∈ 𝐴 ⟨(𝑐‘0), (𝑐‘1)⟩ ∈ V
192	179	fnmpt 6573	. . . . . . 7 ⊢ (∀𝑐 ∈ 𝐴 ⟨(𝑐‘0), (𝑐‘1)⟩ ∈ V → (𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩) Fn 𝐴)
193	191, 192	mp1i 13	. . . . . 6 ⊢ (𝜑 → (𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩) Fn 𝐴)
194		eqidd 2739	. . . . . 6 ⊢ (𝜑 → ran (𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩) = ran (𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩))
195	140	ad2antrr 723	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑛 ∈ 𝐴) ∧ ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑐) = ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑛)) → 𝑐:(0..^3)⟶ℕ)
196	195	ffnd 6601	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑛 ∈ 𝐴) ∧ ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑐) = ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑛)) → 𝑐 Fn (0..^3))
197	19	ad4ant13 748	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑛 ∈ 𝐴) ∧ ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑐) = ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑛)) → 𝑛:(0..^3)⟶ℕ)
198	197	ffnd 6601	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑛 ∈ 𝐴) ∧ ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑐) = ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑛)) → 𝑛 Fn (0..^3))
199		simpr 485	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑛 ∈ 𝐴) ∧ ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑐) = ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑛)) → ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑐) = ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑛))
200	179	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩) = (𝑐 ∈ 𝐴 ↦ ⟨(𝑐‘0), (𝑐‘1)⟩))
201	190	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → ⟨(𝑐‘0), (𝑐‘1)⟩ ∈ V)
202	200, 201	fvmpt2d 6888	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑐) = ⟨(𝑐‘0), (𝑐‘1)⟩)
203	202	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑛 ∈ 𝐴) → ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑐) = ⟨(𝑐‘0), (𝑐‘1)⟩)
204	203	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑛 ∈ 𝐴) ∧ ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑐) = ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑛)) → ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑐) = ⟨(𝑐‘0), (𝑐‘1)⟩)
205		fveq1 6773	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑐 = 𝑛 → (𝑐‘0) = (𝑛‘0))
206		fveq1 6773	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑐 = 𝑛 → (𝑐‘1) = (𝑛‘1))
207	205, 206	opeq12d 4812	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑐 = 𝑛 → ⟨(𝑐‘0), (𝑐‘1)⟩ = ⟨(𝑛‘0), (𝑛‘1)⟩)
208		opex 5379	. . . . . . . . . . . . . . . . . . . 20 ⊢ ⟨(𝑛‘0), (𝑛‘1)⟩ ∈ V
209	208	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → ⟨(𝑛‘0), (𝑛‘1)⟩ ∈ V)
210	179, 207, 17, 209	fvmptd3 6898	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑛) = ⟨(𝑛‘0), (𝑛‘1)⟩)
211	210	adantlr 712	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑛 ∈ 𝐴) → ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑛) = ⟨(𝑛‘0), (𝑛‘1)⟩)
212	211	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑛 ∈ 𝐴) ∧ ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑐) = ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑛)) → ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑛) = ⟨(𝑛‘0), (𝑛‘1)⟩)
213	199, 204, 212	3eqtr3d 2786	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑛 ∈ 𝐴) ∧ ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑐) = ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑛)) → ⟨(𝑐‘0), (𝑐‘1)⟩ = ⟨(𝑛‘0), (𝑛‘1)⟩)
214	183, 184	opth2 5395	. . . . . . . . . . . . . . 15 ⊢ (⟨(𝑐‘0), (𝑐‘1)⟩ = ⟨(𝑛‘0), (𝑛‘1)⟩ ↔ ((𝑐‘0) = (𝑛‘0) ∧ (𝑐‘1) = (𝑛‘1)))
215	213, 214	sylib 217	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑛 ∈ 𝐴) ∧ ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑐) = ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑛)) → ((𝑐‘0) = (𝑛‘0) ∧ (𝑐‘1) = (𝑛‘1)))
216	215	simpld 495	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑛 ∈ 𝐴) ∧ ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑐) = ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑛)) → (𝑐‘0) = (𝑛‘0))
217	216	ad2antrr 723	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑛 ∈ 𝐴) ∧ ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑐) = ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑛)) ∧ 𝑖 ∈ (0..^3)) ∧ 𝑖 = 0) → (𝑐‘0) = (𝑛‘0))
218		simpr 485	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑛 ∈ 𝐴) ∧ ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑐) = ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑛)) ∧ 𝑖 ∈ (0..^3)) ∧ 𝑖 = 0) → 𝑖 = 0)
219	218	fveq2d 6778	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑛 ∈ 𝐴) ∧ ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑐) = ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑛)) ∧ 𝑖 ∈ (0..^3)) ∧ 𝑖 = 0) → (𝑐‘𝑖) = (𝑐‘0))
220	218	fveq2d 6778	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑛 ∈ 𝐴) ∧ ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑐) = ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑛)) ∧ 𝑖 ∈ (0..^3)) ∧ 𝑖 = 0) → (𝑛‘𝑖) = (𝑛‘0))
221	217, 219, 220	3eqtr4d 2788	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑛 ∈ 𝐴) ∧ ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑐) = ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑛)) ∧ 𝑖 ∈ (0..^3)) ∧ 𝑖 = 0) → (𝑐‘𝑖) = (𝑛‘𝑖))
222	215	simprd 496	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑛 ∈ 𝐴) ∧ ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑐) = ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑛)) → (𝑐‘1) = (𝑛‘1))
223	222	ad2antrr 723	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑛 ∈ 𝐴) ∧ ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑐) = ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑛)) ∧ 𝑖 ∈ (0..^3)) ∧ 𝑖 = 1) → (𝑐‘1) = (𝑛‘1))
224		simpr 485	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑛 ∈ 𝐴) ∧ ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑐) = ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑛)) ∧ 𝑖 ∈ (0..^3)) ∧ 𝑖 = 1) → 𝑖 = 1)
225	224	fveq2d 6778	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑛 ∈ 𝐴) ∧ ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑐) = ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑛)) ∧ 𝑖 ∈ (0..^3)) ∧ 𝑖 = 1) → (𝑐‘𝑖) = (𝑐‘1))
226	224	fveq2d 6778	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑛 ∈ 𝐴) ∧ ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑐) = ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑛)) ∧ 𝑖 ∈ (0..^3)) ∧ 𝑖 = 1) → (𝑛‘𝑖) = (𝑛‘1))
227	223, 225, 226	3eqtr4d 2788	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑛 ∈ 𝐴) ∧ ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑐) = ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑛)) ∧ 𝑖 ∈ (0..^3)) ∧ 𝑖 = 1) → (𝑐‘𝑖) = (𝑛‘𝑖))
228	216	ad2antrr 723	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑛 ∈ 𝐴) ∧ ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑐) = ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑛)) ∧ 𝑖 ∈ (0..^3)) ∧ 𝑖 = 2) → (𝑐‘0) = (𝑛‘0))
229	222	ad2antrr 723	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑛 ∈ 𝐴) ∧ ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑐) = ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑛)) ∧ 𝑖 ∈ (0..^3)) ∧ 𝑖 = 2) → (𝑐‘1) = (𝑛‘1))
230	228, 229	oveq12d 7293	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑛 ∈ 𝐴) ∧ ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑐) = ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑛)) ∧ 𝑖 ∈ (0..^3)) ∧ 𝑖 = 2) → ((𝑐‘0) + (𝑐‘1)) = ((𝑛‘0) + (𝑛‘1)))
231	230	oveq2d 7291	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑛 ∈ 𝐴) ∧ ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑐) = ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑛)) ∧ 𝑖 ∈ (0..^3)) ∧ 𝑖 = 2) → (𝑁 − ((𝑐‘0) + (𝑐‘1))) = (𝑁 − ((𝑛‘0) + (𝑛‘1))))
232	22	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑛 ∈ 𝐴) ∧ ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑐) = ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑛)) ∧ 𝑖 ∈ (0..^3)) ∧ 𝑖 = 2) → (0..^3) = {0, 1, 2})
233	232	sumeq1d 15413	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑛 ∈ 𝐴) ∧ ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑐) = ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑛)) ∧ 𝑖 ∈ (0..^3)) ∧ 𝑖 = 2) → Σ𝑗 ∈ (0..^3)(𝑐‘𝑗) = Σ𝑗 ∈ {0, 1, 2} (𝑐‘𝑗))
234		ssidd 3944	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑛 ∈ 𝐴) ∧ ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑐) = ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑛)) ∧ 𝑖 ∈ (0..^3)) ∧ 𝑖 = 2) → ℕ ⊆ ℕ)
235	136	ad4antr 729	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑛 ∈ 𝐴) ∧ ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑐) = ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑛)) ∧ 𝑖 ∈ (0..^3)) ∧ 𝑖 = 2) → 𝑁 ∈ ℤ)
236	3	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑛 ∈ 𝐴) ∧ ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑐) = ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑛)) ∧ 𝑖 ∈ (0..^3)) ∧ 𝑖 = 2) → 3 ∈ ℕ₀)
237	139	ad4antr 729	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑛 ∈ 𝐴) ∧ ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑐) = ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑛)) ∧ 𝑖 ∈ (0..^3)) ∧ 𝑖 = 2) → 𝑐 ∈ (ℕ(repr‘3)𝑁))
238	234, 235, 236, 237	reprsum 32593	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑛 ∈ 𝐴) ∧ ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑐) = ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑛)) ∧ 𝑖 ∈ (0..^3)) ∧ 𝑖 = 2) → Σ𝑗 ∈ (0..^3)(𝑐‘𝑗) = 𝑁)
239		fveq2 6774	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 0 → (𝑐‘𝑗) = (𝑐‘0))
240		fveq2 6774	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 1 → (𝑐‘𝑗) = (𝑐‘1))
241		fveq2 6774	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 2 → (𝑐‘𝑗) = (𝑐‘2))
242	142	nncnd 11989	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (𝑐‘0) ∈ ℂ)
243	242	ad4antr 729	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑛 ∈ 𝐴) ∧ ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑐) = ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑛)) ∧ 𝑖 ∈ (0..^3)) ∧ 𝑖 = 2) → (𝑐‘0) ∈ ℂ)
244	169	nncnd 11989	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (𝑐‘1) ∈ ℂ)
245	244	ad4antr 729	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑛 ∈ 𝐴) ∧ ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑐) = ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑛)) ∧ 𝑖 ∈ (0..^3)) ∧ 𝑖 = 2) → (𝑐‘1) ∈ ℂ)
246	35	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → 2 ∈ (0..^3))
247	140, 246	ffvelrnd 6962	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (𝑐‘2) ∈ ℕ)
248	247	nncnd 11989	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (𝑐‘2) ∈ ℂ)
249	248	ad4antr 729	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑛 ∈ 𝐴) ∧ ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑐) = ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑛)) ∧ 𝑖 ∈ (0..^3)) ∧ 𝑖 = 2) → (𝑐‘2) ∈ ℂ)
250	243, 245, 249	3jca 1127	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑛 ∈ 𝐴) ∧ ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑐) = ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑛)) ∧ 𝑖 ∈ (0..^3)) ∧ 𝑖 = 2) → ((𝑐‘0) ∈ ℂ ∧ (𝑐‘1) ∈ ℂ ∧ (𝑐‘2) ∈ ℂ))
251	20, 27, 33	3pm3.2i 1338	. . . . . . . . . . . . . . . . 17 ⊢ (0 ∈ V ∧ 1 ∈ V ∧ 2 ∈ V)
252	251	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑛 ∈ 𝐴) ∧ ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑐) = ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑛)) ∧ 𝑖 ∈ (0..^3)) ∧ 𝑖 = 2) → (0 ∈ V ∧ 1 ∈ V ∧ 2 ∈ V))
253		0ne1 12044	. . . . . . . . . . . . . . . . 17 ⊢ 0 ≠ 1
254	253	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑛 ∈ 𝐴) ∧ ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑐) = ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑛)) ∧ 𝑖 ∈ (0..^3)) ∧ 𝑖 = 2) → 0 ≠ 1)
255		0ne2 12180	. . . . . . . . . . . . . . . . 17 ⊢ 0 ≠ 2
256	255	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑛 ∈ 𝐴) ∧ ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑐) = ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑛)) ∧ 𝑖 ∈ (0..^3)) ∧ 𝑖 = 2) → 0 ≠ 2)
257		1ne2 12181	. . . . . . . . . . . . . . . . 17 ⊢ 1 ≠ 2
258	257	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑛 ∈ 𝐴) ∧ ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑐) = ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑛)) ∧ 𝑖 ∈ (0..^3)) ∧ 𝑖 = 2) → 1 ≠ 2)
259	239, 240, 241, 250, 252, 254, 256, 258	sumtp 15461	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑛 ∈ 𝐴) ∧ ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑐) = ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑛)) ∧ 𝑖 ∈ (0..^3)) ∧ 𝑖 = 2) → Σ𝑗 ∈ {0, 1, 2} (𝑐‘𝑗) = (((𝑐‘0) + (𝑐‘1)) + (𝑐‘2)))
260	233, 238, 259	3eqtr3rd 2787	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑛 ∈ 𝐴) ∧ ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑐) = ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑛)) ∧ 𝑖 ∈ (0..^3)) ∧ 𝑖 = 2) → (((𝑐‘0) + (𝑐‘1)) + (𝑐‘2)) = 𝑁)
261	243, 245	addcld 10994	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑛 ∈ 𝐴) ∧ ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑐) = ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑛)) ∧ 𝑖 ∈ (0..^3)) ∧ 𝑖 = 2) → ((𝑐‘0) + (𝑐‘1)) ∈ ℂ)
262	100	ad5antr 731	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑛 ∈ 𝐴) ∧ ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑐) = ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑛)) ∧ 𝑖 ∈ (0..^3)) ∧ 𝑖 = 2) → 𝑁 ∈ ℂ)
263	261, 249, 262	addrsub 11392	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑛 ∈ 𝐴) ∧ ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑐) = ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑛)) ∧ 𝑖 ∈ (0..^3)) ∧ 𝑖 = 2) → ((((𝑐‘0) + (𝑐‘1)) + (𝑐‘2)) = 𝑁 ↔ (𝑐‘2) = (𝑁 − ((𝑐‘0) + (𝑐‘1)))))
264	260, 263	mpbid 231	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑛 ∈ 𝐴) ∧ ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑐) = ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑛)) ∧ 𝑖 ∈ (0..^3)) ∧ 𝑖 = 2) → (𝑐‘2) = (𝑁 − ((𝑐‘0) + (𝑐‘1))))
265	232	sumeq1d 15413	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑛 ∈ 𝐴) ∧ ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑐) = ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑛)) ∧ 𝑖 ∈ (0..^3)) ∧ 𝑖 = 2) → Σ𝑗 ∈ (0..^3)(𝑛‘𝑗) = Σ𝑗 ∈ {0, 1, 2} (𝑛‘𝑗))
266	18	ad4ant13 748	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑛 ∈ 𝐴) ∧ ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑐) = ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑛)) → 𝑛 ∈ (ℕ(repr‘3)𝑁))
267	266	ad2antrr 723	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑛 ∈ 𝐴) ∧ ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑐) = ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑛)) ∧ 𝑖 ∈ (0..^3)) ∧ 𝑖 = 2) → 𝑛 ∈ (ℕ(repr‘3)𝑁))
268	234, 235, 236, 267	reprsum 32593	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑛 ∈ 𝐴) ∧ ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑐) = ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑛)) ∧ 𝑖 ∈ (0..^3)) ∧ 𝑖 = 2) → Σ𝑗 ∈ (0..^3)(𝑛‘𝑗) = 𝑁)
269		fveq2 6774	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 0 → (𝑛‘𝑗) = (𝑛‘0))
270		fveq2 6774	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 1 → (𝑛‘𝑗) = (𝑛‘1))
271		fveq2 6774	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 2 → (𝑛‘𝑗) = (𝑛‘2))
272	25	nncnd 11989	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → (𝑛‘0) ∈ ℂ)
273	272	adantlr 712	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑛 ∈ 𝐴) → (𝑛‘0) ∈ ℂ)
274	273	ad3antrrr 727	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑛 ∈ 𝐴) ∧ ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑐) = ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑛)) ∧ 𝑖 ∈ (0..^3)) ∧ 𝑖 = 2) → (𝑛‘0) ∈ ℂ)
275	31	nncnd 11989	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → (𝑛‘1) ∈ ℂ)
276	275	adantlr 712	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑛 ∈ 𝐴) → (𝑛‘1) ∈ ℂ)
277	276	ad3antrrr 727	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑛 ∈ 𝐴) ∧ ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑐) = ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑛)) ∧ 𝑖 ∈ (0..^3)) ∧ 𝑖 = 2) → (𝑛‘1) ∈ ℂ)
278	37	nncnd 11989	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → (𝑛‘2) ∈ ℂ)
279	278	adantlr 712	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑛 ∈ 𝐴) → (𝑛‘2) ∈ ℂ)
280	279	ad3antrrr 727	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑛 ∈ 𝐴) ∧ ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑐) = ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑛)) ∧ 𝑖 ∈ (0..^3)) ∧ 𝑖 = 2) → (𝑛‘2) ∈ ℂ)
281	274, 277, 280	3jca 1127	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑛 ∈ 𝐴) ∧ ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑐) = ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑛)) ∧ 𝑖 ∈ (0..^3)) ∧ 𝑖 = 2) → ((𝑛‘0) ∈ ℂ ∧ (𝑛‘1) ∈ ℂ ∧ (𝑛‘2) ∈ ℂ))
282	269, 270, 271, 281, 252, 254, 256, 258	sumtp 15461	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑛 ∈ 𝐴) ∧ ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑐) = ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑛)) ∧ 𝑖 ∈ (0..^3)) ∧ 𝑖 = 2) → Σ𝑗 ∈ {0, 1, 2} (𝑛‘𝑗) = (((𝑛‘0) + (𝑛‘1)) + (𝑛‘2)))
283	265, 268, 282	3eqtr3rd 2787	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑛 ∈ 𝐴) ∧ ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑐) = ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑛)) ∧ 𝑖 ∈ (0..^3)) ∧ 𝑖 = 2) → (((𝑛‘0) + (𝑛‘1)) + (𝑛‘2)) = 𝑁)
284	274, 277	addcld 10994	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑛 ∈ 𝐴) ∧ ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑐) = ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑛)) ∧ 𝑖 ∈ (0..^3)) ∧ 𝑖 = 2) → ((𝑛‘0) + (𝑛‘1)) ∈ ℂ)
285	284, 280, 262	addrsub 11392	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑛 ∈ 𝐴) ∧ ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑐) = ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑛)) ∧ 𝑖 ∈ (0..^3)) ∧ 𝑖 = 2) → ((((𝑛‘0) + (𝑛‘1)) + (𝑛‘2)) = 𝑁 ↔ (𝑛‘2) = (𝑁 − ((𝑛‘0) + (𝑛‘1)))))
286	283, 285	mpbid 231	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑛 ∈ 𝐴) ∧ ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑐) = ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑛)) ∧ 𝑖 ∈ (0..^3)) ∧ 𝑖 = 2) → (𝑛‘2) = (𝑁 − ((𝑛‘0) + (𝑛‘1))))
287	231, 264, 286	3eqtr4d 2788	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑛 ∈ 𝐴) ∧ ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑐) = ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑛)) ∧ 𝑖 ∈ (0..^3)) ∧ 𝑖 = 2) → (𝑐‘2) = (𝑛‘2))
288		simpr 485	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑛 ∈ 𝐴) ∧ ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑐) = ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑛)) ∧ 𝑖 ∈ (0..^3)) ∧ 𝑖 = 2) → 𝑖 = 2)
289	288	fveq2d 6778	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑛 ∈ 𝐴) ∧ ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑐) = ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑛)) ∧ 𝑖 ∈ (0..^3)) ∧ 𝑖 = 2) → (𝑐‘𝑖) = (𝑐‘2))
290	288	fveq2d 6778	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑛 ∈ 𝐴) ∧ ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑐) = ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑛)) ∧ 𝑖 ∈ (0..^3)) ∧ 𝑖 = 2) → (𝑛‘𝑖) = (𝑛‘2))
291	287, 289, 290	3eqtr4d 2788	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑛 ∈ 𝐴) ∧ ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑐) = ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑛)) ∧ 𝑖 ∈ (0..^3)) ∧ 𝑖 = 2) → (𝑐‘𝑖) = (𝑛‘𝑖))
292		simpr 485	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑛 ∈ 𝐴) ∧ ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑐) = ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑛)) ∧ 𝑖 ∈ (0..^3)) → 𝑖 ∈ (0..^3))
293	292, 22	eleqtrdi 2849	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑛 ∈ 𝐴) ∧ ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑐) = ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑛)) ∧ 𝑖 ∈ (0..^3)) → 𝑖 ∈ {0, 1, 2})
294		vex 3436	. . . . . . . . . . . . 13 ⊢ 𝑖 ∈ V
295	294	eltp 4624	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ {0, 1, 2} ↔ (𝑖 = 0 ∨ 𝑖 = 1 ∨ 𝑖 = 2))
296	293, 295	sylib 217	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑛 ∈ 𝐴) ∧ ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑐) = ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑛)) ∧ 𝑖 ∈ (0..^3)) → (𝑖 = 0 ∨ 𝑖 = 1 ∨ 𝑖 = 2))
297	221, 227, 291, 296	mpjao3dan 1430	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑛 ∈ 𝐴) ∧ ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑐) = ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑛)) ∧ 𝑖 ∈ (0..^3)) → (𝑐‘𝑖) = (𝑛‘𝑖))
298	196, 198, 297	eqfnfvd 6912	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑛 ∈ 𝐴) ∧ ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑐) = ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑛)) → 𝑐 = 𝑛)
299	298	ex 413	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑛 ∈ 𝐴) → (((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑐) = ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑛) → 𝑐 = 𝑛))
300	299	anasss 467	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐴 ∧ 𝑛 ∈ 𝐴)) → (((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑐) = ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑛) → 𝑐 = 𝑛))
301	300	ralrimivva 3123	. . . . . 6 ⊢ (𝜑 → ∀𝑐 ∈ 𝐴 ∀𝑛 ∈ 𝐴 (((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑐) = ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑛) → 𝑐 = 𝑛))
302		dff1o6 7147	. . . . . . 7 ⊢ ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩):𝐴–1-1-onto→ran (𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩) ↔ ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩) Fn 𝐴 ∧ ran (𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩) = ran (𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩) ∧ ∀𝑐 ∈ 𝐴 ∀𝑛 ∈ 𝐴 (((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑐) = ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑛) → 𝑐 = 𝑛)))
303	302	biimpri 227	. . . . . 6 ⊢ (((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩) Fn 𝐴 ∧ ran (𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩) = ran (𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩) ∧ ∀𝑐 ∈ 𝐴 ∀𝑛 ∈ 𝐴 (((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑐) = ((𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)‘𝑛) → 𝑐 = 𝑛)) → (𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩):𝐴–1-1-onto→ran (𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩))
304	193, 194, 301, 303	syl3anc 1370	. . . . 5 ⊢ (𝜑 → (𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩):𝐴–1-1-onto→ran (𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩))
305	181	sselda 3921	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ ran (𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)) → 𝑢 ∈ ((((1...𝑁) ∖ ℙ) ∪ {2}) × (1...𝑁)))
306	305, 124	syldan 591	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ ran (𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)) → (Λ‘(1^st ‘𝑢)) ∈ ℝ)
307	305, 128	syldan 591	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ ran (𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)) → (Λ‘(2^nd ‘𝑢)) ∈ ℝ)
308	306, 307	remulcld 11005	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ ran (𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)) → ((Λ‘(1^st ‘𝑢)) · (Λ‘(2^nd ‘𝑢))) ∈ ℝ)
309	308	recnd 11003	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 ∈ ran (𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)) → ((Λ‘(1^st ‘𝑢)) · (Λ‘(2^nd ‘𝑢))) ∈ ℂ)
310	189, 10, 304, 210, 309	fsumf1o 15435	. . . 4 ⊢ (𝜑 → Σ𝑢 ∈ ran (𝑑 ∈ 𝐴 ↦ ⟨(𝑑‘0), (𝑑‘1)⟩)((Λ‘(1^st ‘𝑢)) · (Λ‘(2^nd ‘𝑢))) = Σ𝑛 ∈ 𝐴 ((Λ‘(𝑛‘0)) · (Λ‘(𝑛‘1))))
311	75	recnd 11003	. . . . . 6 ⊢ (𝜑 → Σ𝑗 ∈ (1...𝑁)(Λ‘𝑗) ∈ ℂ)
312	69	recnd 11003	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (((1...𝑁) ∖ ℙ) ∪ {2})) → (Λ‘𝑖) ∈ ℂ)
313	53, 311, 312	fsummulc1 15497	. . . . 5 ⊢ (𝜑 → (Σ𝑖 ∈ (((1...𝑁) ∖ ℙ) ∪ {2})(Λ‘𝑖) · Σ𝑗 ∈ (1...𝑁)(Λ‘𝑗)) = Σ𝑖 ∈ (((1...𝑁) ∖ ℙ) ∪ {2})((Λ‘𝑖) · Σ𝑗 ∈ (1...𝑁)(Λ‘𝑗)))
314	47	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (((1...𝑁) ∖ ℙ) ∪ {2})) → (1...𝑁) ∈ Fin)
315	74	adantrl 713	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑖 ∈ (((1...𝑁) ∖ ℙ) ∪ {2}) ∧ 𝑗 ∈ (1...𝑁))) → (Λ‘𝑗) ∈ ℝ)
316	315	anassrs 468	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (((1...𝑁) ∖ ℙ) ∪ {2})) ∧ 𝑗 ∈ (1...𝑁)) → (Λ‘𝑗) ∈ ℝ)
317	316	recnd 11003	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (((1...𝑁) ∖ ℙ) ∪ {2})) ∧ 𝑗 ∈ (1...𝑁)) → (Λ‘𝑗) ∈ ℂ)
318	314, 312, 317	fsummulc2 15496	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (((1...𝑁) ∖ ℙ) ∪ {2})) → ((Λ‘𝑖) · Σ𝑗 ∈ (1...𝑁)(Λ‘𝑗)) = Σ𝑗 ∈ (1...𝑁)((Λ‘𝑖) · (Λ‘𝑗)))
319	318	sumeq2dv 15415	. . . . 5 ⊢ (𝜑 → Σ𝑖 ∈ (((1...𝑁) ∖ ℙ) ∪ {2})((Λ‘𝑖) · Σ𝑗 ∈ (1...𝑁)(Λ‘𝑗)) = Σ𝑖 ∈ (((1...𝑁) ∖ ℙ) ∪ {2})Σ𝑗 ∈ (1...𝑁)((Λ‘𝑖) · (Λ‘𝑗)))
320		vex 3436	. . . . . . . . 9 ⊢ 𝑗 ∈ V
321	294, 320	op1std 7841	. . . . . . . 8 ⊢ (𝑢 = ⟨𝑖, 𝑗⟩ → (1^st ‘𝑢) = 𝑖)
322	321	fveq2d 6778	. . . . . . 7 ⊢ (𝑢 = ⟨𝑖, 𝑗⟩ → (Λ‘(1^st ‘𝑢)) = (Λ‘𝑖))
323	294, 320	op2ndd 7842	. . . . . . . 8 ⊢ (𝑢 = ⟨𝑖, 𝑗⟩ → (2^nd ‘𝑢) = 𝑗)
324	323	fveq2d 6778	. . . . . . 7 ⊢ (𝑢 = ⟨𝑖, 𝑗⟩ → (Λ‘(2^nd ‘𝑢)) = (Λ‘𝑗))
325	322, 324	oveq12d 7293	. . . . . 6 ⊢ (𝑢 = ⟨𝑖, 𝑗⟩ → ((Λ‘(1^st ‘𝑢)) · (Λ‘(2^nd ‘𝑢))) = ((Λ‘𝑖) · (Λ‘𝑗)))
326	69	adantrr 714	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑖 ∈ (((1...𝑁) ∖ ℙ) ∪ {2}) ∧ 𝑗 ∈ (1...𝑁))) → (Λ‘𝑖) ∈ ℝ)
327	326, 315	remulcld 11005	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑖 ∈ (((1...𝑁) ∖ ℙ) ∪ {2}) ∧ 𝑗 ∈ (1...𝑁))) → ((Λ‘𝑖) · (Λ‘𝑗)) ∈ ℝ)
328	327	recnd 11003	. . . . . 6 ⊢ ((𝜑 ∧ (𝑖 ∈ (((1...𝑁) ∖ ℙ) ∪ {2}) ∧ 𝑗 ∈ (1...𝑁))) → ((Λ‘𝑖) · (Λ‘𝑗)) ∈ ℂ)
329	325, 53, 71, 328	fsumxp 15484	. . . . 5 ⊢ (𝜑 → Σ𝑖 ∈ (((1...𝑁) ∖ ℙ) ∪ {2})Σ𝑗 ∈ (1...𝑁)((Λ‘𝑖) · (Λ‘𝑗)) = Σ𝑢 ∈ ((((1...𝑁) ∖ ℙ) ∪ {2}) × (1...𝑁))((Λ‘(1^st ‘𝑢)) · (Λ‘(2^nd ‘𝑢))))
330	313, 319, 329	3eqtrrd 2783	. . . 4 ⊢ (𝜑 → Σ𝑢 ∈ ((((1...𝑁) ∖ ℙ) ∪ {2}) × (1...𝑁))((Λ‘(1^st ‘𝑢)) · (Λ‘(2^nd ‘𝑢))) = (Σ𝑖 ∈ (((1...𝑁) ∖ ℙ) ∪ {2})(Λ‘𝑖) · Σ𝑗 ∈ (1...𝑁)(Λ‘𝑗)))
331	182, 310, 330	3brtr3d 5105	. . 3 ⊢ (𝜑 → Σ𝑛 ∈ 𝐴 ((Λ‘(𝑛‘0)) · (Λ‘(𝑛‘1))) ≤ (Σ𝑖 ∈ (((1...𝑁) ∖ ℙ) ∪ {2})(Λ‘𝑖) · Σ𝑗 ∈ (1...𝑁)(Λ‘𝑗)))
332	45, 76, 43, 116, 331	lemul2ad 11915	. 2 ⊢ (𝜑 → ((log‘𝑁) · Σ𝑛 ∈ 𝐴 ((Λ‘(𝑛‘0)) · (Λ‘(𝑛‘1)))) ≤ ((log‘𝑁) · (Σ𝑖 ∈ (((1...𝑁) ∖ ℙ) ∪ {2})(Λ‘𝑖) · Σ𝑗 ∈ (1...𝑁)(Λ‘𝑗))))
333	41, 46, 77, 113, 332	letrd 11132	1 ⊢ (𝜑 → Σ𝑛 ∈ 𝐴 ((Λ‘(𝑛‘0)) · ((Λ‘(𝑛‘1)) · (Λ‘(𝑛‘2)))) ≤ ((log‘𝑁) · (Σ𝑖 ∈ (((1...𝑁) ∖ ℙ) ∪ {2})(Λ‘𝑖) · Σ𝑗 ∈ (1...𝑁)(Λ‘𝑗))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∨ w3o 1085 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 {crab 3068 Vcvv 3432 ∖ cdif 3884 ∪ cun 3885 ∩ cin 3886 ⊆ wss 3887 {csn 4561 {ctp 4565 ⟨cop 4567 class class class wbr 5074 ↦ cmpt 5157 × cxp 5587 ran crn 5590 Fn wfn 6428 ⟶wf 6429 –1-1-onto→wf1o 6432 ‘cfv 6433 (class class class)co 7275 1^st c1st 7829 2^nd c2nd 7830 Fincfn 8733 ℂcc 10869 ℝcr 10870 0cc0 10871 1c1 10872 + caddc 10874 · cmul 10876 ≤ cle 11010 − cmin 11205 ℕcn 11973 2c2 12028 3c3 12029 ℕ₀cn0 12233 ℤcz 12319 ℝ⁺crp 12730 ...cfz 13239 ..^cfzo 13382 Σcsu 15397 ∥ cdvds 15963 ℙcprime 16376 logclog 25710 Λcvma 26241 reprcrepr 32588

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 ax-addf 10950 ax-mulf 10951

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-oadd 8301 df-er 8498 df-map 8617 df-pm 8618 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-fi 9170 df-sup 9201 df-inf 9202 df-oi 9269 df-dju 9659 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-q 12689 df-rp 12731 df-xneg 12848 df-xadd 12849 df-xmul 12850 df-ioo 13083 df-ioc 13084 df-ico 13085 df-icc 13086 df-fz 13240 df-fzo 13383 df-fl 13512 df-mod 13590 df-seq 13722 df-exp 13783 df-fac 13988 df-bc 14017 df-hash 14045 df-shft 14778 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-limsup 15180 df-clim 15197 df-rlim 15198 df-sum 15398 df-ef 15777 df-sin 15779 df-cos 15780 df-pi 15782 df-dvds 15964 df-gcd 16202 df-prm 16377 df-pc 16538 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-starv 16977 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-hom 16986 df-cco 16987 df-rest 17133 df-topn 17134 df-0g 17152 df-gsum 17153 df-topgen 17154 df-pt 17155 df-prds 17158 df-xrs 17213 df-qtop 17218 df-imas 17219 df-xps 17221 df-mre 17295 df-mrc 17296 df-acs 17298 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-submnd 18431 df-mulg 18701 df-cntz 18923 df-cmn 19388 df-psmet 20589 df-xmet 20590 df-met 20591 df-bl 20592 df-mopn 20593 df-fbas 20594 df-fg 20595 df-cnfld 20598 df-top 22043 df-topon 22060 df-topsp 22082 df-bases 22096 df-cld 22170 df-ntr 22171 df-cls 22172 df-nei 22249 df-lp 22287 df-perf 22288 df-cn 22378 df-cnp 22379 df-haus 22466 df-tx 22713 df-hmeo 22906 df-fil 22997 df-fm 23089 df-flim 23090 df-flf 23091 df-xms 23473 df-ms 23474 df-tms 23475 df-cncf 24041 df-limc 25030 df-dv 25031 df-log 25712 df-vma 26247 df-repr 32589

This theorem is referenced by: hgt750leme 32638

W3C validator