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Theorem sumdchr2 27233

Description: Lemma for sumdchr 27235. (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 2747	. 2 ⊢ ((♯‘𝐷) = if(𝐴 = 1 , (♯‘𝐷), 0) → (Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) = (♯‘𝐷) ↔ Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) = if(𝐴 = 1 , (♯‘𝐷), 0)))
2		eqeq2 2747	. 2 ⊢ (0 = if(𝐴 = 1 , (♯‘𝐷), 0) → (Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) = 0 ↔ Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) = if(𝐴 = 1 , (♯‘𝐷), 0)))
3		fveq2 6876	. . . . . 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 27204	. . . . . . . 8 ⊢ 𝐷 ⊆ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld))
8		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ 𝐷)
9	7, 8	sselid 3956	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)))
10		eqid 2735	. . . . . . . . 9 ⊢ (mulGrp‘𝑍) = (mulGrp‘𝑍)
11		sumdchr2.1	. . . . . . . . 9 ⊢ 1 = (1_r‘𝑍)
12	10, 11	ringidval 20143	. . . . . . . 8 ⊢ 1 = (0_g‘(mulGrp‘𝑍))
13		eqid 2735	. . . . . . . . 9 ⊢ (mulGrp‘ℂ_fld) = (mulGrp‘ℂ_fld)
14		cnfld1 21356	. . . . . . . . 9 ⊢ 1 = (1_r‘ℂ_fld)
15	13, 14	ringidval 20143	. . . . . . . 8 ⊢ 1 = (0_g‘(mulGrp‘ℂ_fld))
16	12, 15	mhm0 18772	. . . . . . 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 2792	. . . . 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 15718	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 1 ) → Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) = Σ𝑥 ∈ 𝐷 1)
21		sumdchr2.n	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ)
22	4, 6	dchrfi 27218	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → 𝐷 ∈ Fin)
23	21, 22	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ Fin)
24		ax-1cn 11187	. . . . . 6 ⊢ 1 ∈ ℂ
25		fsumconst 15806	. . . . . 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 14374	. . . . . . . 8 ⊢ (𝐷 ∈ Fin → (♯‘𝐷) ∈ ℕ₀)
28	21, 22, 27	3syl 18	. . . . . . 7 ⊢ (𝜑 → (♯‘𝐷) ∈ ℕ₀)
29	28	nn0cnd 12564	. . . . . 6 ⊢ (𝜑 → (♯‘𝐷) ∈ ℂ)
30	29	mulridd 11252	. . . . 5 ⊢ (𝜑 → ((♯‘𝐷) · 1) = (♯‘𝐷))
31	26, 30	eqtrd 2770	. . . 4 ⊢ (𝜑 → Σ𝑥 ∈ 𝐷 1 = (♯‘𝐷))
32	31	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 1 ) → Σ𝑥 ∈ 𝐷 1 = (♯‘𝐷))
33	20, 32	eqtrd 2770	. 2 ⊢ ((𝜑 ∧ 𝐴 = 1 ) → Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) = (♯‘𝐷))
34		df-ne 2933	. . 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 27230	. . . 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 27205	. . . . . . 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 7075	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) ∧ 𝑥 ∈ 𝐷) → (𝑥‘𝐴) ∈ ℂ)
48	42, 47	fsumcl 15749	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) ∈ ℂ)
49		0cnd 11228	. . . . 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 27205	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → 𝑦:𝐵⟶ℂ)
52	51, 45	ffvelcdmd 7075	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → (𝑦‘𝐴) ∈ ℂ)
53		subcl 11481	. . . . . 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 11509	. . . . . . . 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 2976	. . . . . 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 7413	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑥 → (𝑦(+_g‘𝐺)𝑧) = (𝑦(+_g‘𝐺)𝑥))
61	60	fveq1d 6878	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑥 → ((𝑦(+_g‘𝐺)𝑧)‘𝐴) = ((𝑦(+_g‘𝐺)𝑥)‘𝐴))
62	61	cbvsumv 15712	. . . . . . . . . 10 ⊢ Σ𝑧 ∈ 𝐷 ((𝑦(+_g‘𝐺)𝑧)‘𝐴) = Σ𝑥 ∈ 𝐷 ((𝑦(+_g‘𝐺)𝑥)‘𝐴)
63		eqid 2735	. . . . . . . . . . . . . 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 27211	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) ∧ 𝑥 ∈ 𝐷) → (𝑦(+_g‘𝐺)𝑥) = (𝑦 ∘_f · 𝑥))
66	65	fveq1d 6878	. . . . . . . . . . . 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 6707	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) ∧ 𝑥 ∈ 𝐷) → 𝑦 Fn 𝐵)
69	44	ffnd 6707	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) ∧ 𝑥 ∈ 𝐷) → 𝑥 Fn 𝐵)
70	35	fvexi 6890	. . . . . . . . . . . . . 14 ⊢ 𝐵 ∈ V
71	70	a1i 11	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) ∧ 𝑥 ∈ 𝐷) → 𝐵 ∈ V)
72		fnfvof 7688	. . . . . . . . . . . . 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 2770	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) ∧ 𝑥 ∈ 𝐷) → ((𝑦(+_g‘𝐺)𝑥)‘𝐴) = ((𝑦‘𝐴) · (𝑥‘𝐴)))
75	74	sumeq2dv 15718	. . . . . . . . . 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 2782	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → Σ𝑧 ∈ 𝐷 ((𝑦(+_g‘𝐺)𝑧)‘𝐴) = Σ𝑥 ∈ 𝐷 ((𝑦‘𝐴) · (𝑥‘𝐴)))
77		fveq1 6875	. . . . . . . . . 10 ⊢ (𝑥 = (𝑦(+_g‘𝐺)𝑧) → (𝑥‘𝐴) = ((𝑦(+_g‘𝐺)𝑧)‘𝐴))
78	4	dchrabl 27217	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → 𝐺 ∈ Abel)
79		ablgrp 19766	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
80	41, 78, 79	3syl 18	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → 𝐺 ∈ Grp)
81		eqid 2735	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ 𝐷 ↦ (𝑏 ∈ 𝐷 ↦ (𝑎(+_g‘𝐺)𝑏))) = (𝑎 ∈ 𝐷 ↦ (𝑏 ∈ 𝐷 ↦ (𝑎(+_g‘𝐺)𝑏)))
82	81, 6, 63	grplactf1o 19027	. . . . . . . . . . 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 19025	. . . . . . . . . . 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 15739	. . . . . . . . 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 15800	. . . . . . . . 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 2781	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → ((𝑦‘𝐴) · Σ𝑥 ∈ 𝐷 (𝑥‘𝐴)) = Σ𝑥 ∈ 𝐷 (𝑥‘𝐴))
89	48	mullidd 11253	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → (1 · Σ𝑥 ∈ 𝐷 (𝑥‘𝐴)) = Σ𝑥 ∈ 𝐷 (𝑥‘𝐴))
90	88, 89	oveq12d 7423	. . . . . . 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 11582	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → (Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) − Σ𝑥 ∈ 𝐷 (𝑥‘𝐴)) = 0)
92	90, 91	eqtrd 2770	. . . . . 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 11694	. . . . . 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 11434	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → (((𝑦‘𝐴) − 1) · 0) = 0)
96	92, 94, 95	3eqtr4d 2780	. . . . 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 11872	. . . 4 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) = 0)
98	40, 97	rexlimddv 3147	. . 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 4539	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 1540 ∈ wcel 2108 ≠ wne 2932 Vcvv 3459 ifcif 4500 ↦ cmpt 5201 Fn wfn 6526 ⟶wf 6527 –1-1-onto→wf1o 6530 ‘cfv 6531 (class class class)co 7405 ∘_f cof 7669 Fincfn 8959 ℂcc 11127 0cc0 11129 1c1 11130 · cmul 11134 − cmin 11466 ℕcn 12240 ℕ₀cn0 12501 ♯chash 14348 Σcsu 15702 Basecbs 17228 +_gcplusg 17271 MndHom cmhm 18759 Grpcgrp 18916 Abelcabl 19762 mulGrpcmgp 20100 1_rcur 20141 ℂ_fldccnfld 21315 ℤ/nℤczn 21463 DChrcdchr 27195

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-inf2 9655 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206 ax-pre-sup 11207 ax-addf 11208 ax-mulf 11209

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-tp 4606 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-iin 4970 df-disj 5087 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-se 5607 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-isom 6540 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7671 df-rpss 7717 df-om 7862 df-1st 7988 df-2nd 7989 df-supp 8160 df-tpos 8225 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-oadd 8484 df-omul 8485 df-er 8719 df-ec 8721 df-qs 8725 df-map 8842 df-pm 8843 df-ixp 8912 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-fsupp 9374 df-fi 9423 df-sup 9454 df-inf 9455 df-oi 9524 df-dju 9915 df-card 9953 df-acn 9956 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-div 11895 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12502 df-xnn0 12575 df-z 12589 df-dec 12709 df-uz 12853 df-q 12965 df-rp 13009 df-xneg 13128 df-xadd 13129 df-xmul 13130 df-ioo 13366 df-ioc 13367 df-ico 13368 df-icc 13369 df-fz 13525 df-fzo 13672 df-fl 13809 df-mod 13887 df-seq 14020 df-exp 14080 df-fac 14292 df-bc 14321 df-hash 14349 df-word 14532 df-concat 14589 df-s1 14614 df-shft 15086 df-cj 15118 df-re 15119 df-im 15120 df-sqrt 15254 df-abs 15255 df-limsup 15487 df-clim 15504 df-rlim 15505 df-sum 15703 df-ef 16083 df-sin 16085 df-cos 16086 df-pi 16088 df-dvds 16273 df-gcd 16514 df-prm 16691 df-phi 16785 df-pc 16857 df-struct 17166 df-sets 17183 df-slot 17201 df-ndx 17213 df-base 17229 df-ress 17252 df-plusg 17284 df-mulr 17285 df-starv 17286 df-sca 17287 df-vsca 17288 df-ip 17289 df-tset 17290 df-ple 17291 df-ds 17293 df-unif 17294 df-hom 17295 df-cco 17296 df-rest 17436 df-topn 17437 df-0g 17455 df-gsum 17456 df-topgen 17457 df-pt 17458 df-prds 17461 df-xrs 17516 df-qtop 17521 df-imas 17522 df-qus 17523 df-xps 17524 df-mre 17598 df-mrc 17599 df-acs 17601 df-mgm 18618 df-sgrp 18697 df-mnd 18713 df-mhm 18761 df-submnd 18762 df-grp 18919 df-minusg 18920 df-sbg 18921 df-mulg 19051 df-subg 19106 df-nsg 19107 df-eqg 19108 df-ghm 19196 df-gim 19242 df-ga 19273 df-cntz 19300 df-oppg 19329 df-od 19509 df-gex 19510 df-pgp 19511 df-lsm 19617 df-pj1 19618 df-cmn 19763 df-abl 19764 df-cyg 19859 df-dprd 19978 df-dpj 19979 df-mgp 20101 df-rng 20113 df-ur 20142 df-ring 20195 df-cring 20196 df-oppr 20297 df-dvdsr 20317 df-unit 20318 df-invr 20348 df-rhm 20432 df-subrng 20506 df-subrg 20530 df-lmod 20819 df-lss 20889 df-lsp 20929 df-sra 21131 df-rgmod 21132 df-lidl 21169 df-rsp 21170 df-2idl 21211 df-psmet 21307 df-xmet 21308 df-met 21309 df-bl 21310 df-mopn 21311 df-fbas 21312 df-fg 21313 df-cnfld 21316 df-zring 21408 df-zrh 21464 df-zn 21467 df-top 22832 df-topon 22849 df-topsp 22871 df-bases 22884 df-cld 22957 df-ntr 22958 df-cls 22959 df-nei 23036 df-lp 23074 df-perf 23075 df-cn 23165 df-cnp 23166 df-haus 23253 df-tx 23500 df-hmeo 23693 df-fil 23784 df-fm 23876 df-flim 23877 df-flf 23878 df-xms 24259 df-ms 24260 df-tms 24261 df-cncf 24822 df-0p 25623 df-limc 25819 df-dv 25820 df-ply 26145 df-idp 26146 df-coe 26147 df-dgr 26148 df-quot 26251 df-log 26517 df-cxp 26518 df-dchr 27196

This theorem is referenced by: dchrhash 27234 sumdchr 27235

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