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Theorem sumdchr2 27214

Description: Lemma for sumdchr 27216. (Contributed by Mario Carneiro, 28-Apr-2016.)

**Hypotheses**
Ref	Expression
sumdchr.g	⊢ 𝐺 = (DChr‘𝑁)
sumdchr.d	⊢ 𝐷 = (Base‘𝐺)
sumdchr2.z	⊢ 𝑍 = (ℤ/nℤ‘𝑁)
sumdchr2.1	⊢ 1 = (1_r‘𝑍)
sumdchr2.b	⊢ 𝐵 = (Base‘𝑍)
sumdchr2.n	⊢ (𝜑 → 𝑁 ∈ ℕ)
sumdchr2.x	⊢ (𝜑 → 𝐴 ∈ 𝐵)

**Assertion**
Ref	Expression
sumdchr2	⊢ (𝜑 → Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) = if(𝐴 = 1 , (♯‘𝐷), 0))

Distinct variable groups: 𝑥, 1 𝑥,𝐴 𝑥,𝐷 𝑥,𝑁 𝑥,𝐺 𝜑,𝑥 Allowed substitution hints: 𝐵(𝑥) 𝑍(𝑥)

Proof of Theorem sumdchr2 Dummy variables 𝑦 𝑧 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq2 2743	. 2 ⊢ ((♯‘𝐷) = if(𝐴 = 1 , (♯‘𝐷), 0) → (Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) = (♯‘𝐷) ↔ Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) = if(𝐴 = 1 , (♯‘𝐷), 0)))
2		eqeq2 2743	. 2 ⊢ (0 = if(𝐴 = 1 , (♯‘𝐷), 0) → (Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) = 0 ↔ Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) = if(𝐴 = 1 , (♯‘𝐷), 0)))
3		fveq2 6828	. . . . . 6 ⊢ (𝐴 = 1 → (𝑥‘𝐴) = (𝑥‘ 1 ))
4		sumdchr.g	. . . . . . . . 9 ⊢ 𝐺 = (DChr‘𝑁)
5		sumdchr2.z	. . . . . . . . 9 ⊢ 𝑍 = (ℤ/nℤ‘𝑁)
6		sumdchr.d	. . . . . . . . 9 ⊢ 𝐷 = (Base‘𝐺)
7	4, 5, 6	dchrmhm 27185	. . . . . . . 8 ⊢ 𝐷 ⊆ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld))
8		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ 𝐷)
9	7, 8	sselid 3927	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)))
10		eqid 2731	. . . . . . . . 9 ⊢ (mulGrp‘𝑍) = (mulGrp‘𝑍)
11		sumdchr2.1	. . . . . . . . 9 ⊢ 1 = (1_r‘𝑍)
12	10, 11	ringidval 20107	. . . . . . . 8 ⊢ 1 = (0_g‘(mulGrp‘𝑍))
13		eqid 2731	. . . . . . . . 9 ⊢ (mulGrp‘ℂ_fld) = (mulGrp‘ℂ_fld)
14		cnfld1 21336	. . . . . . . . 9 ⊢ 1 = (1_r‘ℂ_fld)
15	13, 14	ringidval 20107	. . . . . . . 8 ⊢ 1 = (0_g‘(mulGrp‘ℂ_fld))
16	12, 15	mhm0 18708	. . . . . . 7 ⊢ (𝑥 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)) → (𝑥‘ 1 ) = 1)
17	9, 16	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑥‘ 1 ) = 1)
18	3, 17	sylan9eqr 2788	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝐴 = 1 ) → (𝑥‘𝐴) = 1)
19	18	an32s 652	. . . 4 ⊢ (((𝜑 ∧ 𝐴 = 1 ) ∧ 𝑥 ∈ 𝐷) → (𝑥‘𝐴) = 1)
20	19	sumeq2dv 15615	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 1 ) → Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) = Σ𝑥 ∈ 𝐷 1)
21		sumdchr2.n	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ)
22	4, 6	dchrfi 27199	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → 𝐷 ∈ Fin)
23	21, 22	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ Fin)
24		ax-1cn 11070	. . . . . 6 ⊢ 1 ∈ ℂ
25		fsumconst 15703	. . . . . 6 ⊢ ((𝐷 ∈ Fin ∧ 1 ∈ ℂ) → Σ𝑥 ∈ 𝐷 1 = ((♯‘𝐷) · 1))
26	23, 24, 25	sylancl 586	. . . . 5 ⊢ (𝜑 → Σ𝑥 ∈ 𝐷 1 = ((♯‘𝐷) · 1))
27		hashcl 14269	. . . . . . . 8 ⊢ (𝐷 ∈ Fin → (♯‘𝐷) ∈ ℕ₀)
28	21, 22, 27	3syl 18	. . . . . . 7 ⊢ (𝜑 → (♯‘𝐷) ∈ ℕ₀)
29	28	nn0cnd 12450	. . . . . 6 ⊢ (𝜑 → (♯‘𝐷) ∈ ℂ)
30	29	mulridd 11135	. . . . 5 ⊢ (𝜑 → ((♯‘𝐷) · 1) = (♯‘𝐷))
31	26, 30	eqtrd 2766	. . . 4 ⊢ (𝜑 → Σ𝑥 ∈ 𝐷 1 = (♯‘𝐷))
32	31	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 1 ) → Σ𝑥 ∈ 𝐷 1 = (♯‘𝐷))
33	20, 32	eqtrd 2766	. 2 ⊢ ((𝜑 ∧ 𝐴 = 1 ) → Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) = (♯‘𝐷))
34		df-ne 2929	. . 3 ⊢ (𝐴 ≠ 1 ↔ ¬ 𝐴 = 1 )
35		sumdchr2.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑍)
36	21	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 1 ) → 𝑁 ∈ ℕ)
37		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 1 ) → 𝐴 ≠ 1 )
38		sumdchr2.x	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
39	38	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 1 ) → 𝐴 ∈ 𝐵)
40	4, 5, 6, 35, 11, 36, 37, 39	dchrpt 27211	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 1 ) → ∃𝑦 ∈ 𝐷 (𝑦‘𝐴) ≠ 1)
41	36	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → 𝑁 ∈ ℕ)
42	41, 22	syl 17	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → 𝐷 ∈ Fin)
43		simpr 484	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ 𝐷)
44	4, 5, 6, 35, 43	dchrf 27186	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) ∧ 𝑥 ∈ 𝐷) → 𝑥:𝐵⟶ℂ)
45	39	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → 𝐴 ∈ 𝐵)
46	45	adantr 480	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) ∧ 𝑥 ∈ 𝐷) → 𝐴 ∈ 𝐵)
47	44, 46	ffvelcdmd 7024	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) ∧ 𝑥 ∈ 𝐷) → (𝑥‘𝐴) ∈ ℂ)
48	42, 47	fsumcl 15646	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) ∈ ℂ)
49		0cnd 11111	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → 0 ∈ ℂ)
50		simprl 770	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → 𝑦 ∈ 𝐷)
51	4, 5, 6, 35, 50	dchrf 27186	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → 𝑦:𝐵⟶ℂ)
52	51, 45	ffvelcdmd 7024	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → (𝑦‘𝐴) ∈ ℂ)
53		subcl 11365	. . . . . 6 ⊢ (((𝑦‘𝐴) ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑦‘𝐴) − 1) ∈ ℂ)
54	52, 24, 53	sylancl 586	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → ((𝑦‘𝐴) − 1) ∈ ℂ)
55		simprr 772	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → (𝑦‘𝐴) ≠ 1)
56		subeq0 11393	. . . . . . . 8 ⊢ (((𝑦‘𝐴) ∈ ℂ ∧ 1 ∈ ℂ) → (((𝑦‘𝐴) − 1) = 0 ↔ (𝑦‘𝐴) = 1))
57	52, 24, 56	sylancl 586	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → (((𝑦‘𝐴) − 1) = 0 ↔ (𝑦‘𝐴) = 1))
58	57	necon3bid 2972	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → (((𝑦‘𝐴) − 1) ≠ 0 ↔ (𝑦‘𝐴) ≠ 1))
59	55, 58	mpbird 257	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → ((𝑦‘𝐴) − 1) ≠ 0)
60		oveq2 7360	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑥 → (𝑦(+_g‘𝐺)𝑧) = (𝑦(+_g‘𝐺)𝑥))
61	60	fveq1d 6830	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑥 → ((𝑦(+_g‘𝐺)𝑧)‘𝐴) = ((𝑦(+_g‘𝐺)𝑥)‘𝐴))
62	61	cbvsumv 15609	. . . . . . . . . 10 ⊢ Σ𝑧 ∈ 𝐷 ((𝑦(+_g‘𝐺)𝑧)‘𝐴) = Σ𝑥 ∈ 𝐷 ((𝑦(+_g‘𝐺)𝑥)‘𝐴)
63		eqid 2731	. . . . . . . . . . . . . 14 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
64	50	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) ∧ 𝑥 ∈ 𝐷) → 𝑦 ∈ 𝐷)
65	4, 5, 6, 63, 64, 43	dchrmul 27192	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) ∧ 𝑥 ∈ 𝐷) → (𝑦(+_g‘𝐺)𝑥) = (𝑦 ∘_f · 𝑥))
66	65	fveq1d 6830	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) ∧ 𝑥 ∈ 𝐷) → ((𝑦(+_g‘𝐺)𝑥)‘𝐴) = ((𝑦 ∘_f · 𝑥)‘𝐴))
67	51	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) ∧ 𝑥 ∈ 𝐷) → 𝑦:𝐵⟶ℂ)
68	67	ffnd 6658	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) ∧ 𝑥 ∈ 𝐷) → 𝑦 Fn 𝐵)
69	44	ffnd 6658	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) ∧ 𝑥 ∈ 𝐷) → 𝑥 Fn 𝐵)
70	35	fvexi 6842	. . . . . . . . . . . . . 14 ⊢ 𝐵 ∈ V
71	70	a1i 11	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) ∧ 𝑥 ∈ 𝐷) → 𝐵 ∈ V)
72		fnfvof 7633	. . . . . . . . . . . . 13 ⊢ (((𝑦 Fn 𝐵 ∧ 𝑥 Fn 𝐵) ∧ (𝐵 ∈ V ∧ 𝐴 ∈ 𝐵)) → ((𝑦 ∘_f · 𝑥)‘𝐴) = ((𝑦‘𝐴) · (𝑥‘𝐴)))
73	68, 69, 71, 46, 72	syl22anc 838	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) ∧ 𝑥 ∈ 𝐷) → ((𝑦 ∘_f · 𝑥)‘𝐴) = ((𝑦‘𝐴) · (𝑥‘𝐴)))
74	66, 73	eqtrd 2766	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) ∧ 𝑥 ∈ 𝐷) → ((𝑦(+_g‘𝐺)𝑥)‘𝐴) = ((𝑦‘𝐴) · (𝑥‘𝐴)))
75	74	sumeq2dv 15615	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → Σ𝑥 ∈ 𝐷 ((𝑦(+_g‘𝐺)𝑥)‘𝐴) = Σ𝑥 ∈ 𝐷 ((𝑦‘𝐴) · (𝑥‘𝐴)))
76	62, 75	eqtrid 2778	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → Σ𝑧 ∈ 𝐷 ((𝑦(+_g‘𝐺)𝑧)‘𝐴) = Σ𝑥 ∈ 𝐷 ((𝑦‘𝐴) · (𝑥‘𝐴)))
77		fveq1 6827	. . . . . . . . . 10 ⊢ (𝑥 = (𝑦(+_g‘𝐺)𝑧) → (𝑥‘𝐴) = ((𝑦(+_g‘𝐺)𝑧)‘𝐴))
78	4	dchrabl 27198	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → 𝐺 ∈ Abel)
79		ablgrp 19703	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
80	41, 78, 79	3syl 18	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → 𝐺 ∈ Grp)
81		eqid 2731	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ 𝐷 ↦ (𝑏 ∈ 𝐷 ↦ (𝑎(+_g‘𝐺)𝑏))) = (𝑎 ∈ 𝐷 ↦ (𝑏 ∈ 𝐷 ↦ (𝑎(+_g‘𝐺)𝑏)))
82	81, 6, 63	grplactf1o 18963	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝐷) → ((𝑎 ∈ 𝐷 ↦ (𝑏 ∈ 𝐷 ↦ (𝑎(+_g‘𝐺)𝑏)))‘𝑦):𝐷–1-1-onto→𝐷)
83	80, 50, 82	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → ((𝑎 ∈ 𝐷 ↦ (𝑏 ∈ 𝐷 ↦ (𝑎(+_g‘𝐺)𝑏)))‘𝑦):𝐷–1-1-onto→𝐷)
84	81, 6	grplactval 18961	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝐷 ∧ 𝑧 ∈ 𝐷) → (((𝑎 ∈ 𝐷 ↦ (𝑏 ∈ 𝐷 ↦ (𝑎(+_g‘𝐺)𝑏)))‘𝑦)‘𝑧) = (𝑦(+_g‘𝐺)𝑧))
85	50, 84	sylan 580	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) ∧ 𝑧 ∈ 𝐷) → (((𝑎 ∈ 𝐷 ↦ (𝑏 ∈ 𝐷 ↦ (𝑎(+_g‘𝐺)𝑏)))‘𝑦)‘𝑧) = (𝑦(+_g‘𝐺)𝑧))
86	77, 42, 83, 85, 47	fsumf1o 15636	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) = Σ𝑧 ∈ 𝐷 ((𝑦(+_g‘𝐺)𝑧)‘𝐴))
87	42, 52, 47	fsummulc2 15697	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → ((𝑦‘𝐴) · Σ𝑥 ∈ 𝐷 (𝑥‘𝐴)) = Σ𝑥 ∈ 𝐷 ((𝑦‘𝐴) · (𝑥‘𝐴)))
88	76, 86, 87	3eqtr4rd 2777	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → ((𝑦‘𝐴) · Σ𝑥 ∈ 𝐷 (𝑥‘𝐴)) = Σ𝑥 ∈ 𝐷 (𝑥‘𝐴))
89	48	mullidd 11136	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → (1 · Σ𝑥 ∈ 𝐷 (𝑥‘𝐴)) = Σ𝑥 ∈ 𝐷 (𝑥‘𝐴))
90	88, 89	oveq12d 7370	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → (((𝑦‘𝐴) · Σ𝑥 ∈ 𝐷 (𝑥‘𝐴)) − (1 · Σ𝑥 ∈ 𝐷 (𝑥‘𝐴))) = (Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) − Σ𝑥 ∈ 𝐷 (𝑥‘𝐴)))
91	48	subidd 11466	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → (Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) − Σ𝑥 ∈ 𝐷 (𝑥‘𝐴)) = 0)
92	90, 91	eqtrd 2766	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → (((𝑦‘𝐴) · Σ𝑥 ∈ 𝐷 (𝑥‘𝐴)) − (1 · Σ𝑥 ∈ 𝐷 (𝑥‘𝐴))) = 0)
93	24	a1i 11	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → 1 ∈ ℂ)
94	52, 93, 48	subdird 11580	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → (((𝑦‘𝐴) − 1) · Σ𝑥 ∈ 𝐷 (𝑥‘𝐴)) = (((𝑦‘𝐴) · Σ𝑥 ∈ 𝐷 (𝑥‘𝐴)) − (1 · Σ𝑥 ∈ 𝐷 (𝑥‘𝐴))))
95	54	mul01d 11318	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → (((𝑦‘𝐴) − 1) · 0) = 0)
96	92, 94, 95	3eqtr4d 2776	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → (((𝑦‘𝐴) − 1) · Σ𝑥 ∈ 𝐷 (𝑥‘𝐴)) = (((𝑦‘𝐴) − 1) · 0))
97	48, 49, 54, 59, 96	mulcanad 11758	. . . 4 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) = 0)
98	40, 97	rexlimddv 3139	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ 1 ) → Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) = 0)
99	34, 98	sylan2br 595	. 2 ⊢ ((𝜑 ∧ ¬ 𝐴 = 1 ) → Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) = 0)
100	1, 2, 33, 99	ifbothda 4513	1 ⊢ (𝜑 → Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) = if(𝐴 = 1 , (♯‘𝐷), 0))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1541 ∈ wcel 2111 ≠ wne 2928 Vcvv 3436 ifcif 4474 ↦ cmpt 5174 Fn wfn 6482 ⟶wf 6483 –1-1-onto→wf1o 6486 ‘cfv 6487 (class class class)co 7352 ∘_f cof 7614 Fincfn 8875 ℂcc 11010 0cc0 11012 1c1 11013 · cmul 11017 − cmin 11350 ℕcn 12131 ℕ₀cn0 12387 ♯chash 14243 Σcsu 15599 Basecbs 17126 +_gcplusg 17167 MndHom cmhm 18695 Grpcgrp 18852 Abelcabl 19699 mulGrpcmgp 20064 1_rcur 20105 ℂ_fldccnfld 21297 ℤ/nℤczn 21445 DChrcdchr 27176

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5219 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674 ax-inf2 9537 ax-cnex 11068 ax-resscn 11069 ax-1cn 11070 ax-icn 11071 ax-addcl 11072 ax-addrcl 11073 ax-mulcl 11074 ax-mulrcl 11075 ax-mulcom 11076 ax-addass 11077 ax-mulass 11078 ax-distr 11079 ax-i2m1 11080 ax-1ne0 11081 ax-1rid 11082 ax-rnegex 11083 ax-rrecex 11084 ax-cnre 11085 ax-pre-lttri 11086 ax-pre-lttrn 11087 ax-pre-ltadd 11088 ax-pre-mulgt0 11089 ax-pre-sup 11090 ax-addf 11091 ax-mulf 11092

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-tp 4580 df-op 4582 df-uni 4859 df-int 4898 df-iun 4943 df-iin 4944 df-disj 5061 df-br 5094 df-opab 5156 df-mpt 5175 df-tr 5201 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6254 df-ord 6315 df-on 6316 df-lim 6317 df-suc 6318 df-iota 6443 df-fun 6489 df-fn 6490 df-f 6491 df-f1 6492 df-fo 6493 df-f1o 6494 df-fv 6495 df-isom 6496 df-riota 7309 df-ov 7355 df-oprab 7356 df-mpo 7357 df-of 7616 df-rpss 7662 df-om 7803 df-1st 7927 df-2nd 7928 df-supp 8097 df-tpos 8162 df-frecs 8217 df-wrecs 8248 df-recs 8297 df-rdg 8335 df-1o 8391 df-2o 8392 df-oadd 8395 df-omul 8396 df-er 8628 df-ec 8630 df-qs 8634 df-map 8758 df-pm 8759 df-ixp 8828 df-en 8876 df-dom 8877 df-sdom 8878 df-fin 8879 df-fsupp 9252 df-fi 9301 df-sup 9332 df-inf 9333 df-oi 9402 df-dju 9800 df-card 9838 df-acn 9841 df-pnf 11154 df-mnf 11155 df-xr 11156 df-ltxr 11157 df-le 11158 df-sub 11352 df-neg 11353 df-div 11781 df-nn 12132 df-2 12194 df-3 12195 df-4 12196 df-5 12197 df-6 12198 df-7 12199 df-8 12200 df-9 12201 df-n0 12388 df-xnn0 12461 df-z 12475 df-dec 12595 df-uz 12739 df-q 12853 df-rp 12897 df-xneg 13017 df-xadd 13018 df-xmul 13019 df-ioo 13255 df-ioc 13256 df-ico 13257 df-icc 13258 df-fz 13414 df-fzo 13561 df-fl 13702 df-mod 13780 df-seq 13915 df-exp 13975 df-fac 14187 df-bc 14216 df-hash 14244 df-word 14427 df-concat 14484 df-s1 14510 df-shft 14980 df-cj 15012 df-re 15013 df-im 15014 df-sqrt 15148 df-abs 15149 df-limsup 15384 df-clim 15401 df-rlim 15402 df-sum 15600 df-ef 15980 df-sin 15982 df-cos 15983 df-pi 15985 df-dvds 16170 df-gcd 16412 df-prm 16589 df-phi 16683 df-pc 16755 df-struct 17064 df-sets 17081 df-slot 17099 df-ndx 17111 df-base 17127 df-ress 17148 df-plusg 17180 df-mulr 17181 df-starv 17182 df-sca 17183 df-vsca 17184 df-ip 17185 df-tset 17186 df-ple 17187 df-ds 17189 df-unif 17190 df-hom 17191 df-cco 17192 df-rest 17332 df-topn 17333 df-0g 17351 df-gsum 17352 df-topgen 17353 df-pt 17354 df-prds 17357 df-xrs 17412 df-qtop 17417 df-imas 17418 df-qus 17419 df-xps 17420 df-mre 17494 df-mrc 17495 df-acs 17497 df-mgm 18554 df-sgrp 18633 df-mnd 18649 df-mhm 18697 df-submnd 18698 df-grp 18855 df-minusg 18856 df-sbg 18857 df-mulg 18987 df-subg 19042 df-nsg 19043 df-eqg 19044 df-ghm 19131 df-gim 19177 df-ga 19208 df-cntz 19235 df-oppg 19264 df-od 19446 df-gex 19447 df-pgp 19448 df-lsm 19554 df-pj1 19555 df-cmn 19700 df-abl 19701 df-cyg 19796 df-dprd 19915 df-dpj 19916 df-mgp 20065 df-rng 20077 df-ur 20106 df-ring 20159 df-cring 20160 df-oppr 20261 df-dvdsr 20281 df-unit 20282 df-invr 20312 df-rhm 20396 df-subrng 20467 df-subrg 20491 df-lmod 20801 df-lss 20871 df-lsp 20911 df-sra 21113 df-rgmod 21114 df-lidl 21151 df-rsp 21152 df-2idl 21193 df-psmet 21289 df-xmet 21290 df-met 21291 df-bl 21292 df-mopn 21293 df-fbas 21294 df-fg 21295 df-cnfld 21298 df-zring 21390 df-zrh 21446 df-zn 21449 df-top 22815 df-topon 22832 df-topsp 22854 df-bases 22867 df-cld 22940 df-ntr 22941 df-cls 22942 df-nei 23019 df-lp 23057 df-perf 23058 df-cn 23148 df-cnp 23149 df-haus 23236 df-tx 23483 df-hmeo 23676 df-fil 23767 df-fm 23859 df-flim 23860 df-flf 23861 df-xms 24241 df-ms 24242 df-tms 24243 df-cncf 24804 df-0p 25604 df-limc 25800 df-dv 25801 df-ply 26126 df-idp 26127 df-coe 26128 df-dgr 26129 df-quot 26232 df-log 26498 df-cxp 26499 df-dchr 27177

This theorem is referenced by: dchrhash 27215 sumdchr 27216

W3C validator