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Theorem dchrsum2 27156

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 2738	. 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 2738	. 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 6884	. . . . . 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 27128	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝐷 → 𝑁 ∈ ℕ)
11	8, 10	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ)
12	11	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑈) → 𝑁 ∈ ℕ)
13		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑈) → 𝑎 ∈ 𝑈)
14	4, 5, 6, 7, 12, 13	dchr1 27145	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑈) → ( 1 ‘𝑎) = 1)
15	3, 14	sylan9eqr 2788	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑋 = 1 ) → (𝑋‘𝑎) = 1)
16	15	an32s 649	. . . 4 ⊢ (((𝜑 ∧ 𝑋 = 1 ) ∧ 𝑎 ∈ 𝑈) → (𝑋‘𝑎) = 1)
17	16	sumeq2dv 15655	. . 3 ⊢ ((𝜑 ∧ 𝑋 = 1 ) → Σ𝑎 ∈ 𝑈 (𝑋‘𝑎) = Σ𝑎 ∈ 𝑈 1)
18	5, 7	znunithash 21459	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (♯‘𝑈) = (ϕ‘𝑁))
19	11, 18	syl 17	. . . . . . . 8 ⊢ (𝜑 → (♯‘𝑈) = (ϕ‘𝑁))
20	11	phicld 16714	. . . . . . . . 9 ⊢ (𝜑 → (ϕ‘𝑁) ∈ ℕ)
21	20	nnnn0d 12536	. . . . . . . 8 ⊢ (𝜑 → (ϕ‘𝑁) ∈ ℕ₀)
22	19, 21	eqeltrd 2827	. . . . . . 7 ⊢ (𝜑 → (♯‘𝑈) ∈ ℕ₀)
23	7	fvexi 6899	. . . . . . . 8 ⊢ 𝑈 ∈ V
24		hashclb 14323	. . . . . . . 8 ⊢ (𝑈 ∈ V → (𝑈 ∈ Fin ↔ (♯‘𝑈) ∈ ℕ₀))
25	23, 24	ax-mp 5	. . . . . . 7 ⊢ (𝑈 ∈ Fin ↔ (♯‘𝑈) ∈ ℕ₀)
26	22, 25	sylibr 233	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ Fin)
27		ax-1cn 11170	. . . . . 6 ⊢ 1 ∈ ℂ
28		fsumconst 15742	. . . . . 6 ⊢ ((𝑈 ∈ Fin ∧ 1 ∈ ℂ) → Σ𝑎 ∈ 𝑈 1 = ((♯‘𝑈) · 1))
29	26, 27, 28	sylancl 585	. . . . 5 ⊢ (𝜑 → Σ𝑎 ∈ 𝑈 1 = ((♯‘𝑈) · 1))
30	19	oveq1d 7420	. . . . 5 ⊢ (𝜑 → ((♯‘𝑈) · 1) = ((ϕ‘𝑁) · 1))
31	20	nncnd 12232	. . . . . 6 ⊢ (𝜑 → (ϕ‘𝑁) ∈ ℂ)
32	31	mulridd 11235	. . . . 5 ⊢ (𝜑 → ((ϕ‘𝑁) · 1) = (ϕ‘𝑁))
33	29, 30, 32	3eqtrd 2770	. . . 4 ⊢ (𝜑 → Σ𝑎 ∈ 𝑈 1 = (ϕ‘𝑁))
34	33	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑋 = 1 ) → Σ𝑎 ∈ 𝑈 1 = (ϕ‘𝑁))
35	17, 34	eqtrd 2766	. 2 ⊢ ((𝜑 ∧ 𝑋 = 1 ) → Σ𝑎 ∈ 𝑈 (𝑋‘𝑎) = (ϕ‘𝑁))
36	4	dchrabl 27142	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 𝐺 ∈ Abel)
37		ablgrp 19705	. . . . . . . 8 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
38	9, 6	grpidcl 18895	. . . . . . . 8 ⊢ (𝐺 ∈ Grp → 1 ∈ 𝐷)
39	11, 36, 37, 38	4syl 19	. . . . . . 7 ⊢ (𝜑 → 1 ∈ 𝐷)
40	4, 5, 9, 7, 8, 39	dchreq 27146	. . . . . 6 ⊢ (𝜑 → (𝑋 = 1 ↔ ∀𝑘 ∈ 𝑈 (𝑋‘𝑘) = ( 1 ‘𝑘)))
41	40	notbid 318	. . . . 5 ⊢ (𝜑 → (¬ 𝑋 = 1 ↔ ¬ ∀𝑘 ∈ 𝑈 (𝑋‘𝑘) = ( 1 ‘𝑘)))
42		rexnal 3094	. . . . 5 ⊢ (∃𝑘 ∈ 𝑈 ¬ (𝑋‘𝑘) = ( 1 ‘𝑘) ↔ ¬ ∀𝑘 ∈ 𝑈 (𝑋‘𝑘) = ( 1 ‘𝑘))
43	41, 42	bitr4di 289	. . . 4 ⊢ (𝜑 → (¬ 𝑋 = 1 ↔ ∃𝑘 ∈ 𝑈 ¬ (𝑋‘𝑘) = ( 1 ‘𝑘)))
44		df-ne 2935	. . . . . 6 ⊢ ((𝑋‘𝑘) ≠ ( 1 ‘𝑘) ↔ ¬ (𝑋‘𝑘) = ( 1 ‘𝑘))
45	11	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → 𝑁 ∈ ℕ)
46		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → 𝑘 ∈ 𝑈)
47	4, 5, 6, 7, 45, 46	dchr1 27145	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → ( 1 ‘𝑘) = 1)
48	47	neeq2d 2995	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → ((𝑋‘𝑘) ≠ ( 1 ‘𝑘) ↔ (𝑋‘𝑘) ≠ 1))
49	26	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → 𝑈 ∈ Fin)
50		eqid 2726	. . . . . . . . . . . . 13 ⊢ (Base‘𝑍) = (Base‘𝑍)
51	4, 5, 9, 50, 8	dchrf 27130	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋:(Base‘𝑍)⟶ℂ)
52	50, 7	unitss 20278	. . . . . . . . . . . . 13 ⊢ 𝑈 ⊆ (Base‘𝑍)
53	52	sseli 3973	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ 𝑈 → 𝑎 ∈ (Base‘𝑍))
54		ffvelcdm 7077	. . . . . . . . . . . 12 ⊢ ((𝑋:(Base‘𝑍)⟶ℂ ∧ 𝑎 ∈ (Base‘𝑍)) → (𝑋‘𝑎) ∈ ℂ)
55	51, 53, 54	syl2an 595	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑈) → (𝑋‘𝑎) ∈ ℂ)
56	55	adantlr 712	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) ∧ 𝑎 ∈ 𝑈) → (𝑋‘𝑎) ∈ ℂ)
57	49, 56	fsumcl 15685	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → Σ𝑎 ∈ 𝑈 (𝑋‘𝑎) ∈ ℂ)
58		0cnd 11211	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → 0 ∈ ℂ)
59	51	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → 𝑋:(Base‘𝑍)⟶ℂ)
60		simprl 768	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → 𝑘 ∈ 𝑈)
61	52, 60	sselid 3975	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → 𝑘 ∈ (Base‘𝑍))
62	59, 61	ffvelcdmd 7081	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → (𝑋‘𝑘) ∈ ℂ)
63		subcl 11463	. . . . . . . . . 10 ⊢ (((𝑋‘𝑘) ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑋‘𝑘) − 1) ∈ ℂ)
64	62, 27, 63	sylancl 585	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → ((𝑋‘𝑘) − 1) ∈ ℂ)
65		simprr 770	. . . . . . . . . 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 585	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → (((𝑋‘𝑘) − 1) = 0 ↔ (𝑋‘𝑘) = 1))
68	67	necon3bid 2979	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → (((𝑋‘𝑘) − 1) ≠ 0 ↔ (𝑋‘𝑘) ≠ 1))
69	65, 68	mpbird 257	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → ((𝑋‘𝑘) − 1) ≠ 0)
70		oveq2 7413	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑎 → (𝑘(._r‘𝑍)𝑥) = (𝑘(._r‘𝑍)𝑎))
71	70	fveq2d 6889	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑎 → (𝑋‘(𝑘(._r‘𝑍)𝑥)) = (𝑋‘(𝑘(._r‘𝑍)𝑎)))
72	71	cbvsumv 15648	. . . . . . . . . . . . . 14 ⊢ Σ𝑥 ∈ 𝑈 (𝑋‘(𝑘(._r‘𝑍)𝑥)) = Σ𝑎 ∈ 𝑈 (𝑋‘(𝑘(._r‘𝑍)𝑎))
73	4, 5, 9	dchrmhm 27129	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐷 ⊆ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld))
74	73, 8	sselid 3975	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)))
75	74	ad2antrr 723	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) ∧ 𝑎 ∈ 𝑈) → 𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)))
76	61	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) ∧ 𝑎 ∈ 𝑈) → 𝑘 ∈ (Base‘𝑍))
77	53	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) ∧ 𝑎 ∈ 𝑈) → 𝑎 ∈ (Base‘𝑍))
78		eqid 2726	. . . . . . . . . . . . . . . . . 18 ⊢ (mulGrp‘𝑍) = (mulGrp‘𝑍)
79	78, 50	mgpbas 20045	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘𝑍) = (Base‘(mulGrp‘𝑍))
80		eqid 2726	. . . . . . . . . . . . . . . . . 18 ⊢ (._r‘𝑍) = (._r‘𝑍)
81	78, 80	mgpplusg 20043	. . . . . . . . . . . . . . . . 17 ⊢ (._r‘𝑍) = (+_g‘(mulGrp‘𝑍))
82		eqid 2726	. . . . . . . . . . . . . . . . . 18 ⊢ (mulGrp‘ℂ_fld) = (mulGrp‘ℂ_fld)
83		cnfldmul 21248	. . . . . . . . . . . . . . . . . 18 ⊢ · = (._r‘ℂ_fld)
84	82, 83	mgpplusg 20043	. . . . . . . . . . . . . . . . 17 ⊢ · = (+_g‘(mulGrp‘ℂ_fld))
85	79, 81, 84	mhmlin 18723	. . . . . . . . . . . . . . . 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 1368	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) ∧ 𝑎 ∈ 𝑈) → (𝑋‘(𝑘(._r‘𝑍)𝑎)) = ((𝑋‘𝑘) · (𝑋‘𝑎)))
87	86	sumeq2dv 15655	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → Σ𝑎 ∈ 𝑈 (𝑋‘(𝑘(._r‘𝑍)𝑎)) = Σ𝑎 ∈ 𝑈 ((𝑋‘𝑘) · (𝑋‘𝑎)))
88	72, 87	eqtrid 2778	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → Σ𝑥 ∈ 𝑈 (𝑋‘(𝑘(._r‘𝑍)𝑥)) = Σ𝑎 ∈ 𝑈 ((𝑋‘𝑘) · (𝑋‘𝑎)))
89		fveq2 6885	. . . . . . . . . . . . . 14 ⊢ (𝑎 = (𝑘(._r‘𝑍)𝑥) → (𝑋‘𝑎) = (𝑋‘(𝑘(._r‘𝑍)𝑥)))
90	11	nnnn0d 12536	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
91	5	zncrng 21439	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℕ₀ → 𝑍 ∈ CRing)
92		crngring 20150	. . . . . . . . . . . . . . . 16 ⊢ (𝑍 ∈ CRing → 𝑍 ∈ Ring)
93		eqid 2726	. . . . . . . . . . . . . . . . 17 ⊢ ((mulGrp‘𝑍) ↾_s 𝑈) = ((mulGrp‘𝑍) ↾_s 𝑈)
94	7, 93	unitgrp 20285	. . . . . . . . . . . . . . . 16 ⊢ (𝑍 ∈ Ring → ((mulGrp‘𝑍) ↾_s 𝑈) ∈ Grp)
95	90, 91, 92, 94	4syl 19	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((mulGrp‘𝑍) ↾_s 𝑈) ∈ Grp)
96		eqid 2726	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 ∈ 𝑈 ↦ (𝑐 ∈ 𝑈 ↦ (𝑏(._r‘𝑍)𝑐))) = (𝑏 ∈ 𝑈 ↦ (𝑐 ∈ 𝑈 ↦ (𝑏(._r‘𝑍)𝑐)))
97	7, 93	unitgrpbas 20284	. . . . . . . . . . . . . . . 16 ⊢ 𝑈 = (Base‘((mulGrp‘𝑍) ↾_s 𝑈))
98	93, 81	ressplusg 17244	. . . . . . . . . . . . . . . . 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 18972	. . . . . . . . . . . . . . 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 682	. . . . . . . . . . . . . 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 18970	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) → (((𝑏 ∈ 𝑈 ↦ (𝑐 ∈ 𝑈 ↦ (𝑏(._r‘𝑍)𝑐)))‘𝑘)‘𝑥) = (𝑘(._r‘𝑍)𝑥))
103	60, 102	sylan 579	. . . . . . . . . . . . . 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 15675	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → Σ𝑎 ∈ 𝑈 (𝑋‘𝑎) = Σ𝑥 ∈ 𝑈 (𝑋‘(𝑘(._r‘𝑍)𝑥)))
105	49, 62, 56	fsummulc2 15736	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → ((𝑋‘𝑘) · Σ𝑎 ∈ 𝑈 (𝑋‘𝑎)) = Σ𝑎 ∈ 𝑈 ((𝑋‘𝑘) · (𝑋‘𝑎)))
106	88, 104, 105	3eqtr4rd 2777	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → ((𝑋‘𝑘) · Σ𝑎 ∈ 𝑈 (𝑋‘𝑎)) = Σ𝑎 ∈ 𝑈 (𝑋‘𝑎))
107	57	mullidd 11236	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → (1 · Σ𝑎 ∈ 𝑈 (𝑋‘𝑎)) = Σ𝑎 ∈ 𝑈 (𝑋‘𝑎))
108	106, 107	oveq12d 7423	. . . . . . . . . . 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 2766	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → (((𝑋‘𝑘) · Σ𝑎 ∈ 𝑈 (𝑋‘𝑎)) − (1 · Σ𝑎 ∈ 𝑈 (𝑋‘𝑎))) = 0)
111		1cnd 11213	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → 1 ∈ ℂ)
112	62, 111, 57	subdird 11675	. . . . . . . . . 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 11417	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → (((𝑋‘𝑘) − 1) · 0) = 0)
114	110, 112, 113	3eqtr4d 2776	. . . . . . . . 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 11853	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → Σ𝑎 ∈ 𝑈 (𝑋‘𝑎) = 0)
116	115	expr 456	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → ((𝑋‘𝑘) ≠ 1 → Σ𝑎 ∈ 𝑈 (𝑋‘𝑎) = 0))
117	48, 116	sylbid 239	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → ((𝑋‘𝑘) ≠ ( 1 ‘𝑘) → Σ𝑎 ∈ 𝑈 (𝑋‘𝑎) = 0))
118	44, 117	biimtrrid 242	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → (¬ (𝑋‘𝑘) = ( 1 ‘𝑘) → Σ𝑎 ∈ 𝑈 (𝑋‘𝑎) = 0))
119	118	rexlimdva 3149	. . . 4 ⊢ (𝜑 → (∃𝑘 ∈ 𝑈 ¬ (𝑋‘𝑘) = ( 1 ‘𝑘) → Σ𝑎 ∈ 𝑈 (𝑋‘𝑎) = 0))
120	43, 119	sylbid 239	. . 3 ⊢ (𝜑 → (¬ 𝑋 = 1 → Σ𝑎 ∈ 𝑈 (𝑋‘𝑎) = 0))
121	120	imp 406	. 2 ⊢ ((𝜑 ∧ ¬ 𝑋 = 1 ) → Σ𝑎 ∈ 𝑈 (𝑋‘𝑎) = 0)
122	1, 2, 35, 121	ifbothda 4561	1 ⊢ (𝜑 → Σ𝑎 ∈ 𝑈 (𝑋‘𝑎) = if(𝑋 = 1 , (ϕ‘𝑁), 0))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ≠ wne 2934 ∀wral 3055 ∃wrex 3064 Vcvv 3468 ifcif 4523 ↦ cmpt 5224 ⟶wf 6533 –1-1-onto→wf1o 6536 ‘cfv 6537 (class class class)co 7405 Fincfn 8941 ℂcc 11110 0cc0 11112 1c1 11113 · cmul 11117 − cmin 11448 ℕcn 12216 ℕ₀cn0 12476 ♯chash 14295 Σcsu 15638 ϕcphi 16706 Basecbs 17153 ↾_s cress 17182 +_gcplusg 17206 ._rcmulr 17207 0_gc0g 17394 MndHom cmhm 18711 Grpcgrp 18863 Abelcabl 19701 mulGrpcmgp 20039 Ringcrg 20138 CRingccrg 20139 Unitcui 20257 ℂ_fldccnfld 21240 ℤ/nℤczn 21389 DChrcdchr 27120

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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 ax-addf 11191 ax-mulf 11192

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7667 df-om 7853 df-1st 7974 df-2nd 7975 df-tpos 8212 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-oadd 8471 df-er 8705 df-ec 8707 df-qs 8711 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 df-inf 9440 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-xnn0 12549 df-z 12563 df-dec 12682 df-uz 12827 df-rp 12981 df-fz 13491 df-fzo 13634 df-fl 13763 df-mod 13841 df-seq 13973 df-exp 14033 df-hash 14296 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-clim 15438 df-sum 15639 df-dvds 16205 df-gcd 16443 df-phi 16708 df-struct 17089 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-mulr 17220 df-starv 17221 df-sca 17222 df-vsca 17223 df-ip 17224 df-tset 17225 df-ple 17226 df-ds 17228 df-unif 17229 df-0g 17396 df-imas 17463 df-qus 17464 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-mhm 18713 df-grp 18866 df-minusg 18867 df-sbg 18868 df-mulg 18996 df-subg 19050 df-nsg 19051 df-eqg 19052 df-ghm 19139 df-cmn 19702 df-abl 19703 df-mgp 20040 df-rng 20058 df-ur 20087 df-ring 20140 df-cring 20141 df-oppr 20236 df-dvdsr 20259 df-unit 20260 df-invr 20290 df-rhm 20374 df-subrng 20446 df-subrg 20471 df-lmod 20708 df-lss 20779 df-lsp 20819 df-sra 21021 df-rgmod 21022 df-lidl 21067 df-rsp 21068 df-2idl 21107 df-cnfld 21241 df-zring 21334 df-zrh 21390 df-zn 21393 df-dchr 27121

This theorem is referenced by: dchrsum 27157

W3C validator