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Theorem sumdchr2 27242

Description: Lemma for sumdchr 27244. (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 2749	. 2 ⊢ ((♯‘𝐷) = if(𝐴 = 1 , (♯‘𝐷), 0) → (Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) = (♯‘𝐷) ↔ Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) = if(𝐴 = 1 , (♯‘𝐷), 0)))
2		eqeq2 2749	. 2 ⊢ (0 = if(𝐴 = 1 , (♯‘𝐷), 0) → (Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) = 0 ↔ Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) = if(𝐴 = 1 , (♯‘𝐷), 0)))
3		fveq2 6835	. . . . . 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 27213	. . . . . . . 8 ⊢ 𝐷 ⊆ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld))
8		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ 𝐷)
9	7, 8	sselid 3932	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)))
10		eqid 2737	. . . . . . . . 9 ⊢ (mulGrp‘𝑍) = (mulGrp‘𝑍)
11		sumdchr2.1	. . . . . . . . 9 ⊢ 1 = (1_r‘𝑍)
12	10, 11	ringidval 20123	. . . . . . . 8 ⊢ 1 = (0_g‘(mulGrp‘𝑍))
13		eqid 2737	. . . . . . . . 9 ⊢ (mulGrp‘ℂ_fld) = (mulGrp‘ℂ_fld)
14		cnfld1 21353	. . . . . . . . 9 ⊢ 1 = (1_r‘ℂ_fld)
15	13, 14	ringidval 20123	. . . . . . . 8 ⊢ 1 = (0_g‘(mulGrp‘ℂ_fld))
16	12, 15	mhm0 18724	. . . . . . 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 2794	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝐴 = 1 ) → (𝑥‘𝐴) = 1)
19	18	an32s 653	. . . 4 ⊢ (((𝜑 ∧ 𝐴 = 1 ) ∧ 𝑥 ∈ 𝐷) → (𝑥‘𝐴) = 1)
20	19	sumeq2dv 15630	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 1 ) → Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) = Σ𝑥 ∈ 𝐷 1)
21		sumdchr2.n	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ)
22	4, 6	dchrfi 27227	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → 𝐷 ∈ Fin)
23	21, 22	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ Fin)
24		ax-1cn 11089	. . . . . 6 ⊢ 1 ∈ ℂ
25		fsumconst 15718	. . . . . 6 ⊢ ((𝐷 ∈ Fin ∧ 1 ∈ ℂ) → Σ𝑥 ∈ 𝐷 1 = ((♯‘𝐷) · 1))
26	23, 24, 25	sylancl 587	. . . . 5 ⊢ (𝜑 → Σ𝑥 ∈ 𝐷 1 = ((♯‘𝐷) · 1))
27		hashcl 14284	. . . . . . . 8 ⊢ (𝐷 ∈ Fin → (♯‘𝐷) ∈ ℕ₀)
28	21, 22, 27	3syl 18	. . . . . . 7 ⊢ (𝜑 → (♯‘𝐷) ∈ ℕ₀)
29	28	nn0cnd 12469	. . . . . 6 ⊢ (𝜑 → (♯‘𝐷) ∈ ℂ)
30	29	mulridd 11154	. . . . 5 ⊢ (𝜑 → ((♯‘𝐷) · 1) = (♯‘𝐷))
31	26, 30	eqtrd 2772	. . . 4 ⊢ (𝜑 → Σ𝑥 ∈ 𝐷 1 = (♯‘𝐷))
32	31	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 1 ) → Σ𝑥 ∈ 𝐷 1 = (♯‘𝐷))
33	20, 32	eqtrd 2772	. 2 ⊢ ((𝜑 ∧ 𝐴 = 1 ) → Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) = (♯‘𝐷))
34		df-ne 2934	. . 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 27239	. . . 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 27214	. . . . . . 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 7032	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) ∧ 𝑥 ∈ 𝐷) → (𝑥‘𝐴) ∈ ℂ)
48	42, 47	fsumcl 15661	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) ∈ ℂ)
49		0cnd 11130	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → 0 ∈ ℂ)
50		simprl 771	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → 𝑦 ∈ 𝐷)
51	4, 5, 6, 35, 50	dchrf 27214	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → 𝑦:𝐵⟶ℂ)
52	51, 45	ffvelcdmd 7032	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → (𝑦‘𝐴) ∈ ℂ)
53		subcl 11384	. . . . . 6 ⊢ (((𝑦‘𝐴) ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑦‘𝐴) − 1) ∈ ℂ)
54	52, 24, 53	sylancl 587	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → ((𝑦‘𝐴) − 1) ∈ ℂ)
55		simprr 773	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → (𝑦‘𝐴) ≠ 1)
56		subeq0 11412	. . . . . . . 8 ⊢ (((𝑦‘𝐴) ∈ ℂ ∧ 1 ∈ ℂ) → (((𝑦‘𝐴) − 1) = 0 ↔ (𝑦‘𝐴) = 1))
57	52, 24, 56	sylancl 587	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → (((𝑦‘𝐴) − 1) = 0 ↔ (𝑦‘𝐴) = 1))
58	57	necon3bid 2977	. . . . . 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 7369	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑥 → (𝑦(+_g‘𝐺)𝑧) = (𝑦(+_g‘𝐺)𝑥))
61	60	fveq1d 6837	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑥 → ((𝑦(+_g‘𝐺)𝑧)‘𝐴) = ((𝑦(+_g‘𝐺)𝑥)‘𝐴))
62	61	cbvsumv 15624	. . . . . . . . . 10 ⊢ Σ𝑧 ∈ 𝐷 ((𝑦(+_g‘𝐺)𝑧)‘𝐴) = Σ𝑥 ∈ 𝐷 ((𝑦(+_g‘𝐺)𝑥)‘𝐴)
63		eqid 2737	. . . . . . . . . . . . . 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 27220	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) ∧ 𝑥 ∈ 𝐷) → (𝑦(+_g‘𝐺)𝑥) = (𝑦 ∘_f · 𝑥))
66	65	fveq1d 6837	. . . . . . . . . . . 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 6664	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) ∧ 𝑥 ∈ 𝐷) → 𝑦 Fn 𝐵)
69	44	ffnd 6664	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) ∧ 𝑥 ∈ 𝐷) → 𝑥 Fn 𝐵)
70	35	fvexi 6849	. . . . . . . . . . . . . 14 ⊢ 𝐵 ∈ V
71	70	a1i 11	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) ∧ 𝑥 ∈ 𝐷) → 𝐵 ∈ V)
72		fnfvof 7642	. . . . . . . . . . . . 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 839	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) ∧ 𝑥 ∈ 𝐷) → ((𝑦 ∘_f · 𝑥)‘𝐴) = ((𝑦‘𝐴) · (𝑥‘𝐴)))
74	66, 73	eqtrd 2772	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) ∧ 𝑥 ∈ 𝐷) → ((𝑦(+_g‘𝐺)𝑥)‘𝐴) = ((𝑦‘𝐴) · (𝑥‘𝐴)))
75	74	sumeq2dv 15630	. . . . . . . . . 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 2784	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → Σ𝑧 ∈ 𝐷 ((𝑦(+_g‘𝐺)𝑧)‘𝐴) = Σ𝑥 ∈ 𝐷 ((𝑦‘𝐴) · (𝑥‘𝐴)))
77		fveq1 6834	. . . . . . . . . 10 ⊢ (𝑥 = (𝑦(+_g‘𝐺)𝑧) → (𝑥‘𝐴) = ((𝑦(+_g‘𝐺)𝑧)‘𝐴))
78	4	dchrabl 27226	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → 𝐺 ∈ Abel)
79		ablgrp 19719	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
80	41, 78, 79	3syl 18	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → 𝐺 ∈ Grp)
81		eqid 2737	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ 𝐷 ↦ (𝑏 ∈ 𝐷 ↦ (𝑎(+_g‘𝐺)𝑏))) = (𝑎 ∈ 𝐷 ↦ (𝑏 ∈ 𝐷 ↦ (𝑎(+_g‘𝐺)𝑏)))
82	81, 6, 63	grplactf1o 18979	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝐷) → ((𝑎 ∈ 𝐷 ↦ (𝑏 ∈ 𝐷 ↦ (𝑎(+_g‘𝐺)𝑏)))‘𝑦):𝐷–1-1-onto→𝐷)
83	80, 50, 82	syl2anc 585	. . . . . . . . . 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 18977	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝐷 ∧ 𝑧 ∈ 𝐷) → (((𝑎 ∈ 𝐷 ↦ (𝑏 ∈ 𝐷 ↦ (𝑎(+_g‘𝐺)𝑏)))‘𝑦)‘𝑧) = (𝑦(+_g‘𝐺)𝑧))
85	50, 84	sylan 581	. . . . . . . . . 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 15651	. . . . . . . . 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 15712	. . . . . . . . 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 2783	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → ((𝑦‘𝐴) · Σ𝑥 ∈ 𝐷 (𝑥‘𝐴)) = Σ𝑥 ∈ 𝐷 (𝑥‘𝐴))
89	48	mullidd 11155	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → (1 · Σ𝑥 ∈ 𝐷 (𝑥‘𝐴)) = Σ𝑥 ∈ 𝐷 (𝑥‘𝐴))
90	88, 89	oveq12d 7379	. . . . . . 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 11485	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → (Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) − Σ𝑥 ∈ 𝐷 (𝑥‘𝐴)) = 0)
92	90, 91	eqtrd 2772	. . . . . 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 11599	. . . . . 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 11337	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → (((𝑦‘𝐴) − 1) · 0) = 0)
96	92, 94, 95	3eqtr4d 2782	. . . . 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 11777	. . . 4 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) = 0)
98	40, 97	rexlimddv 3144	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ 1 ) → Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) = 0)
99	34, 98	sylan2br 596	. 2 ⊢ ((𝜑 ∧ ¬ 𝐴 = 1 ) → Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) = 0)
100	1, 2, 33, 99	ifbothda 4519	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 1542 ∈ wcel 2114 ≠ wne 2933 Vcvv 3441 ifcif 4480 ↦ cmpt 5180 Fn wfn 6488 ⟶wf 6489 –1-1-onto→wf1o 6492 ‘cfv 6493 (class class class)co 7361 ∘_f cof 7623 Fincfn 8888 ℂcc 11029 0cc0 11031 1c1 11032 · cmul 11036 − cmin 11369 ℕcn 12150 ℕ₀cn0 12406 ♯chash 14258 Σcsu 15614 Basecbs 17141 +_gcplusg 17182 MndHom cmhm 18711 Grpcgrp 18868 Abelcabl 19715 mulGrpcmgp 20080 1_rcur 20121 ℂ_fldccnfld 21314 ℤ/nℤczn 21462 DChrcdchr 27204

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 5225 ax-sep 5242 ax-nul 5252 ax-pow 5311 ax-pr 5378 ax-un 7683 ax-inf2 9555 ax-cnex 11087 ax-resscn 11088 ax-1cn 11089 ax-icn 11090 ax-addcl 11091 ax-addrcl 11092 ax-mulcl 11093 ax-mulrcl 11094 ax-mulcom 11095 ax-addass 11096 ax-mulass 11097 ax-distr 11098 ax-i2m1 11099 ax-1ne0 11100 ax-1rid 11101 ax-rnegex 11102 ax-rrecex 11103 ax-cnre 11104 ax-pre-lttri 11105 ax-pre-lttrn 11106 ax-pre-ltadd 11107 ax-pre-mulgt0 11108 ax-pre-sup 11109 ax-addf 11110 ax-mulf 11111

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 3062 df-rmo 3351 df-reu 3352 df-rab 3401 df-v 3443 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4287 df-if 4481 df-pw 4557 df-sn 4582 df-pr 4584 df-tp 4586 df-op 4588 df-uni 4865 df-int 4904 df-iun 4949 df-iin 4950 df-disj 5067 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7318 df-ov 7364 df-oprab 7365 df-mpo 7366 df-of 7625 df-rpss 7671 df-om 7812 df-1st 7936 df-2nd 7937 df-supp 8106 df-tpos 8171 df-frecs 8226 df-wrecs 8257 df-recs 8306 df-rdg 8344 df-1o 8400 df-2o 8401 df-oadd 8404 df-omul 8405 df-er 8638 df-ec 8640 df-qs 8644 df-map 8770 df-pm 8771 df-ixp 8841 df-en 8889 df-dom 8890 df-sdom 8891 df-fin 8892 df-fsupp 9270 df-fi 9319 df-sup 9350 df-inf 9351 df-oi 9420 df-dju 9818 df-card 9856 df-acn 9859 df-pnf 11173 df-mnf 11174 df-xr 11175 df-ltxr 11176 df-le 11177 df-sub 11371 df-neg 11372 df-div 11800 df-nn 12151 df-2 12213 df-3 12214 df-4 12215 df-5 12216 df-6 12217 df-7 12218 df-8 12219 df-9 12220 df-n0 12407 df-xnn0 12480 df-z 12494 df-dec 12613 df-uz 12757 df-q 12867 df-rp 12911 df-xneg 13031 df-xadd 13032 df-xmul 13033 df-ioo 13270 df-ioc 13271 df-ico 13272 df-icc 13273 df-fz 13429 df-fzo 13576 df-fl 13717 df-mod 13795 df-seq 13930 df-exp 13990 df-fac 14202 df-bc 14231 df-hash 14259 df-word 14442 df-concat 14499 df-s1 14525 df-shft 14995 df-cj 15027 df-re 15028 df-im 15029 df-sqrt 15163 df-abs 15164 df-limsup 15399 df-clim 15416 df-rlim 15417 df-sum 15615 df-ef 15995 df-sin 15997 df-cos 15998 df-pi 16000 df-dvds 16185 df-gcd 16427 df-prm 16604 df-phi 16698 df-pc 16770 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17142 df-ress 17163 df-plusg 17195 df-mulr 17196 df-starv 17197 df-sca 17198 df-vsca 17199 df-ip 17200 df-tset 17201 df-ple 17202 df-ds 17204 df-unif 17205 df-hom 17206 df-cco 17207 df-rest 17347 df-topn 17348 df-0g 17366 df-gsum 17367 df-topgen 17368 df-pt 17369 df-prds 17372 df-xrs 17428 df-qtop 17433 df-imas 17434 df-qus 17435 df-xps 17436 df-mre 17510 df-mrc 17511 df-acs 17513 df-mgm 18570 df-sgrp 18649 df-mnd 18665 df-mhm 18713 df-submnd 18714 df-grp 18871 df-minusg 18872 df-sbg 18873 df-mulg 19003 df-subg 19058 df-nsg 19059 df-eqg 19060 df-ghm 19147 df-gim 19193 df-ga 19224 df-cntz 19251 df-oppg 19280 df-od 19462 df-gex 19463 df-pgp 19464 df-lsm 19570 df-pj1 19571 df-cmn 19716 df-abl 19717 df-cyg 19812 df-dprd 19931 df-dpj 19932 df-mgp 20081 df-rng 20093 df-ur 20122 df-ring 20175 df-cring 20176 df-oppr 20278 df-dvdsr 20298 df-unit 20299 df-invr 20329 df-rhm 20413 df-subrng 20484 df-subrg 20508 df-lmod 20818 df-lss 20888 df-lsp 20928 df-sra 21130 df-rgmod 21131 df-lidl 21168 df-rsp 21169 df-2idl 21210 df-psmet 21306 df-xmet 21307 df-met 21308 df-bl 21309 df-mopn 21310 df-fbas 21311 df-fg 21312 df-cnfld 21315 df-zring 21407 df-zrh 21463 df-zn 21466 df-top 22843 df-topon 22860 df-topsp 22882 df-bases 22895 df-cld 22968 df-ntr 22969 df-cls 22970 df-nei 23047 df-lp 23085 df-perf 23086 df-cn 23176 df-cnp 23177 df-haus 23264 df-tx 23511 df-hmeo 23704 df-fil 23795 df-fm 23887 df-flim 23888 df-flf 23889 df-xms 24269 df-ms 24270 df-tms 24271 df-cncf 24832 df-0p 25632 df-limc 25828 df-dv 25829 df-ply 26154 df-idp 26155 df-coe 26156 df-dgr 26157 df-quot 26260 df-log 26526 df-cxp 26527 df-dchr 27205

This theorem is referenced by: dchrhash 27243 sumdchr 27244

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