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Theorem sumdchr2 27560

Description: Lemma for sumdchr 27562. (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 2772	. 2 ⊢ ((♯‘𝐷) = if(𝐴 = 1 , (♯‘𝐷), 0) → (Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) = (♯‘𝐷) ↔ Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) = if(𝐴 = 1 , (♯‘𝐷), 0)))
2		eqeq2 2772	. 2 ⊢ (0 = if(𝐴 = 1 , (♯‘𝐷), 0) → (Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) = 0 ↔ Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) = if(𝐴 = 1 , (♯‘𝐷), 0)))
3		fveq2 6873	. . . . . 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 27531	. . . . . . . 8 ⊢ 𝐷 ⊆ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld))
8		simpr 490	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ 𝐷)
9	7, 8	sselid 3928	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)))
10		eqid 2760	. . . . . . . . 9 ⊢ (mulGrp‘𝑍) = (mulGrp‘𝑍)
11		sumdchr2.1	. . . . . . . . 9 ⊢ 1 = (1_r‘𝑍)
12	10, 11	ringidval 20370	. . . . . . . 8 ⊢ 1 = (0_g‘(mulGrp‘𝑍))
13		eqid 2760	. . . . . . . . 9 ⊢ (mulGrp‘ℂ_fld) = (mulGrp‘ℂ_fld)
14		cnfld1 21664	. . . . . . . . 9 ⊢ 1 = (1_r‘ℂ_fld)
15	13, 14	ringidval 20370	. . . . . . . 8 ⊢ 1 = (0_g‘(mulGrp‘ℂ_fld))
16	12, 15	mhm0 18950	. . . . . . 7 ⊢ (𝑥 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)) → (𝑥‘ 1 ) = 1)
17	9, 16	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑥‘ 1 ) = 1)
18	3, 17	sylan9eqr 2817	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝐴 = 1 ) → (𝑥‘𝐴) = 1)
19	18	an32s 665	. . . 4 ⊢ (((𝜑 ∧ 𝐴 = 1 ) ∧ 𝑥 ∈ 𝐷) → (𝑥‘𝐴) = 1)
20	19	sumeq2dv 15836	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 1 ) → Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) = Σ𝑥 ∈ 𝐷 1)
21		sumdchr2.n	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ)
22	4, 6	dchrfi 27545	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → 𝐷 ∈ Fin)
23	21, 22	syl 18	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ Fin)
24		ax-1cn 11229	. . . . . 6 ⊢ 1 ∈ ℂ
25		fsumconst 15923	. . . . . 6 ⊢ ((𝐷 ∈ Fin ∧ 1 ∈ ℂ) → Σ𝑥 ∈ 𝐷 1 = ((♯‘𝐷) · 1))
26	23, 24, 25	sylancl 598	. . . . 5 ⊢ (𝜑 → Σ𝑥 ∈ 𝐷 1 = ((♯‘𝐷) · 1))
27		hashcl 14467	. . . . . . . 8 ⊢ (𝐷 ∈ Fin → (♯‘𝐷) ∈ ℕ₀)
28	21, 22, 27	3syl 19	. . . . . . 7 ⊢ (𝜑 → (♯‘𝐷) ∈ ℕ₀)
29	28	nn0cnd 12638	. . . . . 6 ⊢ (𝜑 → (♯‘𝐷) ∈ ℂ)
30	29	mulridd 11297	. . . . 5 ⊢ (𝜑 → ((♯‘𝐷) · 1) = (♯‘𝐷))
31	26, 30	eqtrd 2795	. . . 4 ⊢ (𝜑 → Σ𝑥 ∈ 𝐷 1 = (♯‘𝐷))
32	31	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 1 ) → Σ𝑥 ∈ 𝐷 1 = (♯‘𝐷))
33	20, 32	eqtrd 2795	. 2 ⊢ ((𝜑 ∧ 𝐴 = 1 ) → Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) = (♯‘𝐷))
34		df-ne 2956	. . 3 ⊢ (𝐴 ≠ 1 ↔ ¬ 𝐴 = 1 )
35		sumdchr2.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑍)
36	21	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 1 ) → 𝑁 ∈ ℕ)
37		simpr 490	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 1 ) → 𝐴 ≠ 1 )
38		sumdchr2.x	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
39	38	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 1 ) → 𝐴 ∈ 𝐵)
40	4, 5, 6, 35, 11, 36, 37, 39	dchrpt 27557	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 1 ) → ∃𝑦 ∈ 𝐷 (𝑦‘𝐴) ≠ 1)
41	36	adantr 486	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → 𝑁 ∈ ℕ)
42	41, 22	syl 18	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → 𝐷 ∈ Fin)
43		simpr 490	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ 𝐷)
44	4, 5, 6, 35, 43	dchrf 27532	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) ∧ 𝑥 ∈ 𝐷) → 𝑥:𝐵⟶ℂ)
45	39	adantr 486	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → 𝐴 ∈ 𝐵)
46	45	adantr 486	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) ∧ 𝑥 ∈ 𝐷) → 𝐴 ∈ 𝐵)
47	44, 46	ffvelcdmd 7073	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) ∧ 𝑥 ∈ 𝐷) → (𝑥‘𝐴) ∈ ℂ)
48	42, 47	fsumcl 15866	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) ∈ ℂ)
49		0cnd 11270	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → 0 ∈ ℂ)
50		simprl 783	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → 𝑦 ∈ 𝐷)
51	4, 5, 6, 35, 50	dchrf 27532	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → 𝑦:𝐵⟶ℂ)
52	51, 45	ffvelcdmd 7073	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → (𝑦‘𝐴) ∈ ℂ)
53		subcl 11527	. . . . . 6 ⊢ (((𝑦‘𝐴) ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑦‘𝐴) − 1) ∈ ℂ)
54	52, 24, 53	sylancl 598	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → ((𝑦‘𝐴) − 1) ∈ ℂ)
55		simprr 785	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → (𝑦‘𝐴) ≠ 1)
56		subeq0 11555	. . . . . . . 8 ⊢ (((𝑦‘𝐴) ∈ ℂ ∧ 1 ∈ ℂ) → (((𝑦‘𝐴) − 1) = 0 ↔ (𝑦‘𝐴) = 1))
57	52, 24, 56	sylancl 598	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → (((𝑦‘𝐴) − 1) = 0 ↔ (𝑦‘𝐴) = 1))
58	57	necon3bid 2999	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → (((𝑦‘𝐴) − 1) ≠ 0 ↔ (𝑦‘𝐴) ≠ 1))
59	55, 58	mpbird 260	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → ((𝑦‘𝐴) − 1) ≠ 0)
60		oveq2 7416	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑥 → (𝑦(+_g‘𝐺)𝑧) = (𝑦(+_g‘𝐺)𝑥))
61	60	fveq1d 6875	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑥 → ((𝑦(+_g‘𝐺)𝑧)‘𝐴) = ((𝑦(+_g‘𝐺)𝑥)‘𝐴))
62	61	cbvsumv 15830	. . . . . . . . . 10 ⊢ Σ𝑧 ∈ 𝐷 ((𝑦(+_g‘𝐺)𝑧)‘𝐴) = Σ𝑥 ∈ 𝐷 ((𝑦(+_g‘𝐺)𝑥)‘𝐴)
63		eqid 2760	. . . . . . . . . . . . . 14 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
64	50	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) ∧ 𝑥 ∈ 𝐷) → 𝑦 ∈ 𝐷)
65	4, 5, 6, 63, 64, 43	dchrmul 27538	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) ∧ 𝑥 ∈ 𝐷) → (𝑦(+_g‘𝐺)𝑥) = (𝑦 ∘_f · 𝑥))
66	65	fveq1d 6875	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) ∧ 𝑥 ∈ 𝐷) → ((𝑦(+_g‘𝐺)𝑥)‘𝐴) = ((𝑦 ∘_f · 𝑥)‘𝐴))
67	51	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) ∧ 𝑥 ∈ 𝐷) → 𝑦:𝐵⟶ℂ)
68	67	ffnd 6698	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) ∧ 𝑥 ∈ 𝐷) → 𝑦 Fn 𝐵)
69	44	ffnd 6698	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) ∧ 𝑥 ∈ 𝐷) → 𝑥 Fn 𝐵)
70	35	fvexi 6887	. . . . . . . . . . . . . 14 ⊢ 𝐵 ∈ V
71	70	a1i 11	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) ∧ 𝑥 ∈ 𝐷) → 𝐵 ∈ V)
72		fnfvof 7693	. . . . . . . . . . . . 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 852	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) ∧ 𝑥 ∈ 𝐷) → ((𝑦 ∘_f · 𝑥)‘𝐴) = ((𝑦‘𝐴) · (𝑥‘𝐴)))
74	66, 73	eqtrd 2795	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) ∧ 𝑥 ∈ 𝐷) → ((𝑦(+_g‘𝐺)𝑥)‘𝐴) = ((𝑦‘𝐴) · (𝑥‘𝐴)))
75	74	sumeq2dv 15836	. . . . . . . . . 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 2807	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → Σ𝑧 ∈ 𝐷 ((𝑦(+_g‘𝐺)𝑧)‘𝐴) = Σ𝑥 ∈ 𝐷 ((𝑦‘𝐴) · (𝑥‘𝐴)))
77		fveq1 6872	. . . . . . . . . 10 ⊢ (𝑥 = (𝑦(+_g‘𝐺)𝑧) → (𝑥‘𝐴) = ((𝑦(+_g‘𝐺)𝑧)‘𝐴))
78	4	dchrabl 27544	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → 𝐺 ∈ Abel)
79		ablgrp 19960	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
80	41, 78, 79	3syl 19	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → 𝐺 ∈ Grp)
81		eqid 2760	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ 𝐷 ↦ (𝑏 ∈ 𝐷 ↦ (𝑎(+_g‘𝐺)𝑏))) = (𝑎 ∈ 𝐷 ↦ (𝑏 ∈ 𝐷 ↦ (𝑎(+_g‘𝐺)𝑏)))
82	81, 6, 63	grplactf1o 19215	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝐷) → ((𝑎 ∈ 𝐷 ↦ (𝑏 ∈ 𝐷 ↦ (𝑎(+_g‘𝐺)𝑏)))‘𝑦):𝐷–1-1-onto→𝐷)
83	80, 50, 82	syl2anc 596	. . . . . . . . . 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 19213	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝐷 ∧ 𝑧 ∈ 𝐷) → (((𝑎 ∈ 𝐷 ↦ (𝑏 ∈ 𝐷 ↦ (𝑎(+_g‘𝐺)𝑏)))‘𝑦)‘𝑧) = (𝑦(+_g‘𝐺)𝑧))
85	50, 84	sylan 592	. . . . . . . . . 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 15856	. . . . . . . . 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 15917	. . . . . . . . 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 2806	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → ((𝑦‘𝐴) · Σ𝑥 ∈ 𝐷 (𝑥‘𝐴)) = Σ𝑥 ∈ 𝐷 (𝑥‘𝐴))
89	48	mullidd 11298	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → (1 · Σ𝑥 ∈ 𝐷 (𝑥‘𝐴)) = Σ𝑥 ∈ 𝐷 (𝑥‘𝐴))
90	88, 89	oveq12d 7426	. . . . . . 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 11628	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → (Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) − Σ𝑥 ∈ 𝐷 (𝑥‘𝐴)) = 0)
92	90, 91	eqtrd 2795	. . . . . 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 11742	. . . . . 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 11480	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → (((𝑦‘𝐴) − 1) · 0) = 0)
96	92, 94, 95	3eqtr4d 2805	. . . . 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 11920	. . . 4 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) = 0)
98	40, 97	rexlimddv 3169	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ 1 ) → Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) = 0)
99	34, 98	sylan2br 607	. 2 ⊢ ((𝜑 ∧ ¬ 𝐴 = 1 ) → Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) = 0)
100	1, 2, 33, 99	ifbothda 4520	1 ⊢ (𝜑 → Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) = if(𝐴 = 1 , (♯‘𝐷), 0))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ≠ wne 2955 Vcvv 3450 ifcif 4481 ↦ cmpt 5185 Fn wfn 6522 ⟶wf 6523 –1-1-onto→wf1o 6526 ‘cfv 6527 (class class class)co 7408 ∘_f cof 7674 Fincfn 8951 ℂcc 11169 0cc0 11171 1c1 11172 · cmul 11176 − cmin 11512 ℕcn 12304 ℕ₀cn0 12575 ♯chash 14441 Σcsu 15820 Basecbs 17348 +_gcplusg 17389 MndHom cmhm 18937 Grpcgrp 19105 Abelcabl 19956 mulGrpcmgp 20321 1_rcur 20368 ℂ_fldccnfld 21639 ℤ/nℤczn 21769 DChrcdchr 27522

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7734 ax-inf2 9620 ax-cnex 11227 ax-resscn 11228 ax-1cn 11229 ax-icn 11230 ax-addcl 11231 ax-addrcl 11232 ax-mulcl 11233 ax-mulrcl 11234 ax-mulcom 11235 ax-addass 11236 ax-mulass 11237 ax-distr 11238 ax-i2m1 11239 ax-1ne0 11240 ax-1rid 11241 ax-rnegex 11242 ax-rrecex 11243 ax-cnre 11244 ax-pre-lttri 11245 ax-pre-lttrn 11246 ax-pre-ltadd 11247 ax-pre-mulgt0 11248 ax-pre-sup 11249 ax-addf 11250 ax-mulf 11251

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-tp 4588 df-op 4590 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-disj 5070 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-se 5601 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-pred 6293 df-ord 6354 df-on 6355 df-lim 6356 df-suc 6357 df-iota 6483 df-fun 6529 df-fn 6530 df-f 6531 df-f1 6532 df-fo 6533 df-f1o 6534 df-fv 6535 df-isom 6536 df-riota 7365 df-ov 7411 df-oprab 7412 df-mpo 7413 df-of 7676 df-rpss 7722 df-om 7861 df-1st 7984 df-2nd 7985 df-supp 8156 df-tpos 8221 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-1o 8454 df-2o 8455 df-oadd 8458 df-omul 8459 df-er 8695 df-ec 8697 df-qs 8701 df-map 8827 df-pm 8828 df-ixp 8904 df-en 8952 df-dom 8953 df-sdom 8954 df-fin 8955 df-fsupp 9332 df-fi 9381 df-sup 9412 df-inf 9413 df-oi 9482 df-dju 9953 df-card 9991 df-acn 9994 df-pnf 11316 df-mnf 11317 df-xr 11318 df-ltxr 11319 df-le 11320 df-sub 11514 df-neg 11515 df-div 11943 df-nn 12305 df-2 12374 df-3 12375 df-4 12376 df-5 12377 df-6 12378 df-7 12379 df-8 12380 df-9 12381 df-n0 12576 df-xnn0 12649 df-z 12663 df-dec 12784 df-uz 12935 df-q 13045 df-rp 13090 df-xneg 13210 df-xadd 13211 df-xmul 13212 df-ioo 13449 df-ioc 13450 df-ico 13451 df-icc 13452 df-fz 13609 df-fzo 13757 df-fl 13900 df-mod 13978 df-seq 14113 df-exp 14173 df-fac 14385 df-bc 14414 df-hash 14442 df-word 14626 df-concat 14683 df-s1 14710 df-shft 15187 df-cj 15233 df-re 15234 df-im 15235 df-sqrt 15369 df-abs 15370 df-limsup 15605 df-clim 15622 df-rlim 15623 df-sum 15821 df-ef 16200 df-sin 16202 df-cos 16203 df-pi 16205 df-dvds 16390 df-gcd 16632 df-prm 16809 df-phi 16904 df-pc 16976 df-struct 17286 df-sets 17303 df-slot 17321 df-ndx 17333 df-base 17349 df-ress 17370 df-plusg 17402 df-mulr 17403 df-starv 17404 df-sca 17405 df-vsca 17406 df-ip 17407 df-tset 17408 df-ple 17409 df-ds 17411 df-unif 17412 df-hom 17413 df-cco 17414 df-rest 17554 df-topn 17555 df-0g 17573 df-gsum 17574 df-topgen 17575 df-pt 17576 df-prds 17579 df-xrs 17635 df-qtop 17640 df-imas 17641 df-qus 17642 df-xps 17643 df-mre 17717 df-mrc 17718 df-acs 17720 df-mgm 18777 df-sgrp 18869 df-mnd 18885 df-mhm 18939 df-submnd 18940 df-grp 19108 df-minusg 19109 df-sbg 19110 df-mulg 19239 df-subg 19294 df-nsg 19295 df-eqg 19296 df-ghm 19389 df-gim 19434 df-ga 19465 df-cntz 19492 df-oppg 19521 df-od 19703 df-gex 19704 df-pgp 19705 df-lsm 19811 df-pj1 19812 df-cmn 19957 df-abl 19958 df-cyg 20053 df-dprd 20172 df-dpj 20173 df-mgp 20322 df-rng 20336 df-ur 20369 df-ring 20422 df-cring 20423 df-oppr 20528 df-dvdsr 20548 df-unit 20549 df-invr 20579 df-rhm 20663 df-subrng 20759 df-subrg 20783 df-lmod 21098 df-lss 21168 df-lsp 21208 df-sra 21409 df-rgmod 21410 df-lidl 21447 df-rsp 21448 df-2idl 21504 df-psmet 21631 df-xmet 21632 df-met 21633 df-bl 21634 df-mopn 21635 df-fbas 21636 df-fg 21637 df-cnfld 21640 df-zring 21714 df-zrh 21770 df-zn 21773 df-top 23173 df-topon 23190 df-topsp 23212 df-bases 23225 df-cld 23298 df-ntr 23299 df-cls 23300 df-nei 23377 df-lp 23415 df-perf 23416 df-cn 23506 df-cnp 23507 df-haus 23594 df-tx 23842 df-hmeo 24035 df-fil 24126 df-fm 24218 df-flim 24219 df-flf 24220 df-xms 24600 df-ms 24601 df-tms 24602 df-cncf 25160 df-0p 25952 df-limc 26147 df-dv 26148 df-ply 26467 df-idp 26468 df-coe 26469 df-dgr 26470 df-quot 26575 df-log 26847 df-cxp 26848 df-dchr 27523

This theorem is used by: dchrhash 27561 sumdchr 27562

W3C validator