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Theorem dchrsum2 27443

Description: An orthogonality relation for Dirichlet characters: the sum of all the values of a Dirichlet character 𝑋 is 0 if 𝑋 is non-principal and ϕ(𝑛) otherwise. Part of Theorem 6.5.1 of [Shapiro] p. 230. (Contributed by Mario Carneiro, 28-Apr-2016.)

**Hypotheses**
Ref	Expression
dchrsum.g	⊢ 𝐺 = (DChr‘𝑁)
dchrsum.z	⊢ 𝑍 = (ℤ/nℤ‘𝑁)
dchrsum.d	⊢ 𝐷 = (Base‘𝐺)
dchrsum.1	⊢ 1 = (0_g‘𝐺)
dchrsum.x	⊢ (𝜑 → 𝑋 ∈ 𝐷)
dchrsum2.u	⊢ 𝑈 = (Unit‘𝑍)

**Assertion**
Ref	Expression
dchrsum2	⊢ (𝜑 → Σ𝑎 ∈ 𝑈 (𝑋‘𝑎) = if(𝑋 = 1 , (ϕ‘𝑁), 0))

Distinct variable groups: 1 ,𝑎 𝜑,𝑎 𝑈,𝑎 𝑋,𝑎 𝑍,𝑎 Allowed substitution hints: 𝐷(𝑎) 𝐺(𝑎) 𝑁(𝑎)

Proof of Theorem dchrsum2 Dummy variables 𝑘 𝑥 𝑏 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq2 2774	. 2 ⊢ ((ϕ‘𝑁) = if(𝑋 = 1 , (ϕ‘𝑁), 0) → (Σ𝑎 ∈ 𝑈 (𝑋‘𝑎) = (ϕ‘𝑁) ↔ Σ𝑎 ∈ 𝑈 (𝑋‘𝑎) = if(𝑋 = 1 , (ϕ‘𝑁), 0)))
2		eqeq2 2774	. 2 ⊢ (0 = if(𝑋 = 1 , (ϕ‘𝑁), 0) → (Σ𝑎 ∈ 𝑈 (𝑋‘𝑎) = 0 ↔ Σ𝑎 ∈ 𝑈 (𝑋‘𝑎) = if(𝑋 = 1 , (ϕ‘𝑁), 0)))
3		fveq1 6880	. . . . . 6 ⊢ (𝑋 = 1 → (𝑋‘𝑎) = ( 1 ‘𝑎))
4		dchrsum.g	. . . . . . 7 ⊢ 𝐺 = (DChr‘𝑁)
5		dchrsum.z	. . . . . . 7 ⊢ 𝑍 = (ℤ/nℤ‘𝑁)
6		dchrsum.1	. . . . . . 7 ⊢ 1 = (0_g‘𝐺)
7		dchrsum2.u	. . . . . . 7 ⊢ 𝑈 = (Unit‘𝑍)
8		dchrsum.x	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ 𝐷)
9		dchrsum.d	. . . . . . . . . 10 ⊢ 𝐷 = (Base‘𝐺)
10	4, 9	dchrrcl 27415	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝐷 → 𝑁 ∈ ℕ)
11	8, 10	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ)
12	11	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑈) → 𝑁 ∈ ℕ)
13		simpr 489	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑈) → 𝑎 ∈ 𝑈)
14	4, 5, 6, 7, 12, 13	dchr1 27432	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑈) → ( 1 ‘𝑎) = 1)
15	3, 14	sylan9eqr 2819	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑋 = 1 ) → (𝑋‘𝑎) = 1)
16	15	an32s 664	. . . 4 ⊢ (((𝜑 ∧ 𝑋 = 1 ) ∧ 𝑎 ∈ 𝑈) → (𝑋‘𝑎) = 1)
17	16	sumeq2dv 15760	. . 3 ⊢ ((𝜑 ∧ 𝑋 = 1 ) → Σ𝑎 ∈ 𝑈 (𝑋‘𝑎) = Σ𝑎 ∈ 𝑈 1)
18	5, 7	znunithash 21725	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (♯‘𝑈) = (ϕ‘𝑁))
19	11, 18	syl 18	. . . . . . . 8 ⊢ (𝜑 → (♯‘𝑈) = (ϕ‘𝑁))
20	11	phicld 16837	. . . . . . . . 9 ⊢ (𝜑 → (ϕ‘𝑁) ∈ ℕ)
21	20	nnnn0d 12571	. . . . . . . 8 ⊢ (𝜑 → (ϕ‘𝑁) ∈ ℕ₀)
22	19, 21	eqeltrd 2862	. . . . . . 7 ⊢ (𝜑 → (♯‘𝑈) ∈ ℕ₀)
23	7	fvexi 6895	. . . . . . . 8 ⊢ 𝑈 ∈ V
24		hashclb 14401	. . . . . . . 8 ⊢ (𝑈 ∈ V → (𝑈 ∈ Fin ↔ (♯‘𝑈) ∈ ℕ₀))
25	23, 24	ax-mp 5	. . . . . . 7 ⊢ (𝑈 ∈ Fin ↔ (♯‘𝑈) ∈ ℕ₀)
26	22, 25	sylibr 237	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ Fin)
27		ax-1cn 11164	. . . . . 6 ⊢ 1 ∈ ℂ
28		fsumconst 15848	. . . . . 6 ⊢ ((𝑈 ∈ Fin ∧ 1 ∈ ℂ) → Σ𝑎 ∈ 𝑈 1 = ((♯‘𝑈) · 1))
29	26, 27, 28	sylancl 597	. . . . 5 ⊢ (𝜑 → Σ𝑎 ∈ 𝑈 1 = ((♯‘𝑈) · 1))
30	19	oveq1d 7427	. . . . 5 ⊢ (𝜑 → ((♯‘𝑈) · 1) = ((ϕ‘𝑁) · 1))
31	20	nncnd 12255	. . . . . 6 ⊢ (𝜑 → (ϕ‘𝑁) ∈ ℂ)
32	31	mulridd 11232	. . . . 5 ⊢ (𝜑 → ((ϕ‘𝑁) · 1) = (ϕ‘𝑁))
33	29, 30, 32	3eqtrd 2801	. . . 4 ⊢ (𝜑 → Σ𝑎 ∈ 𝑈 1 = (ϕ‘𝑁))
34	33	adantr 485	. . 3 ⊢ ((𝜑 ∧ 𝑋 = 1 ) → Σ𝑎 ∈ 𝑈 1 = (ϕ‘𝑁))
35	17, 34	eqtrd 2797	. 2 ⊢ ((𝜑 ∧ 𝑋 = 1 ) → Σ𝑎 ∈ 𝑈 (𝑋‘𝑎) = (ϕ‘𝑁))
36	4	dchrabl 27429	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 𝐺 ∈ Abel)
37		ablgrp 19861	. . . . . . . 8 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
38	9, 6	grpidcl 19038	. . . . . . . 8 ⊢ (𝐺 ∈ Grp → 1 ∈ 𝐷)
39	11, 36, 37, 38	4syl 20	. . . . . . 7 ⊢ (𝜑 → 1 ∈ 𝐷)
40	4, 5, 9, 7, 8, 39	dchreq 27433	. . . . . 6 ⊢ (𝜑 → (𝑋 = 1 ↔ ∀𝑘 ∈ 𝑈 (𝑋‘𝑘) = ( 1 ‘𝑘)))
41	40	notbid 321	. . . . 5 ⊢ (𝜑 → (¬ 𝑋 = 1 ↔ ¬ ∀𝑘 ∈ 𝑈 (𝑋‘𝑘) = ( 1 ‘𝑘)))
42		rexnal 3116	. . . . 5 ⊢ (∃𝑘 ∈ 𝑈 ¬ (𝑋‘𝑘) = ( 1 ‘𝑘) ↔ ¬ ∀𝑘 ∈ 𝑈 (𝑋‘𝑘) = ( 1 ‘𝑘))
43	41, 42	bitr4di 292	. . . 4 ⊢ (𝜑 → (¬ 𝑋 = 1 ↔ ∃𝑘 ∈ 𝑈 ¬ (𝑋‘𝑘) = ( 1 ‘𝑘)))
44		df-ne 2958	. . . . . 6 ⊢ ((𝑋‘𝑘) ≠ ( 1 ‘𝑘) ↔ ¬ (𝑋‘𝑘) = ( 1 ‘𝑘))
45	11	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → 𝑁 ∈ ℕ)
46		simpr 489	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → 𝑘 ∈ 𝑈)
47	4, 5, 6, 7, 45, 46	dchr1 27432	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → ( 1 ‘𝑘) = 1)
48	47	neeq2d 3017	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → ((𝑋‘𝑘) ≠ ( 1 ‘𝑘) ↔ (𝑋‘𝑘) ≠ 1))
49	26	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → 𝑈 ∈ Fin)
50		eqid 2762	. . . . . . . . . . . . 13 ⊢ (Base‘𝑍) = (Base‘𝑍)
51	4, 5, 9, 50, 8	dchrf 27417	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋:(Base‘𝑍)⟶ℂ)
52	50, 7	unitss 20465	. . . . . . . . . . . . 13 ⊢ 𝑈 ⊆ (Base‘𝑍)
53	52	sseli 3932	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ 𝑈 → 𝑎 ∈ (Base‘𝑍))
54		ffvelcdm 7076	. . . . . . . . . . . 12 ⊢ ((𝑋:(Base‘𝑍)⟶ℂ ∧ 𝑎 ∈ (Base‘𝑍)) → (𝑋‘𝑎) ∈ ℂ)
55	51, 53, 54	syl2an 607	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑈) → (𝑋‘𝑎) ∈ ℂ)
56	55	adantlr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) ∧ 𝑎 ∈ 𝑈) → (𝑋‘𝑎) ∈ ℂ)
57	49, 56	fsumcl 15791	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → Σ𝑎 ∈ 𝑈 (𝑋‘𝑎) ∈ ℂ)
58		0cnd 11205	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → 0 ∈ ℂ)
59	51	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → 𝑋:(Base‘𝑍)⟶ℂ)
60		simprl 782	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → 𝑘 ∈ 𝑈)
61	52, 60	sselid 3934	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → 𝑘 ∈ (Base‘𝑍))
62	59, 61	ffvelcdmd 7080	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → (𝑋‘𝑘) ∈ ℂ)
63		subcl 11462	. . . . . . . . . 10 ⊢ (((𝑋‘𝑘) ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑋‘𝑘) − 1) ∈ ℂ)
64	62, 27, 63	sylancl 597	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → ((𝑋‘𝑘) − 1) ∈ ℂ)
65		simprr 784	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → (𝑋‘𝑘) ≠ 1)
66		subeq0 11490	. . . . . . . . . . . 12 ⊢ (((𝑋‘𝑘) ∈ ℂ ∧ 1 ∈ ℂ) → (((𝑋‘𝑘) − 1) = 0 ↔ (𝑋‘𝑘) = 1))
67	62, 27, 66	sylancl 597	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → (((𝑋‘𝑘) − 1) = 0 ↔ (𝑋‘𝑘) = 1))
68	67	necon3bid 3001	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → (((𝑋‘𝑘) − 1) ≠ 0 ↔ (𝑋‘𝑘) ≠ 1))
69	65, 68	mpbird 260	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → ((𝑋‘𝑘) − 1) ≠ 0)
70		oveq2 7420	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑎 → (𝑘(._r‘𝑍)𝑥) = (𝑘(._r‘𝑍)𝑎))
71	70	fveq2d 6885	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑎 → (𝑋‘(𝑘(._r‘𝑍)𝑥)) = (𝑋‘(𝑘(._r‘𝑍)𝑎)))
72	71	cbvsumv 15754	. . . . . . . . . . . . . 14 ⊢ Σ𝑥 ∈ 𝑈 (𝑋‘(𝑘(._r‘𝑍)𝑥)) = Σ𝑎 ∈ 𝑈 (𝑋‘(𝑘(._r‘𝑍)𝑎))
73	4, 5, 9	dchrmhm 27416	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐷 ⊆ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld))
74	73, 8	sselid 3934	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)))
75	74	ad2antrr 738	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) ∧ 𝑎 ∈ 𝑈) → 𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)))
76	61	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) ∧ 𝑎 ∈ 𝑈) → 𝑘 ∈ (Base‘𝑍))
77	53	adantl 486	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) ∧ 𝑎 ∈ 𝑈) → 𝑎 ∈ (Base‘𝑍))
78		eqid 2762	. . . . . . . . . . . . . . . . . 18 ⊢ (mulGrp‘𝑍) = (mulGrp‘𝑍)
79	78, 50	mgpbas 20227	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘𝑍) = (Base‘(mulGrp‘𝑍))
80		eqid 2762	. . . . . . . . . . . . . . . . . 18 ⊢ (._r‘𝑍) = (._r‘𝑍)
81	78, 80	mgpplusg 20226	. . . . . . . . . . . . . . . . 17 ⊢ (._r‘𝑍) = (+_g‘(mulGrp‘𝑍))
82		eqid 2762	. . . . . . . . . . . . . . . . . 18 ⊢ (mulGrp‘ℂ_fld) = (mulGrp‘ℂ_fld)
83		cnfldmul 21541	. . . . . . . . . . . . . . . . . 18 ⊢ · = (._r‘ℂ_fld)
84	82, 83	mgpplusg 20226	. . . . . . . . . . . . . . . . 17 ⊢ · = (+_g‘(mulGrp‘ℂ_fld))
85	79, 81, 84	mhmlin 18857	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)) ∧ 𝑘 ∈ (Base‘𝑍) ∧ 𝑎 ∈ (Base‘𝑍)) → (𝑋‘(𝑘(._r‘𝑍)𝑎)) = ((𝑋‘𝑘) · (𝑋‘𝑎)))
86	75, 76, 77, 85	syl3anc 1397	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) ∧ 𝑎 ∈ 𝑈) → (𝑋‘(𝑘(._r‘𝑍)𝑎)) = ((𝑋‘𝑘) · (𝑋‘𝑎)))
87	86	sumeq2dv 15760	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → Σ𝑎 ∈ 𝑈 (𝑋‘(𝑘(._r‘𝑍)𝑎)) = Σ𝑎 ∈ 𝑈 ((𝑋‘𝑘) · (𝑋‘𝑎)))
88	72, 87	eqtrid 2809	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → Σ𝑥 ∈ 𝑈 (𝑋‘(𝑘(._r‘𝑍)𝑥)) = Σ𝑎 ∈ 𝑈 ((𝑋‘𝑘) · (𝑋‘𝑎)))
89		fveq2 6881	. . . . . . . . . . . . . 14 ⊢ (𝑎 = (𝑘(._r‘𝑍)𝑥) → (𝑋‘𝑎) = (𝑋‘(𝑘(._r‘𝑍)𝑥)))
90	11	nnnn0d 12571	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
91	5	zncrng 21705	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℕ₀ → 𝑍 ∈ CRing)
92		crngring 20333	. . . . . . . . . . . . . . . 16 ⊢ (𝑍 ∈ CRing → 𝑍 ∈ Ring)
93		eqid 2762	. . . . . . . . . . . . . . . . 17 ⊢ ((mulGrp‘𝑍) ↾_s 𝑈) = ((mulGrp‘𝑍) ↾_s 𝑈)
94	7, 93	unitgrp 20472	. . . . . . . . . . . . . . . 16 ⊢ (𝑍 ∈ Ring → ((mulGrp‘𝑍) ↾_s 𝑈) ∈ Grp)
95	90, 91, 92, 94	4syl 20	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((mulGrp‘𝑍) ↾_s 𝑈) ∈ Grp)
96		eqid 2762	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 ∈ 𝑈 ↦ (𝑐 ∈ 𝑈 ↦ (𝑏(._r‘𝑍)𝑐))) = (𝑏 ∈ 𝑈 ↦ (𝑐 ∈ 𝑈 ↦ (𝑏(._r‘𝑍)𝑐)))
97	7, 93	unitgrpbas 20471	. . . . . . . . . . . . . . . 16 ⊢ 𝑈 = (Base‘((mulGrp‘𝑍) ↾_s 𝑈))
98	93, 81	ressplusg 17350	. . . . . . . . . . . . . . . . 17 ⊢ (𝑈 ∈ V → (._r‘𝑍) = (+_g‘((mulGrp‘𝑍) ↾_s 𝑈)))
99	23, 98	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ (._r‘𝑍) = (+_g‘((mulGrp‘𝑍) ↾_s 𝑈))
100	96, 97, 99	grplactf1o 19116	. . . . . . . . . . . . . . 15 ⊢ ((((mulGrp‘𝑍) ↾_s 𝑈) ∈ Grp ∧ 𝑘 ∈ 𝑈) → ((𝑏 ∈ 𝑈 ↦ (𝑐 ∈ 𝑈 ↦ (𝑏(._r‘𝑍)𝑐)))‘𝑘):𝑈–1-1-onto→𝑈)
101	95, 60, 100	syl2an2r 697	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → ((𝑏 ∈ 𝑈 ↦ (𝑐 ∈ 𝑈 ↦ (𝑏(._r‘𝑍)𝑐)))‘𝑘):𝑈–1-1-onto→𝑈)
102	96, 97	grplactval 19114	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) → (((𝑏 ∈ 𝑈 ↦ (𝑐 ∈ 𝑈 ↦ (𝑏(._r‘𝑍)𝑐)))‘𝑘)‘𝑥) = (𝑘(._r‘𝑍)𝑥))
103	60, 102	sylan 591	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) ∧ 𝑥 ∈ 𝑈) → (((𝑏 ∈ 𝑈 ↦ (𝑐 ∈ 𝑈 ↦ (𝑏(._r‘𝑍)𝑐)))‘𝑘)‘𝑥) = (𝑘(._r‘𝑍)𝑥))
104	89, 49, 101, 103, 56	fsumf1o 15781	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → Σ𝑎 ∈ 𝑈 (𝑋‘𝑎) = Σ𝑥 ∈ 𝑈 (𝑋‘(𝑘(._r‘𝑍)𝑥)))
105	49, 62, 56	fsummulc2 15842	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → ((𝑋‘𝑘) · Σ𝑎 ∈ 𝑈 (𝑋‘𝑎)) = Σ𝑎 ∈ 𝑈 ((𝑋‘𝑘) · (𝑋‘𝑎)))
106	88, 104, 105	3eqtr4rd 2808	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → ((𝑋‘𝑘) · Σ𝑎 ∈ 𝑈 (𝑋‘𝑎)) = Σ𝑎 ∈ 𝑈 (𝑋‘𝑎))
107	57	mullidd 11233	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → (1 · Σ𝑎 ∈ 𝑈 (𝑋‘𝑎)) = Σ𝑎 ∈ 𝑈 (𝑋‘𝑎))
108	106, 107	oveq12d 7430	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → (((𝑋‘𝑘) · Σ𝑎 ∈ 𝑈 (𝑋‘𝑎)) − (1 · Σ𝑎 ∈ 𝑈 (𝑋‘𝑎))) = (Σ𝑎 ∈ 𝑈 (𝑋‘𝑎) − Σ𝑎 ∈ 𝑈 (𝑋‘𝑎)))
109	57	subidd 11563	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → (Σ𝑎 ∈ 𝑈 (𝑋‘𝑎) − Σ𝑎 ∈ 𝑈 (𝑋‘𝑎)) = 0)
110	108, 109	eqtrd 2797	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → (((𝑋‘𝑘) · Σ𝑎 ∈ 𝑈 (𝑋‘𝑎)) − (1 · Σ𝑎 ∈ 𝑈 (𝑋‘𝑎))) = 0)
111		1cnd 11208	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → 1 ∈ ℂ)
112	62, 111, 57	subdird 11677	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → (((𝑋‘𝑘) − 1) · Σ𝑎 ∈ 𝑈 (𝑋‘𝑎)) = (((𝑋‘𝑘) · Σ𝑎 ∈ 𝑈 (𝑋‘𝑎)) − (1 · Σ𝑎 ∈ 𝑈 (𝑋‘𝑎))))
113	64	mul01d 11415	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → (((𝑋‘𝑘) − 1) · 0) = 0)
114	110, 112, 113	3eqtr4d 2807	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → (((𝑋‘𝑘) − 1) · Σ𝑎 ∈ 𝑈 (𝑋‘𝑎)) = (((𝑋‘𝑘) − 1) · 0))
115	57, 58, 64, 69, 114	mulcanad 11855	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → Σ𝑎 ∈ 𝑈 (𝑋‘𝑎) = 0)
116	115	expr 461	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → ((𝑋‘𝑘) ≠ 1 → Σ𝑎 ∈ 𝑈 (𝑋‘𝑎) = 0))
117	48, 116	sylbid 243	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → ((𝑋‘𝑘) ≠ ( 1 ‘𝑘) → Σ𝑎 ∈ 𝑈 (𝑋‘𝑎) = 0))
118	44, 117	biimtrrid 246	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → (¬ (𝑋‘𝑘) = ( 1 ‘𝑘) → Σ𝑎 ∈ 𝑈 (𝑋‘𝑎) = 0))
119	118	rexlimdva 3165	. . . 4 ⊢ (𝜑 → (∃𝑘 ∈ 𝑈 ¬ (𝑋‘𝑘) = ( 1 ‘𝑘) → Σ𝑎 ∈ 𝑈 (𝑋‘𝑎) = 0))
120	43, 119	sylbid 243	. . 3 ⊢ (𝜑 → (¬ 𝑋 = 1 → Σ𝑎 ∈ 𝑈 (𝑋‘𝑎) = 0))
121	120	imp 411	. 2 ⊢ ((𝜑 ∧ ¬ 𝑋 = 1 ) → Σ𝑎 ∈ 𝑈 (𝑋‘𝑎) = 0)
122	1, 2, 35, 121	ifbothda 4525	1 ⊢ (𝜑 → Σ𝑎 ∈ 𝑈 (𝑋‘𝑎) = if(𝑋 = 1 , (ϕ‘𝑁), 0))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1569 ∈ wcel 2142 ≠ wne 2957 ∀wral 3078 ∃wrex 3088 Vcvv 3454 ifcif 4486 ↦ cmpt 5191 ⟶wf 6532 –1-1-onto→wf1o 6535 ‘cfv 6536 (class class class)co 7412 Fincfn 8941 ℂcc 11104 0cc0 11106 1c1 11107 · cmul 11111 − cmin 11447 ℕcn 12239 ℕ₀cn0 12510 ♯chash 14373 Σcsu 15744 ϕcphi 16829 Basecbs 17275 ↾_s cress 17296 +_gcplusg 17316 ._rcmulr 17317 0_gc0g 17498 MndHom cmhm 18845 Grpcgrp 19006 Abelcabl 19857 mulGrpcmgp 20222 Ringcrg 20321 CRingccrg 20322 Unitcui 20444 ℂ_fldccnfld 21533 ℤ/nℤczn 21663 DChrcdchr 27407

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1824 ax-4 1838 ax-5 1939 ax-6 1996 ax-7 2037 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5237 ax-sep 5256 ax-nul 5268 ax-pow 5335 ax-pr 5403 ax-un 7734 ax-inf2 9608 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185 ax-mulf 11186

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1103 df-3an 1104 df-tru 1572 df-fal 1582 df-ex 1809 df-nf 1813 df-sb 2096 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3456 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4595 df-uni 4872 df-int 4912 df-iun 4957 df-br 5109 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5555 df-eprel 5560 df-po 5568 df-so 5569 df-fr 5613 df-se 5614 df-we 5615 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-rn 5671 df-res 5672 df-ima 5673 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7676 df-om 7861 df-1st 7984 df-2nd 7985 df-tpos 8220 df-frecs 8276 df-wrecs 8307 df-recs 8356 df-rdg 8395 df-1o 8451 df-oadd 8455 df-er 8692 df-ec 8694 df-qs 8698 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9400 df-inf 9401 df-oi 9470 df-card 9932 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11449 df-neg 11450 df-div 11878 df-nn 12240 df-2 12309 df-3 12310 df-4 12311 df-5 12312 df-6 12313 df-7 12314 df-8 12315 df-9 12316 df-n0 12511 df-xnn0 12584 df-z 12598 df-dec 12718 df-uz 12869 df-rp 13023 df-fz 13542 df-fzo 13690 df-fl 13832 df-mod 13910 df-seq 14045 df-exp 14105 df-hash 14374 df-cj 15157 df-re 15158 df-im 15159 df-sqrt 15293 df-abs 15294 df-clim 15546 df-sum 15745 df-dvds 16317 df-gcd 16559 df-phi 16831 df-struct 17213 df-sets 17230 df-slot 17248 df-ndx 17260 df-base 17276 df-ress 17297 df-plusg 17329 df-mulr 17330 df-starv 17331 df-sca 17332 df-vsca 17333 df-ip 17334 df-tset 17335 df-ple 17336 df-ds 17338 df-unif 17339 df-0g 17500 df-imas 17568 df-qus 17569 df-mgm 18704 df-sgrp 18783 df-mnd 18799 df-mhm 18847 df-grp 19009 df-minusg 19010 df-sbg 19011 df-mulg 19140 df-subg 19195 df-nsg 19196 df-eqg 19197 df-ghm 19290 df-cmn 19858 df-abl 19859 df-mgp 20223 df-rng 20237 df-ur 20270 df-ring 20323 df-cring 20324 df-oppr 20426 df-dvdsr 20446 df-unit 20447 df-invr 20477 df-rhm 20561 df-subrng 20656 df-subrg 20680 df-lmod 20994 df-lss 21064 df-lsp 21104 df-sra 21305 df-rgmod 21306 df-lidl 21343 df-rsp 21344 df-2idl 21400 df-cnfld 21534 df-zring 21608 df-zrh 21664 df-zn 21667 df-dchr 27408

This theorem is used by: dchrsum 27444

W3C validator