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Theorem dchrsum2 26321

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 2750	. 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 2750	. 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 6755	. . . . . 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 26293	. . . . . . . . 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 26310	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑈) → ( 1 ‘𝑎) = 1)
15	3, 14	sylan9eqr 2801	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑋 = 1 ) → (𝑋‘𝑎) = 1)
16	15	an32s 648	. . . 4 ⊢ (((𝜑 ∧ 𝑋 = 1 ) ∧ 𝑎 ∈ 𝑈) → (𝑋‘𝑎) = 1)
17	16	sumeq2dv 15343	. . 3 ⊢ ((𝜑 ∧ 𝑋 = 1 ) → Σ𝑎 ∈ 𝑈 (𝑋‘𝑎) = Σ𝑎 ∈ 𝑈 1)
18	5, 7	znunithash 20684	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (♯‘𝑈) = (ϕ‘𝑁))
19	11, 18	syl 17	. . . . . . . 8 ⊢ (𝜑 → (♯‘𝑈) = (ϕ‘𝑁))
20	11	phicld 16401	. . . . . . . . 9 ⊢ (𝜑 → (ϕ‘𝑁) ∈ ℕ)
21	20	nnnn0d 12223	. . . . . . . 8 ⊢ (𝜑 → (ϕ‘𝑁) ∈ ℕ₀)
22	19, 21	eqeltrd 2839	. . . . . . 7 ⊢ (𝜑 → (♯‘𝑈) ∈ ℕ₀)
23	7	fvexi 6770	. . . . . . . 8 ⊢ 𝑈 ∈ V
24		hashclb 14001	. . . . . . . 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 10860	. . . . . 6 ⊢ 1 ∈ ℂ
28		fsumconst 15430	. . . . . 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 7270	. . . . 5 ⊢ (𝜑 → ((♯‘𝑈) · 1) = ((ϕ‘𝑁) · 1))
31	20	nncnd 11919	. . . . . 6 ⊢ (𝜑 → (ϕ‘𝑁) ∈ ℂ)
32	31	mulid1d 10923	. . . . 5 ⊢ (𝜑 → ((ϕ‘𝑁) · 1) = (ϕ‘𝑁))
33	29, 30, 32	3eqtrd 2782	. . . 4 ⊢ (𝜑 → Σ𝑎 ∈ 𝑈 1 = (ϕ‘𝑁))
34	33	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑋 = 1 ) → Σ𝑎 ∈ 𝑈 1 = (ϕ‘𝑁))
35	17, 34	eqtrd 2778	. 2 ⊢ ((𝜑 ∧ 𝑋 = 1 ) → Σ𝑎 ∈ 𝑈 (𝑋‘𝑎) = (ϕ‘𝑁))
36	4	dchrabl 26307	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 𝐺 ∈ Abel)
37		ablgrp 19306	. . . . . . . 8 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
38	9, 6	grpidcl 18522	. . . . . . . 8 ⊢ (𝐺 ∈ Grp → 1 ∈ 𝐷)
39	11, 36, 37, 38	4syl 19	. . . . . . 7 ⊢ (𝜑 → 1 ∈ 𝐷)
40	4, 5, 9, 7, 8, 39	dchreq 26311	. . . . . 6 ⊢ (𝜑 → (𝑋 = 1 ↔ ∀𝑘 ∈ 𝑈 (𝑋‘𝑘) = ( 1 ‘𝑘)))
41	40	notbid 317	. . . . 5 ⊢ (𝜑 → (¬ 𝑋 = 1 ↔ ¬ ∀𝑘 ∈ 𝑈 (𝑋‘𝑘) = ( 1 ‘𝑘)))
42		rexnal 3165	. . . . 5 ⊢ (∃𝑘 ∈ 𝑈 ¬ (𝑋‘𝑘) = ( 1 ‘𝑘) ↔ ¬ ∀𝑘 ∈ 𝑈 (𝑋‘𝑘) = ( 1 ‘𝑘))
43	41, 42	bitr4di 288	. . . 4 ⊢ (𝜑 → (¬ 𝑋 = 1 ↔ ∃𝑘 ∈ 𝑈 ¬ (𝑋‘𝑘) = ( 1 ‘𝑘)))
44		df-ne 2943	. . . . . 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 26310	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → ( 1 ‘𝑘) = 1)
48	47	neeq2d 3003	. . . . . . 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 2738	. . . . . . . . . . . . 13 ⊢ (Base‘𝑍) = (Base‘𝑍)
51	4, 5, 9, 50, 8	dchrf 26295	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋:(Base‘𝑍)⟶ℂ)
52	50, 7	unitss 19817	. . . . . . . . . . . . 13 ⊢ 𝑈 ⊆ (Base‘𝑍)
53	52	sseli 3913	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ 𝑈 → 𝑎 ∈ (Base‘𝑍))
54		ffvelrn 6941	. . . . . . . . . . . 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 711	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) ∧ 𝑎 ∈ 𝑈) → (𝑋‘𝑎) ∈ ℂ)
57	49, 56	fsumcl 15373	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → Σ𝑎 ∈ 𝑈 (𝑋‘𝑎) ∈ ℂ)
58		0cnd 10899	. . . . . . . . 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 767	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → 𝑘 ∈ 𝑈)
61	52, 60	sselid 3915	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → 𝑘 ∈ (Base‘𝑍))
62	59, 61	ffvelrnd 6944	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → (𝑋‘𝑘) ∈ ℂ)
63		subcl 11150	. . . . . . . . . 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 769	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → (𝑋‘𝑘) ≠ 1)
66		subeq0 11177	. . . . . . . . . . . 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 2987	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → (((𝑋‘𝑘) − 1) ≠ 0 ↔ (𝑋‘𝑘) ≠ 1))
69	65, 68	mpbird 256	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → ((𝑋‘𝑘) − 1) ≠ 0)
70		oveq2 7263	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑎 → (𝑘(._r‘𝑍)𝑥) = (𝑘(._r‘𝑍)𝑎))
71	70	fveq2d 6760	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑎 → (𝑋‘(𝑘(._r‘𝑍)𝑥)) = (𝑋‘(𝑘(._r‘𝑍)𝑎)))
72	71	cbvsumv 15336	. . . . . . . . . . . . . 14 ⊢ Σ𝑥 ∈ 𝑈 (𝑋‘(𝑘(._r‘𝑍)𝑥)) = Σ𝑎 ∈ 𝑈 (𝑋‘(𝑘(._r‘𝑍)𝑎))
73	4, 5, 9	dchrmhm 26294	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐷 ⊆ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld))
74	73, 8	sselid 3915	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)))
75	74	ad2antrr 722	. . . . . . . . . . . . . . . 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 2738	. . . . . . . . . . . . . . . . . 18 ⊢ (mulGrp‘𝑍) = (mulGrp‘𝑍)
79	78, 50	mgpbas 19641	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘𝑍) = (Base‘(mulGrp‘𝑍))
80		eqid 2738	. . . . . . . . . . . . . . . . . 18 ⊢ (._r‘𝑍) = (._r‘𝑍)
81	78, 80	mgpplusg 19639	. . . . . . . . . . . . . . . . 17 ⊢ (._r‘𝑍) = (+_g‘(mulGrp‘𝑍))
82		eqid 2738	. . . . . . . . . . . . . . . . . 18 ⊢ (mulGrp‘ℂ_fld) = (mulGrp‘ℂ_fld)
83		cnfldmul 20516	. . . . . . . . . . . . . . . . . 18 ⊢ · = (._r‘ℂ_fld)
84	82, 83	mgpplusg 19639	. . . . . . . . . . . . . . . . 17 ⊢ · = (+_g‘(mulGrp‘ℂ_fld))
85	79, 81, 84	mhmlin 18352	. . . . . . . . . . . . . . . 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 1369	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) ∧ 𝑎 ∈ 𝑈) → (𝑋‘(𝑘(._r‘𝑍)𝑎)) = ((𝑋‘𝑘) · (𝑋‘𝑎)))
87	86	sumeq2dv 15343	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → Σ𝑎 ∈ 𝑈 (𝑋‘(𝑘(._r‘𝑍)𝑎)) = Σ𝑎 ∈ 𝑈 ((𝑋‘𝑘) · (𝑋‘𝑎)))
88	72, 87	syl5eq 2791	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → Σ𝑥 ∈ 𝑈 (𝑋‘(𝑘(._r‘𝑍)𝑥)) = Σ𝑎 ∈ 𝑈 ((𝑋‘𝑘) · (𝑋‘𝑎)))
89		fveq2 6756	. . . . . . . . . . . . . 14 ⊢ (𝑎 = (𝑘(._r‘𝑍)𝑥) → (𝑋‘𝑎) = (𝑋‘(𝑘(._r‘𝑍)𝑥)))
90	11	nnnn0d 12223	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
91	5	zncrng 20664	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℕ₀ → 𝑍 ∈ CRing)
92		crngring 19710	. . . . . . . . . . . . . . . 16 ⊢ (𝑍 ∈ CRing → 𝑍 ∈ Ring)
93		eqid 2738	. . . . . . . . . . . . . . . . 17 ⊢ ((mulGrp‘𝑍) ↾_s 𝑈) = ((mulGrp‘𝑍) ↾_s 𝑈)
94	7, 93	unitgrp 19824	. . . . . . . . . . . . . . . 16 ⊢ (𝑍 ∈ Ring → ((mulGrp‘𝑍) ↾_s 𝑈) ∈ Grp)
95	90, 91, 92, 94	4syl 19	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((mulGrp‘𝑍) ↾_s 𝑈) ∈ Grp)
96		eqid 2738	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 ∈ 𝑈 ↦ (𝑐 ∈ 𝑈 ↦ (𝑏(._r‘𝑍)𝑐))) = (𝑏 ∈ 𝑈 ↦ (𝑐 ∈ 𝑈 ↦ (𝑏(._r‘𝑍)𝑐)))
97	7, 93	unitgrpbas 19823	. . . . . . . . . . . . . . . 16 ⊢ 𝑈 = (Base‘((mulGrp‘𝑍) ↾_s 𝑈))
98	93, 81	ressplusg 16926	. . . . . . . . . . . . . . . . 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 18594	. . . . . . . . . . . . . . 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 681	. . . . . . . . . . . . . 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 18592	. . . . . . . . . . . . . . 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 15363	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → Σ𝑎 ∈ 𝑈 (𝑋‘𝑎) = Σ𝑥 ∈ 𝑈 (𝑋‘(𝑘(._r‘𝑍)𝑥)))
105	49, 62, 56	fsummulc2 15424	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → ((𝑋‘𝑘) · Σ𝑎 ∈ 𝑈 (𝑋‘𝑎)) = Σ𝑎 ∈ 𝑈 ((𝑋‘𝑘) · (𝑋‘𝑎)))
106	88, 104, 105	3eqtr4rd 2789	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → ((𝑋‘𝑘) · Σ𝑎 ∈ 𝑈 (𝑋‘𝑎)) = Σ𝑎 ∈ 𝑈 (𝑋‘𝑎))
107	57	mulid2d 10924	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → (1 · Σ𝑎 ∈ 𝑈 (𝑋‘𝑎)) = Σ𝑎 ∈ 𝑈 (𝑋‘𝑎))
108	106, 107	oveq12d 7273	. . . . . . . . . . 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 11250	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → (Σ𝑎 ∈ 𝑈 (𝑋‘𝑎) − Σ𝑎 ∈ 𝑈 (𝑋‘𝑎)) = 0)
110	108, 109	eqtrd 2778	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → (((𝑋‘𝑘) · Σ𝑎 ∈ 𝑈 (𝑋‘𝑎)) − (1 · Σ𝑎 ∈ 𝑈 (𝑋‘𝑎))) = 0)
111		1cnd 10901	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → 1 ∈ ℂ)
112	62, 111, 57	subdird 11362	. . . . . . . . . 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 11104	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → (((𝑋‘𝑘) − 1) · 0) = 0)
114	110, 112, 113	3eqtr4d 2788	. . . . . . . . 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 11540	. . . . . . . 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	syl5bir 242	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → (¬ (𝑋‘𝑘) = ( 1 ‘𝑘) → Σ𝑎 ∈ 𝑈 (𝑋‘𝑎) = 0))
119	118	rexlimdva 3212	. . . 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 4494	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 1539 ∈ wcel 2108 ≠ wne 2942 ∀wral 3063 ∃wrex 3064 Vcvv 3422 ifcif 4456 ↦ cmpt 5153 ⟶wf 6414 –1-1-onto→wf1o 6417 ‘cfv 6418 (class class class)co 7255 Fincfn 8691 ℂcc 10800 0cc0 10802 1c1 10803 · cmul 10807 − cmin 11135 ℕcn 11903 ℕ₀cn0 12163 ♯chash 13972 Σcsu 15325 ϕcphi 16393 Basecbs 16840 ↾_s cress 16867 +_gcplusg 16888 ._rcmulr 16889 0_gc0g 17067 MndHom cmhm 18343 Grpcgrp 18492 Abelcabl 19302 mulGrpcmgp 19635 Ringcrg 19698 CRingccrg 19699 Unitcui 19796 ℂ_fldccnfld 20510 ℤ/nℤczn 20616 DChrcdchr 26285

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880 ax-addf 10881 ax-mulf 10882

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-om 7688 df-1st 7804 df-2nd 7805 df-tpos 8013 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-oadd 8271 df-er 8456 df-ec 8458 df-qs 8462 df-map 8575 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-sup 9131 df-inf 9132 df-oi 9199 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-xnn0 12236 df-z 12250 df-dec 12367 df-uz 12512 df-rp 12660 df-fz 13169 df-fzo 13312 df-fl 13440 df-mod 13518 df-seq 13650 df-exp 13711 df-hash 13973 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-clim 15125 df-sum 15326 df-dvds 15892 df-gcd 16130 df-phi 16395 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-starv 16903 df-sca 16904 df-vsca 16905 df-ip 16906 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-0g 17069 df-imas 17136 df-qus 17137 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-mhm 18345 df-grp 18495 df-minusg 18496 df-sbg 18497 df-mulg 18616 df-subg 18667 df-nsg 18668 df-eqg 18669 df-ghm 18747 df-cmn 19303 df-abl 19304 df-mgp 19636 df-ur 19653 df-ring 19700 df-cring 19701 df-oppr 19777 df-dvdsr 19798 df-unit 19799 df-invr 19829 df-rnghom 19874 df-subrg 19937 df-lmod 20040 df-lss 20109 df-lsp 20149 df-sra 20349 df-rgmod 20350 df-lidl 20351 df-rsp 20352 df-2idl 20416 df-cnfld 20511 df-zring 20583 df-zrh 20617 df-zn 20620 df-dchr 26286

This theorem is referenced by: dchrsum 26322

W3C validator