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Theorem dchrsum2 27231

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 2749	. 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 2749	. 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 6840	. . . . . 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 27203	. . . . . . . . 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 27220	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑈) → ( 1 ‘𝑎) = 1)
15	3, 14	sylan9eqr 2794	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑋 = 1 ) → (𝑋‘𝑎) = 1)
16	15	an32s 653	. . . 4 ⊢ (((𝜑 ∧ 𝑋 = 1 ) ∧ 𝑎 ∈ 𝑈) → (𝑋‘𝑎) = 1)
17	16	sumeq2dv 15664	. . 3 ⊢ ((𝜑 ∧ 𝑋 = 1 ) → Σ𝑎 ∈ 𝑈 (𝑋‘𝑎) = Σ𝑎 ∈ 𝑈 1)
18	5, 7	znunithash 21544	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (♯‘𝑈) = (ϕ‘𝑁))
19	11, 18	syl 17	. . . . . . . 8 ⊢ (𝜑 → (♯‘𝑈) = (ϕ‘𝑁))
20	11	phicld 16742	. . . . . . . . 9 ⊢ (𝜑 → (ϕ‘𝑁) ∈ ℕ)
21	20	nnnn0d 12498	. . . . . . . 8 ⊢ (𝜑 → (ϕ‘𝑁) ∈ ℕ₀)
22	19, 21	eqeltrd 2837	. . . . . . 7 ⊢ (𝜑 → (♯‘𝑈) ∈ ℕ₀)
23	7	fvexi 6855	. . . . . . . 8 ⊢ 𝑈 ∈ V
24		hashclb 14320	. . . . . . . 8 ⊢ (𝑈 ∈ V → (𝑈 ∈ Fin ↔ (♯‘𝑈) ∈ ℕ₀))
25	23, 24	ax-mp 5	. . . . . . 7 ⊢ (𝑈 ∈ Fin ↔ (♯‘𝑈) ∈ ℕ₀)
26	22, 25	sylibr 234	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ Fin)
27		ax-1cn 11096	. . . . . 6 ⊢ 1 ∈ ℂ
28		fsumconst 15752	. . . . . 6 ⊢ ((𝑈 ∈ Fin ∧ 1 ∈ ℂ) → Σ𝑎 ∈ 𝑈 1 = ((♯‘𝑈) · 1))
29	26, 27, 28	sylancl 587	. . . . 5 ⊢ (𝜑 → Σ𝑎 ∈ 𝑈 1 = ((♯‘𝑈) · 1))
30	19	oveq1d 7382	. . . . 5 ⊢ (𝜑 → ((♯‘𝑈) · 1) = ((ϕ‘𝑁) · 1))
31	20	nncnd 12190	. . . . . 6 ⊢ (𝜑 → (ϕ‘𝑁) ∈ ℂ)
32	31	mulridd 11162	. . . . 5 ⊢ (𝜑 → ((ϕ‘𝑁) · 1) = (ϕ‘𝑁))
33	29, 30, 32	3eqtrd 2776	. . . 4 ⊢ (𝜑 → Σ𝑎 ∈ 𝑈 1 = (ϕ‘𝑁))
34	33	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑋 = 1 ) → Σ𝑎 ∈ 𝑈 1 = (ϕ‘𝑁))
35	17, 34	eqtrd 2772	. 2 ⊢ ((𝜑 ∧ 𝑋 = 1 ) → Σ𝑎 ∈ 𝑈 (𝑋‘𝑎) = (ϕ‘𝑁))
36	4	dchrabl 27217	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 𝐺 ∈ Abel)
37		ablgrp 19760	. . . . . . . 8 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
38	9, 6	grpidcl 18941	. . . . . . . 8 ⊢ (𝐺 ∈ Grp → 1 ∈ 𝐷)
39	11, 36, 37, 38	4syl 19	. . . . . . 7 ⊢ (𝜑 → 1 ∈ 𝐷)
40	4, 5, 9, 7, 8, 39	dchreq 27221	. . . . . 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 3090	. . . . 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 2934	. . . . . 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 27220	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → ( 1 ‘𝑘) = 1)
48	47	neeq2d 2993	. . . . . . 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 2737	. . . . . . . . . . . . 13 ⊢ (Base‘𝑍) = (Base‘𝑍)
51	4, 5, 9, 50, 8	dchrf 27205	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋:(Base‘𝑍)⟶ℂ)
52	50, 7	unitss 20356	. . . . . . . . . . . . 13 ⊢ 𝑈 ⊆ (Base‘𝑍)
53	52	sseli 3918	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ 𝑈 → 𝑎 ∈ (Base‘𝑍))
54		ffvelcdm 7034	. . . . . . . . . . . 12 ⊢ ((𝑋:(Base‘𝑍)⟶ℂ ∧ 𝑎 ∈ (Base‘𝑍)) → (𝑋‘𝑎) ∈ ℂ)
55	51, 53, 54	syl2an 597	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑈) → (𝑋‘𝑎) ∈ ℂ)
56	55	adantlr 716	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) ∧ 𝑎 ∈ 𝑈) → (𝑋‘𝑎) ∈ ℂ)
57	49, 56	fsumcl 15695	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → Σ𝑎 ∈ 𝑈 (𝑋‘𝑎) ∈ ℂ)
58		0cnd 11137	. . . . . . . . 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 771	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → 𝑘 ∈ 𝑈)
61	52, 60	sselid 3920	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → 𝑘 ∈ (Base‘𝑍))
62	59, 61	ffvelcdmd 7038	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → (𝑋‘𝑘) ∈ ℂ)
63		subcl 11392	. . . . . . . . . 10 ⊢ (((𝑋‘𝑘) ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑋‘𝑘) − 1) ∈ ℂ)
64	62, 27, 63	sylancl 587	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → ((𝑋‘𝑘) − 1) ∈ ℂ)
65		simprr 773	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → (𝑋‘𝑘) ≠ 1)
66		subeq0 11420	. . . . . . . . . . . 12 ⊢ (((𝑋‘𝑘) ∈ ℂ ∧ 1 ∈ ℂ) → (((𝑋‘𝑘) − 1) = 0 ↔ (𝑋‘𝑘) = 1))
67	62, 27, 66	sylancl 587	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → (((𝑋‘𝑘) − 1) = 0 ↔ (𝑋‘𝑘) = 1))
68	67	necon3bid 2977	. . . . . . . . . 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 7375	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑎 → (𝑘(._r‘𝑍)𝑥) = (𝑘(._r‘𝑍)𝑎))
71	70	fveq2d 6845	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑎 → (𝑋‘(𝑘(._r‘𝑍)𝑥)) = (𝑋‘(𝑘(._r‘𝑍)𝑎)))
72	71	cbvsumv 15658	. . . . . . . . . . . . . 14 ⊢ Σ𝑥 ∈ 𝑈 (𝑋‘(𝑘(._r‘𝑍)𝑥)) = Σ𝑎 ∈ 𝑈 (𝑋‘(𝑘(._r‘𝑍)𝑎))
73	4, 5, 9	dchrmhm 27204	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐷 ⊆ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld))
74	73, 8	sselid 3920	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)))
75	74	ad2antrr 727	. . . . . . . . . . . . . . . 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 2737	. . . . . . . . . . . . . . . . . 18 ⊢ (mulGrp‘𝑍) = (mulGrp‘𝑍)
79	78, 50	mgpbas 20126	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘𝑍) = (Base‘(mulGrp‘𝑍))
80		eqid 2737	. . . . . . . . . . . . . . . . . 18 ⊢ (._r‘𝑍) = (._r‘𝑍)
81	78, 80	mgpplusg 20125	. . . . . . . . . . . . . . . . 17 ⊢ (._r‘𝑍) = (+_g‘(mulGrp‘𝑍))
82		eqid 2737	. . . . . . . . . . . . . . . . . 18 ⊢ (mulGrp‘ℂ_fld) = (mulGrp‘ℂ_fld)
83		cnfldmul 21360	. . . . . . . . . . . . . . . . . 18 ⊢ · = (._r‘ℂ_fld)
84	82, 83	mgpplusg 20125	. . . . . . . . . . . . . . . . 17 ⊢ · = (+_g‘(mulGrp‘ℂ_fld))
85	79, 81, 84	mhmlin 18761	. . . . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) ∧ 𝑎 ∈ 𝑈) → (𝑋‘(𝑘(._r‘𝑍)𝑎)) = ((𝑋‘𝑘) · (𝑋‘𝑎)))
87	86	sumeq2dv 15664	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → Σ𝑎 ∈ 𝑈 (𝑋‘(𝑘(._r‘𝑍)𝑎)) = Σ𝑎 ∈ 𝑈 ((𝑋‘𝑘) · (𝑋‘𝑎)))
88	72, 87	eqtrid 2784	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → Σ𝑥 ∈ 𝑈 (𝑋‘(𝑘(._r‘𝑍)𝑥)) = Σ𝑎 ∈ 𝑈 ((𝑋‘𝑘) · (𝑋‘𝑎)))
89		fveq2 6841	. . . . . . . . . . . . . 14 ⊢ (𝑎 = (𝑘(._r‘𝑍)𝑥) → (𝑋‘𝑎) = (𝑋‘(𝑘(._r‘𝑍)𝑥)))
90	11	nnnn0d 12498	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
91	5	zncrng 21524	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℕ₀ → 𝑍 ∈ CRing)
92		crngring 20226	. . . . . . . . . . . . . . . 16 ⊢ (𝑍 ∈ CRing → 𝑍 ∈ Ring)
93		eqid 2737	. . . . . . . . . . . . . . . . 17 ⊢ ((mulGrp‘𝑍) ↾_s 𝑈) = ((mulGrp‘𝑍) ↾_s 𝑈)
94	7, 93	unitgrp 20363	. . . . . . . . . . . . . . . 16 ⊢ (𝑍 ∈ Ring → ((mulGrp‘𝑍) ↾_s 𝑈) ∈ Grp)
95	90, 91, 92, 94	4syl 19	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((mulGrp‘𝑍) ↾_s 𝑈) ∈ Grp)
96		eqid 2737	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 ∈ 𝑈 ↦ (𝑐 ∈ 𝑈 ↦ (𝑏(._r‘𝑍)𝑐))) = (𝑏 ∈ 𝑈 ↦ (𝑐 ∈ 𝑈 ↦ (𝑏(._r‘𝑍)𝑐)))
97	7, 93	unitgrpbas 20362	. . . . . . . . . . . . . . . 16 ⊢ 𝑈 = (Base‘((mulGrp‘𝑍) ↾_s 𝑈))
98	93, 81	ressplusg 17254	. . . . . . . . . . . . . . . . 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 19020	. . . . . . . . . . . . . . 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 686	. . . . . . . . . . . . . 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 19018	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) → (((𝑏 ∈ 𝑈 ↦ (𝑐 ∈ 𝑈 ↦ (𝑏(._r‘𝑍)𝑐)))‘𝑘)‘𝑥) = (𝑘(._r‘𝑍)𝑥))
103	60, 102	sylan 581	. . . . . . . . . . . . . 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 15685	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → Σ𝑎 ∈ 𝑈 (𝑋‘𝑎) = Σ𝑥 ∈ 𝑈 (𝑋‘(𝑘(._r‘𝑍)𝑥)))
105	49, 62, 56	fsummulc2 15746	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → ((𝑋‘𝑘) · Σ𝑎 ∈ 𝑈 (𝑋‘𝑎)) = Σ𝑎 ∈ 𝑈 ((𝑋‘𝑘) · (𝑋‘𝑎)))
106	88, 104, 105	3eqtr4rd 2783	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → ((𝑋‘𝑘) · Σ𝑎 ∈ 𝑈 (𝑋‘𝑎)) = Σ𝑎 ∈ 𝑈 (𝑋‘𝑎))
107	57	mullidd 11163	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → (1 · Σ𝑎 ∈ 𝑈 (𝑋‘𝑎)) = Σ𝑎 ∈ 𝑈 (𝑋‘𝑎))
108	106, 107	oveq12d 7385	. . . . . . . . . . 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 11493	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → (Σ𝑎 ∈ 𝑈 (𝑋‘𝑎) − Σ𝑎 ∈ 𝑈 (𝑋‘𝑎)) = 0)
110	108, 109	eqtrd 2772	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → (((𝑋‘𝑘) · Σ𝑎 ∈ 𝑈 (𝑋‘𝑎)) − (1 · Σ𝑎 ∈ 𝑈 (𝑋‘𝑎))) = 0)
111		1cnd 11139	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → 1 ∈ ℂ)
112	62, 111, 57	subdird 11607	. . . . . . . . . 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 11345	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → (((𝑋‘𝑘) − 1) · 0) = 0)
114	110, 112, 113	3eqtr4d 2782	. . . . . . . . 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 11785	. . . . . . . 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 240	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → ((𝑋‘𝑘) ≠ ( 1 ‘𝑘) → Σ𝑎 ∈ 𝑈 (𝑋‘𝑎) = 0))
118	44, 117	biimtrrid 243	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → (¬ (𝑋‘𝑘) = ( 1 ‘𝑘) → Σ𝑎 ∈ 𝑈 (𝑋‘𝑎) = 0))
119	118	rexlimdva 3139	. . . 4 ⊢ (𝜑 → (∃𝑘 ∈ 𝑈 ¬ (𝑋‘𝑘) = ( 1 ‘𝑘) → Σ𝑎 ∈ 𝑈 (𝑋‘𝑎) = 0))
120	43, 119	sylbid 240	. . 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 4506	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 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∀wral 3052 ∃wrex 3062 Vcvv 3430 ifcif 4467 ↦ cmpt 5167 ⟶wf 6495 –1-1-onto→wf1o 6498 ‘cfv 6499 (class class class)co 7367 Fincfn 8893 ℂcc 11036 0cc0 11038 1c1 11039 · cmul 11043 − cmin 11377 ℕcn 12174 ℕ₀cn0 12437 ♯chash 14292 Σcsu 15648 ϕcphi 16734 Basecbs 17179 ↾_s cress 17200 +_gcplusg 17220 ._rcmulr 17221 0_gc0g 17402 MndHom cmhm 18749 Grpcgrp 18909 Abelcabl 19756 mulGrpcmgp 20121 Ringcrg 20214 CRingccrg 20215 Unitcui 20335 ℂ_fldccnfld 21352 ℤ/nℤczn 21482 DChrcdchr 27195

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5308 ax-pr 5376 ax-un 7689 ax-inf2 9562 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117 ax-mulf 11118

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6266 df-ord 6327 df-on 6328 df-lim 6329 df-suc 6330 df-iota 6455 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-isom 6508 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-of 7631 df-om 7818 df-1st 7942 df-2nd 7943 df-tpos 8176 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-oadd 8409 df-er 8643 df-ec 8645 df-qs 8649 df-map 8775 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-sup 9355 df-inf 9356 df-oi 9425 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-xnn0 12511 df-z 12525 df-dec 12645 df-uz 12789 df-rp 12943 df-fz 13462 df-fzo 13609 df-fl 13751 df-mod 13829 df-seq 13964 df-exp 14024 df-hash 14293 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-clim 15450 df-sum 15649 df-dvds 16222 df-gcd 16464 df-phi 16736 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-0g 17404 df-imas 17472 df-qus 17473 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-mhm 18751 df-grp 18912 df-minusg 18913 df-sbg 18914 df-mulg 19044 df-subg 19099 df-nsg 19100 df-eqg 19101 df-ghm 19188 df-cmn 19757 df-abl 19758 df-mgp 20122 df-rng 20134 df-ur 20163 df-ring 20216 df-cring 20217 df-oppr 20317 df-dvdsr 20337 df-unit 20338 df-invr 20368 df-rhm 20452 df-subrng 20523 df-subrg 20547 df-lmod 20857 df-lss 20927 df-lsp 20967 df-sra 21168 df-rgmod 21169 df-lidl 21206 df-rsp 21207 df-2idl 21248 df-cnfld 21353 df-zring 21427 df-zrh 21483 df-zn 21486 df-dchr 27196

This theorem is referenced by: dchrsum 27232

W3C validator