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Theorem dchrsum2 27298

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 2764	. 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 2764	. 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 6851	. . . . . 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 27270	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝐷 → 𝑁 ∈ ℕ)
11	8, 10	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ)
12	11	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑈) → 𝑁 ∈ ℕ)
13		simpr 487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑈) → 𝑎 ∈ 𝑈)
14	4, 5, 6, 7, 12, 13	dchr1 27287	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑈) → ( 1 ‘𝑎) = 1)
15	3, 14	sylan9eqr 2809	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑋 = 1 ) → (𝑋‘𝑎) = 1)
16	15	an32s 660	. . . 4 ⊢ (((𝜑 ∧ 𝑋 = 1 ) ∧ 𝑎 ∈ 𝑈) → (𝑋‘𝑎) = 1)
17	16	sumeq2dv 15701	. . 3 ⊢ ((𝜑 ∧ 𝑋 = 1 ) → Σ𝑎 ∈ 𝑈 (𝑋‘𝑎) = Σ𝑎 ∈ 𝑈 1)
18	5, 7	znunithash 21585	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (♯‘𝑈) = (ϕ‘𝑁))
19	11, 18	syl 17	. . . . . . . 8 ⊢ (𝜑 → (♯‘𝑈) = (ϕ‘𝑁))
20	11	phicld 16779	. . . . . . . . 9 ⊢ (𝜑 → (ϕ‘𝑁) ∈ ℕ)
21	20	nnnn0d 12528	. . . . . . . 8 ⊢ (𝜑 → (ϕ‘𝑁) ∈ ℕ₀)
22	19, 21	eqeltrd 2852	. . . . . . 7 ⊢ (𝜑 → (♯‘𝑈) ∈ ℕ₀)
23	7	fvexi 6866	. . . . . . . 8 ⊢ 𝑈 ∈ V
24		hashclb 14357	. . . . . . . 8 ⊢ (𝑈 ∈ V → (𝑈 ∈ Fin ↔ (♯‘𝑈) ∈ ℕ₀))
25	23, 24	ax-mp 5	. . . . . . 7 ⊢ (𝑈 ∈ Fin ↔ (♯‘𝑈) ∈ ℕ₀)
26	22, 25	sylibr 236	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ Fin)
27		ax-1cn 11117	. . . . . 6 ⊢ 1 ∈ ℂ
28		fsumconst 15789	. . . . . 6 ⊢ ((𝑈 ∈ Fin ∧ 1 ∈ ℂ) → Σ𝑎 ∈ 𝑈 1 = ((♯‘𝑈) · 1))
29	26, 27, 28	sylancl 594	. . . . 5 ⊢ (𝜑 → Σ𝑎 ∈ 𝑈 1 = ((♯‘𝑈) · 1))
30	19	oveq1d 7396	. . . . 5 ⊢ (𝜑 → ((♯‘𝑈) · 1) = ((ϕ‘𝑁) · 1))
31	20	nncnd 12212	. . . . . 6 ⊢ (𝜑 → (ϕ‘𝑁) ∈ ℂ)
32	31	mulridd 11185	. . . . 5 ⊢ (𝜑 → ((ϕ‘𝑁) · 1) = (ϕ‘𝑁))
33	29, 30, 32	3eqtrd 2791	. . . 4 ⊢ (𝜑 → Σ𝑎 ∈ 𝑈 1 = (ϕ‘𝑁))
34	33	adantr 483	. . 3 ⊢ ((𝜑 ∧ 𝑋 = 1 ) → Σ𝑎 ∈ 𝑈 1 = (ϕ‘𝑁))
35	17, 34	eqtrd 2787	. 2 ⊢ ((𝜑 ∧ 𝑋 = 1 ) → Σ𝑎 ∈ 𝑈 (𝑋‘𝑎) = (ϕ‘𝑁))
36	4	dchrabl 27284	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 𝐺 ∈ Abel)
37		ablgrp 19797	. . . . . . . 8 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
38	9, 6	grpidcl 18979	. . . . . . . 8 ⊢ (𝐺 ∈ Grp → 1 ∈ 𝐷)
39	11, 36, 37, 38	4syl 19	. . . . . . 7 ⊢ (𝜑 → 1 ∈ 𝐷)
40	4, 5, 9, 7, 8, 39	dchreq 27288	. . . . . 6 ⊢ (𝜑 → (𝑋 = 1 ↔ ∀𝑘 ∈ 𝑈 (𝑋‘𝑘) = ( 1 ‘𝑘)))
41	40	notbid 320	. . . . 5 ⊢ (𝜑 → (¬ 𝑋 = 1 ↔ ¬ ∀𝑘 ∈ 𝑈 (𝑋‘𝑘) = ( 1 ‘𝑘)))
42		rexnal 3104	. . . . 5 ⊢ (∃𝑘 ∈ 𝑈 ¬ (𝑋‘𝑘) = ( 1 ‘𝑘) ↔ ¬ ∀𝑘 ∈ 𝑈 (𝑋‘𝑘) = ( 1 ‘𝑘))
43	41, 42	bitr4di 291	. . . 4 ⊢ (𝜑 → (¬ 𝑋 = 1 ↔ ∃𝑘 ∈ 𝑈 ¬ (𝑋‘𝑘) = ( 1 ‘𝑘)))
44		df-ne 2948	. . . . . 6 ⊢ ((𝑋‘𝑘) ≠ ( 1 ‘𝑘) ↔ ¬ (𝑋‘𝑘) = ( 1 ‘𝑘))
45	11	adantr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → 𝑁 ∈ ℕ)
46		simpr 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → 𝑘 ∈ 𝑈)
47	4, 5, 6, 7, 45, 46	dchr1 27287	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → ( 1 ‘𝑘) = 1)
48	47	neeq2d 3007	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → ((𝑋‘𝑘) ≠ ( 1 ‘𝑘) ↔ (𝑋‘𝑘) ≠ 1))
49	26	adantr 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → 𝑈 ∈ Fin)
50		eqid 2752	. . . . . . . . . . . . 13 ⊢ (Base‘𝑍) = (Base‘𝑍)
51	4, 5, 9, 50, 8	dchrf 27272	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋:(Base‘𝑍)⟶ℂ)
52	50, 7	unitss 20393	. . . . . . . . . . . . 13 ⊢ 𝑈 ⊆ (Base‘𝑍)
53	52	sseli 3923	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ 𝑈 → 𝑎 ∈ (Base‘𝑍))
54		ffvelcdm 7047	. . . . . . . . . . . 12 ⊢ ((𝑋:(Base‘𝑍)⟶ℂ ∧ 𝑎 ∈ (Base‘𝑍)) → (𝑋‘𝑎) ∈ ℂ)
55	51, 53, 54	syl2an 604	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑈) → (𝑋‘𝑎) ∈ ℂ)
56	55	adantlr 723	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) ∧ 𝑎 ∈ 𝑈) → (𝑋‘𝑎) ∈ ℂ)
57	49, 56	fsumcl 15732	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → Σ𝑎 ∈ 𝑈 (𝑋‘𝑎) ∈ ℂ)
58		0cnd 11158	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → 0 ∈ ℂ)
59	51	adantr 483	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → 𝑋:(Base‘𝑍)⟶ℂ)
60		simprl 778	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → 𝑘 ∈ 𝑈)
61	52, 60	sselid 3925	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → 𝑘 ∈ (Base‘𝑍))
62	59, 61	ffvelcdmd 7051	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → (𝑋‘𝑘) ∈ ℂ)
63		subcl 11415	. . . . . . . . . 10 ⊢ (((𝑋‘𝑘) ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑋‘𝑘) − 1) ∈ ℂ)
64	62, 27, 63	sylancl 594	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → ((𝑋‘𝑘) − 1) ∈ ℂ)
65		simprr 780	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → (𝑋‘𝑘) ≠ 1)
66		subeq0 11443	. . . . . . . . . . . 12 ⊢ (((𝑋‘𝑘) ∈ ℂ ∧ 1 ∈ ℂ) → (((𝑋‘𝑘) − 1) = 0 ↔ (𝑋‘𝑘) = 1))
67	62, 27, 66	sylancl 594	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → (((𝑋‘𝑘) − 1) = 0 ↔ (𝑋‘𝑘) = 1))
68	67	necon3bid 2991	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → (((𝑋‘𝑘) − 1) ≠ 0 ↔ (𝑋‘𝑘) ≠ 1))
69	65, 68	mpbird 259	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → ((𝑋‘𝑘) − 1) ≠ 0)
70		oveq2 7389	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑎 → (𝑘(._r‘𝑍)𝑥) = (𝑘(._r‘𝑍)𝑎))
71	70	fveq2d 6856	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑎 → (𝑋‘(𝑘(._r‘𝑍)𝑥)) = (𝑋‘(𝑘(._r‘𝑍)𝑎)))
72	71	cbvsumv 15695	. . . . . . . . . . . . . 14 ⊢ Σ𝑥 ∈ 𝑈 (𝑋‘(𝑘(._r‘𝑍)𝑥)) = Σ𝑎 ∈ 𝑈 (𝑋‘(𝑘(._r‘𝑍)𝑎))
73	4, 5, 9	dchrmhm 27271	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐷 ⊆ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld))
74	73, 8	sselid 3925	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)))
75	74	ad2antrr 734	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) ∧ 𝑎 ∈ 𝑈) → 𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)))
76	61	adantr 483	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) ∧ 𝑎 ∈ 𝑈) → 𝑘 ∈ (Base‘𝑍))
77	53	adantl 484	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) ∧ 𝑎 ∈ 𝑈) → 𝑎 ∈ (Base‘𝑍))
78		eqid 2752	. . . . . . . . . . . . . . . . . 18 ⊢ (mulGrp‘𝑍) = (mulGrp‘𝑍)
79	78, 50	mgpbas 20163	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘𝑍) = (Base‘(mulGrp‘𝑍))
80		eqid 2752	. . . . . . . . . . . . . . . . . 18 ⊢ (._r‘𝑍) = (._r‘𝑍)
81	78, 80	mgpplusg 20162	. . . . . . . . . . . . . . . . 17 ⊢ (._r‘𝑍) = (+_g‘(mulGrp‘𝑍))
82		eqid 2752	. . . . . . . . . . . . . . . . . 18 ⊢ (mulGrp‘ℂ_fld) = (mulGrp‘ℂ_fld)
83		cnfldmul 21401	. . . . . . . . . . . . . . . . . 18 ⊢ · = (._r‘ℂ_fld)
84	82, 83	mgpplusg 20162	. . . . . . . . . . . . . . . . 17 ⊢ · = (+_g‘(mulGrp‘ℂ_fld))
85	79, 81, 84	mhmlin 18799	. . . . . . . . . . . . . . . 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 1382	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) ∧ 𝑎 ∈ 𝑈) → (𝑋‘(𝑘(._r‘𝑍)𝑎)) = ((𝑋‘𝑘) · (𝑋‘𝑎)))
87	86	sumeq2dv 15701	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → Σ𝑎 ∈ 𝑈 (𝑋‘(𝑘(._r‘𝑍)𝑎)) = Σ𝑎 ∈ 𝑈 ((𝑋‘𝑘) · (𝑋‘𝑎)))
88	72, 87	eqtrid 2799	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → Σ𝑥 ∈ 𝑈 (𝑋‘(𝑘(._r‘𝑍)𝑥)) = Σ𝑎 ∈ 𝑈 ((𝑋‘𝑘) · (𝑋‘𝑎)))
89		fveq2 6852	. . . . . . . . . . . . . 14 ⊢ (𝑎 = (𝑘(._r‘𝑍)𝑥) → (𝑋‘𝑎) = (𝑋‘(𝑘(._r‘𝑍)𝑥)))
90	11	nnnn0d 12528	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
91	5	zncrng 21565	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℕ₀ → 𝑍 ∈ CRing)
92		crngring 20263	. . . . . . . . . . . . . . . 16 ⊢ (𝑍 ∈ CRing → 𝑍 ∈ Ring)
93		eqid 2752	. . . . . . . . . . . . . . . . 17 ⊢ ((mulGrp‘𝑍) ↾_s 𝑈) = ((mulGrp‘𝑍) ↾_s 𝑈)
94	7, 93	unitgrp 20400	. . . . . . . . . . . . . . . 16 ⊢ (𝑍 ∈ Ring → ((mulGrp‘𝑍) ↾_s 𝑈) ∈ Grp)
95	90, 91, 92, 94	4syl 19	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((mulGrp‘𝑍) ↾_s 𝑈) ∈ Grp)
96		eqid 2752	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 ∈ 𝑈 ↦ (𝑐 ∈ 𝑈 ↦ (𝑏(._r‘𝑍)𝑐))) = (𝑏 ∈ 𝑈 ↦ (𝑐 ∈ 𝑈 ↦ (𝑏(._r‘𝑍)𝑐)))
97	7, 93	unitgrpbas 20399	. . . . . . . . . . . . . . . 16 ⊢ 𝑈 = (Base‘((mulGrp‘𝑍) ↾_s 𝑈))
98	93, 81	ressplusg 17292	. . . . . . . . . . . . . . . . 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 19058	. . . . . . . . . . . . . . 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 693	. . . . . . . . . . . . . 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 19056	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) → (((𝑏 ∈ 𝑈 ↦ (𝑐 ∈ 𝑈 ↦ (𝑏(._r‘𝑍)𝑐)))‘𝑘)‘𝑥) = (𝑘(._r‘𝑍)𝑥))
103	60, 102	sylan 588	. . . . . . . . . . . . . 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 15722	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → Σ𝑎 ∈ 𝑈 (𝑋‘𝑎) = Σ𝑥 ∈ 𝑈 (𝑋‘(𝑘(._r‘𝑍)𝑥)))
105	49, 62, 56	fsummulc2 15783	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → ((𝑋‘𝑘) · Σ𝑎 ∈ 𝑈 (𝑋‘𝑎)) = Σ𝑎 ∈ 𝑈 ((𝑋‘𝑘) · (𝑋‘𝑎)))
106	88, 104, 105	3eqtr4rd 2798	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → ((𝑋‘𝑘) · Σ𝑎 ∈ 𝑈 (𝑋‘𝑎)) = Σ𝑎 ∈ 𝑈 (𝑋‘𝑎))
107	57	mullidd 11186	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → (1 · Σ𝑎 ∈ 𝑈 (𝑋‘𝑎)) = Σ𝑎 ∈ 𝑈 (𝑋‘𝑎))
108	106, 107	oveq12d 7399	. . . . . . . . . . 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 11516	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → (Σ𝑎 ∈ 𝑈 (𝑋‘𝑎) − Σ𝑎 ∈ 𝑈 (𝑋‘𝑎)) = 0)
110	108, 109	eqtrd 2787	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → (((𝑋‘𝑘) · Σ𝑎 ∈ 𝑈 (𝑋‘𝑎)) − (1 · Σ𝑎 ∈ 𝑈 (𝑋‘𝑎))) = 0)
111		1cnd 11161	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → 1 ∈ ℂ)
112	62, 111, 57	subdird 11630	. . . . . . . . . 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 11368	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → (((𝑋‘𝑘) − 1) · 0) = 0)
114	110, 112, 113	3eqtr4d 2797	. . . . . . . . 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 11808	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → Σ𝑎 ∈ 𝑈 (𝑋‘𝑎) = 0)
116	115	expr 459	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → ((𝑋‘𝑘) ≠ 1 → Σ𝑎 ∈ 𝑈 (𝑋‘𝑎) = 0))
117	48, 116	sylbid 242	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → ((𝑋‘𝑘) ≠ ( 1 ‘𝑘) → Σ𝑎 ∈ 𝑈 (𝑋‘𝑎) = 0))
118	44, 117	biimtrrid 245	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → (¬ (𝑋‘𝑘) = ( 1 ‘𝑘) → Σ𝑎 ∈ 𝑈 (𝑋‘𝑎) = 0))
119	118	rexlimdva 3153	. . . 4 ⊢ (𝜑 → (∃𝑘 ∈ 𝑈 ¬ (𝑋‘𝑘) = ( 1 ‘𝑘) → Σ𝑎 ∈ 𝑈 (𝑋‘𝑎) = 0))
120	43, 119	sylbid 242	. . 3 ⊢ (𝜑 → (¬ 𝑋 = 1 → Σ𝑎 ∈ 𝑈 (𝑋‘𝑎) = 0))
121	120	imp 409	. 2 ⊢ ((𝜑 ∧ ¬ 𝑋 = 1 ) → Σ𝑎 ∈ 𝑈 (𝑋‘𝑎) = 0)
122	1, 2, 35, 121	ifbothda 4509	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 208 ∧ wa 398 = wceq 1550 ∈ wcel 2132 ≠ wne 2947 ∀wral 3066 ∃wrex 3076 Vcvv 3444 ifcif 4470 ↦ cmpt 5171 ⟶wf 6502 –1-1-onto→wf1o 6505 ‘cfv 6506 (class class class)co 7381 Fincfn 8912 ℂcc 11057 0cc0 11059 1c1 11060 · cmul 11064 − cmin 11400 ℕcn 12196 ℕ₀cn0 12467 ♯chash 14329 Σcsu 15685 ϕcphi 16771 Basecbs 17217 ↾_s cress 17238 +_gcplusg 17258 ._rcmulr 17259 0_gc0g 17440 MndHom cmhm 18787 Grpcgrp 18947 Abelcabl 19793 mulGrpcmgp 20158 Ringcrg 20251 CRingccrg 20252 Unitcui 20372 ℂ_fldccnfld 21393 ℤ/nℤczn 21523 DChrcdchr 27262

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1805 ax-4 1819 ax-5 1920 ax-6 1977 ax-7 2018 ax-8 2134 ax-9 2142 ax-10 2165 ax-11 2181 ax-12 2202 ax-ext 2724 ax-rep 5217 ax-sep 5236 ax-nul 5246 ax-pow 5312 ax-pr 5380 ax-un 7703 ax-inf2 9582 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 ax-pre-sup 11137 ax-addf 11138 ax-mulf 11139

This theorem depends on definitions: df-bi 209 df-an 399 df-or 857 df-3or 1096 df-3an 1097 df-tru 1553 df-fal 1563 df-ex 1790 df-nf 1794 df-sb 2081 df-mo 2556 df-eu 2586 df-clab 2731 df-cleq 2744 df-clel 2827 df-nfc 2901 df-ne 2948 df-nel 3052 df-ral 3067 df-rex 3077 df-rmo 3357 df-reu 3358 df-rab 3405 df-v 3446 df-sbc 3736 df-csb 3844 df-dif 3898 df-un 3900 df-in 3902 df-ss 3912 df-pss 3915 df-nul 4277 df-if 4471 df-pw 4547 df-sn 4573 df-pr 4575 df-tp 4577 df-op 4579 df-uni 4856 df-int 4896 df-iun 4941 df-br 5091 df-opab 5153 df-mpt 5172 df-tr 5198 df-id 5531 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5589 df-se 5590 df-we 5591 df-xp 5642 df-rel 5643 df-cnv 5644 df-co 5645 df-dm 5646 df-rn 5647 df-res 5648 df-ima 5649 df-pred 6273 df-ord 6334 df-on 6335 df-lim 6336 df-suc 6337 df-iota 6462 df-fun 6508 df-fn 6509 df-f 6510 df-f1 6511 df-fo 6512 df-f1o 6513 df-fv 6514 df-isom 6515 df-riota 7338 df-ov 7384 df-oprab 7385 df-mpo 7386 df-of 7645 df-om 7832 df-1st 7955 df-2nd 7956 df-tpos 8190 df-frecs 8246 df-wrecs 8277 df-recs 8326 df-rdg 8365 df-1o 8421 df-oadd 8425 df-er 8662 df-ec 8664 df-qs 8668 df-map 8794 df-en 8913 df-dom 8914 df-sdom 8915 df-fin 8916 df-sup 9374 df-inf 9375 df-oi 9444 df-card 9883 df-pnf 11204 df-mnf 11205 df-xr 11206 df-ltxr 11207 df-le 11208 df-sub 11402 df-neg 11403 df-div 11831 df-nn 12197 df-2 12266 df-3 12267 df-4 12268 df-5 12269 df-6 12270 df-7 12271 df-8 12272 df-9 12273 df-n0 12468 df-xnn0 12541 df-z 12555 df-dec 12675 df-uz 12826 df-rp 12980 df-fz 13499 df-fzo 13646 df-fl 13788 df-mod 13866 df-seq 14001 df-exp 14061 df-hash 14330 df-cj 15098 df-re 15099 df-im 15100 df-sqrt 15234 df-abs 15235 df-clim 15487 df-sum 15686 df-dvds 16259 df-gcd 16501 df-phi 16773 df-struct 17155 df-sets 17172 df-slot 17190 df-ndx 17202 df-base 17218 df-ress 17239 df-plusg 17271 df-mulr 17272 df-starv 17273 df-sca 17274 df-vsca 17275 df-ip 17276 df-tset 17277 df-ple 17278 df-ds 17280 df-unif 17281 df-0g 17442 df-imas 17510 df-qus 17511 df-mgm 18646 df-sgrp 18725 df-mnd 18741 df-mhm 18789 df-grp 18950 df-minusg 18951 df-sbg 18952 df-mulg 19082 df-subg 19137 df-nsg 19138 df-eqg 19139 df-ghm 19226 df-cmn 19794 df-abl 19795 df-mgp 20159 df-rng 20171 df-ur 20200 df-ring 20253 df-cring 20254 df-oppr 20354 df-dvdsr 20374 df-unit 20375 df-invr 20405 df-rhm 20489 df-subrng 20564 df-subrg 20588 df-lmod 20898 df-lss 20968 df-lsp 21008 df-sra 21209 df-rgmod 21210 df-lidl 21247 df-rsp 21248 df-2idl 21289 df-cnfld 21394 df-zring 21468 df-zrh 21524 df-zn 21527 df-dchr 27263

This theorem is referenced by: dchrsum 27299

W3C validator