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Theorem sumdchr2 25528

Description: Lemma for sumdchr 25530. (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 2806	. 2 ⊢ ((♯‘𝐷) = if(𝐴 = 1 , (♯‘𝐷), 0) → (Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) = (♯‘𝐷) ↔ Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) = if(𝐴 = 1 , (♯‘𝐷), 0)))
2		eqeq2 2806	. 2 ⊢ (0 = if(𝐴 = 1 , (♯‘𝐷), 0) → (Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) = 0 ↔ Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) = if(𝐴 = 1 , (♯‘𝐷), 0)))
3		fveq2 6538	. . . . . 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 25499	. . . . . . . 8 ⊢ 𝐷 ⊆ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld))
8		simpr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ 𝐷)
9	7, 8	sseldi 3887	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)))
10		eqid 2795	. . . . . . . . 9 ⊢ (mulGrp‘𝑍) = (mulGrp‘𝑍)
11		sumdchr2.1	. . . . . . . . 9 ⊢ 1 = (1_r‘𝑍)
12	10, 11	ringidval 18943	. . . . . . . 8 ⊢ 1 = (0_g‘(mulGrp‘𝑍))
13		eqid 2795	. . . . . . . . 9 ⊢ (mulGrp‘ℂ_fld) = (mulGrp‘ℂ_fld)
14		cnfld1 20252	. . . . . . . . 9 ⊢ 1 = (1_r‘ℂ_fld)
15	13, 14	ringidval 18943	. . . . . . . 8 ⊢ 1 = (0_g‘(mulGrp‘ℂ_fld))
16	12, 15	mhm0 17782	. . . . . . 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 2853	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝐴 = 1 ) → (𝑥‘𝐴) = 1)
19	18	an32s 648	. . . 4 ⊢ (((𝜑 ∧ 𝐴 = 1 ) ∧ 𝑥 ∈ 𝐷) → (𝑥‘𝐴) = 1)
20	19	sumeq2dv 14893	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 1 ) → Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) = Σ𝑥 ∈ 𝐷 1)
21		sumdchr2.n	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ)
22	4, 6	dchrfi 25513	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → 𝐷 ∈ Fin)
23	21, 22	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ Fin)
24		ax-1cn 10441	. . . . . 6 ⊢ 1 ∈ ℂ
25		fsumconst 14978	. . . . . 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 13567	. . . . . . . 8 ⊢ (𝐷 ∈ Fin → (♯‘𝐷) ∈ ℕ₀)
28	21, 22, 27	3syl 18	. . . . . . 7 ⊢ (𝜑 → (♯‘𝐷) ∈ ℕ₀)
29	28	nn0cnd 11805	. . . . . 6 ⊢ (𝜑 → (♯‘𝐷) ∈ ℂ)
30	29	mulid1d 10504	. . . . 5 ⊢ (𝜑 → ((♯‘𝐷) · 1) = (♯‘𝐷))
31	26, 30	eqtrd 2831	. . . 4 ⊢ (𝜑 → Σ𝑥 ∈ 𝐷 1 = (♯‘𝐷))
32	31	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 1 ) → Σ𝑥 ∈ 𝐷 1 = (♯‘𝐷))
33	20, 32	eqtrd 2831	. 2 ⊢ ((𝜑 ∧ 𝐴 = 1 ) → Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) = (♯‘𝐷))
34		df-ne 2985	. . 3 ⊢ (𝐴 ≠ 1 ↔ ¬ 𝐴 = 1 )
35		sumdchr2.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑍)
36	21	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 1 ) → 𝑁 ∈ ℕ)
37		simpr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 1 ) → 𝐴 ≠ 1 )
38		sumdchr2.x	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
39	38	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 1 ) → 𝐴 ∈ 𝐵)
40	4, 5, 6, 35, 11, 36, 37, 39	dchrpt 25525	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 1 ) → ∃𝑦 ∈ 𝐷 (𝑦‘𝐴) ≠ 1)
41	36	adantr 481	. . . . . . 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 485	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ 𝐷)
44	4, 5, 6, 35, 43	dchrf 25500	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) ∧ 𝑥 ∈ 𝐷) → 𝑥:𝐵⟶ℂ)
45	39	adantr 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → 𝐴 ∈ 𝐵)
46	45	adantr 481	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) ∧ 𝑥 ∈ 𝐷) → 𝐴 ∈ 𝐵)
47	44, 46	ffvelrnd 6717	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) ∧ 𝑥 ∈ 𝐷) → (𝑥‘𝐴) ∈ ℂ)
48	42, 47	fsumcl 14923	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) ∈ ℂ)
49		0cnd 10480	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → 0 ∈ ℂ)
50		simprl 767	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → 𝑦 ∈ 𝐷)
51	4, 5, 6, 35, 50	dchrf 25500	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → 𝑦:𝐵⟶ℂ)
52	51, 45	ffvelrnd 6717	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → (𝑦‘𝐴) ∈ ℂ)
53		subcl 10732	. . . . . 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 769	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → (𝑦‘𝐴) ≠ 1)
56		subeq0 10760	. . . . . . . 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 3028	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → (((𝑦‘𝐴) − 1) ≠ 0 ↔ (𝑦‘𝐴) ≠ 1))
59	55, 58	mpbird 258	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → ((𝑦‘𝐴) − 1) ≠ 0)
60		oveq2 7024	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑥 → (𝑦(+_g‘𝐺)𝑧) = (𝑦(+_g‘𝐺)𝑥))
61	60	fveq1d 6540	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑥 → ((𝑦(+_g‘𝐺)𝑧)‘𝐴) = ((𝑦(+_g‘𝐺)𝑥)‘𝐴))
62	61	cbvsumv 14886	. . . . . . . . . 10 ⊢ Σ𝑧 ∈ 𝐷 ((𝑦(+_g‘𝐺)𝑧)‘𝐴) = Σ𝑥 ∈ 𝐷 ((𝑦(+_g‘𝐺)𝑥)‘𝐴)
63		eqid 2795	. . . . . . . . . . . . . 14 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
64	50	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) ∧ 𝑥 ∈ 𝐷) → 𝑦 ∈ 𝐷)
65	4, 5, 6, 63, 64, 43	dchrmul 25506	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) ∧ 𝑥 ∈ 𝐷) → (𝑦(+_g‘𝐺)𝑥) = (𝑦 ∘_𝑓 · 𝑥))
66	65	fveq1d 6540	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) ∧ 𝑥 ∈ 𝐷) → ((𝑦(+_g‘𝐺)𝑥)‘𝐴) = ((𝑦 ∘_𝑓 · 𝑥)‘𝐴))
67	51	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) ∧ 𝑥 ∈ 𝐷) → 𝑦:𝐵⟶ℂ)
68	67	ffnd 6383	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) ∧ 𝑥 ∈ 𝐷) → 𝑦 Fn 𝐵)
69	44	ffnd 6383	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) ∧ 𝑥 ∈ 𝐷) → 𝑥 Fn 𝐵)
70	35	fvexi 6552	. . . . . . . . . . . . . 14 ⊢ 𝐵 ∈ V
71	70	a1i 11	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) ∧ 𝑥 ∈ 𝐷) → 𝐵 ∈ V)
72		fnfvof 7281	. . . . . . . . . . . . 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 835	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) ∧ 𝑥 ∈ 𝐷) → ((𝑦 ∘_𝑓 · 𝑥)‘𝐴) = ((𝑦‘𝐴) · (𝑥‘𝐴)))
74	66, 73	eqtrd 2831	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) ∧ 𝑥 ∈ 𝐷) → ((𝑦(+_g‘𝐺)𝑥)‘𝐴) = ((𝑦‘𝐴) · (𝑥‘𝐴)))
75	74	sumeq2dv 14893	. . . . . . . . . 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	syl5eq 2843	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → Σ𝑧 ∈ 𝐷 ((𝑦(+_g‘𝐺)𝑧)‘𝐴) = Σ𝑥 ∈ 𝐷 ((𝑦‘𝐴) · (𝑥‘𝐴)))
77		fveq1 6537	. . . . . . . . . 10 ⊢ (𝑥 = (𝑦(+_g‘𝐺)𝑧) → (𝑥‘𝐴) = ((𝑦(+_g‘𝐺)𝑧)‘𝐴))
78	4	dchrabl 25512	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → 𝐺 ∈ Abel)
79		ablgrp 18638	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
80	41, 78, 79	3syl 18	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → 𝐺 ∈ Grp)
81		eqid 2795	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ 𝐷 ↦ (𝑏 ∈ 𝐷 ↦ (𝑎(+_g‘𝐺)𝑏))) = (𝑎 ∈ 𝐷 ↦ (𝑏 ∈ 𝐷 ↦ (𝑎(+_g‘𝐺)𝑏)))
82	81, 6, 63	grplactf1o 17960	. . . . . . . . . . 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 17958	. . . . . . . . . . 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 14913	. . . . . . . . 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 14972	. . . . . . . . 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 2842	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → ((𝑦‘𝐴) · Σ𝑥 ∈ 𝐷 (𝑥‘𝐴)) = Σ𝑥 ∈ 𝐷 (𝑥‘𝐴))
89	48	mulid2d 10505	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → (1 · Σ𝑥 ∈ 𝐷 (𝑥‘𝐴)) = Σ𝑥 ∈ 𝐷 (𝑥‘𝐴))
90	88, 89	oveq12d 7034	. . . . . . 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 10833	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → (Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) − Σ𝑥 ∈ 𝐷 (𝑥‘𝐴)) = 0)
92	90, 91	eqtrd 2831	. . . . . 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 10945	. . . . . 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 10686	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → (((𝑦‘𝐴) − 1) · 0) = 0)
96	92, 94, 95	3eqtr4d 2841	. . . . 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 11123	. . . 4 ⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) = 0)
98	40, 97	rexlimddv 3254	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ 1 ) → Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) = 0)
99	34, 98	sylan2br 594	. 2 ⊢ ((𝜑 ∧ ¬ 𝐴 = 1 ) → Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) = 0)
100	1, 2, 33, 99	ifbothda 4418	1 ⊢ (𝜑 → Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) = if(𝐴 = 1 , (♯‘𝐷), 0))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1522 ∈ wcel 2081 ≠ wne 2984 Vcvv 3437 ifcif 4381 ↦ cmpt 5041 Fn wfn 6220 ⟶wf 6221 –1-1-onto→wf1o 6224 ‘cfv 6225 (class class class)co 7016 ∘_𝑓 cof 7265 Fincfn 8357 ℂcc 10381 0cc0 10383 1c1 10384 · cmul 10388 − cmin 10717 ℕcn 11486 ℕ₀cn0 11745 ♯chash 13540 Σcsu 14876 Basecbs 16312 +_gcplusg 16394 MndHom cmhm 17772 Grpcgrp 17861 Abelcabl 18634 mulGrpcmgp 18929 1_rcur 18941 ℂ_fldccnfld 20227 ℤ/nℤczn 20332 DChrcdchr 25490

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5081 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-inf2 8950 ax-cnex 10439 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460 ax-pre-sup 10461 ax-addf 10462 ax-mulf 10463

This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-fal 1535 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-int 4783 df-iun 4827 df-iin 4828 df-disj 4931 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-se 5403 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-isom 6234 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-of 7267 df-rpss 7307 df-om 7437 df-1st 7545 df-2nd 7546 df-supp 7682 df-tpos 7743 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-1o 7953 df-2o 7954 df-oadd 7957 df-omul 7958 df-er 8139 df-ec 8141 df-qs 8145 df-map 8258 df-pm 8259 df-ixp 8311 df-en 8358 df-dom 8359 df-sdom 8360 df-fin 8361 df-fsupp 8680 df-fi 8721 df-sup 8752 df-inf 8753 df-oi 8820 df-dju 9176 df-card 9214 df-acn 9217 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 df-div 11146 df-nn 11487 df-2 11548 df-3 11549 df-4 11550 df-5 11551 df-6 11552 df-7 11553 df-8 11554 df-9 11555 df-n0 11746 df-xnn0 11816 df-z 11830 df-dec 11948 df-uz 12094 df-q 12198 df-rp 12240 df-xneg 12357 df-xadd 12358 df-xmul 12359 df-ioo 12592 df-ioc 12593 df-ico 12594 df-icc 12595 df-fz 12743 df-fzo 12884 df-fl 13012 df-mod 13088 df-seq 13220 df-exp 13280 df-fac 13484 df-bc 13513 df-hash 13541 df-word 13708 df-concat 13769 df-s1 13794 df-shft 14260 df-cj 14292 df-re 14293 df-im 14294 df-sqrt 14428 df-abs 14429 df-limsup 14662 df-clim 14679 df-rlim 14680 df-sum 14877 df-ef 15254 df-sin 15256 df-cos 15257 df-pi 15259 df-dvds 15441 df-gcd 15677 df-prm 15845 df-phi 15932 df-pc 16003 df-struct 16314 df-ndx 16315 df-slot 16316 df-base 16318 df-sets 16319 df-ress 16320 df-plusg 16407 df-mulr 16408 df-starv 16409 df-sca 16410 df-vsca 16411 df-ip 16412 df-tset 16413 df-ple 16414 df-ds 16416 df-unif 16417 df-hom 16418 df-cco 16419 df-rest 16525 df-topn 16526 df-0g 16544 df-gsum 16545 df-topgen 16546 df-pt 16547 df-prds 16550 df-xrs 16604 df-qtop 16609 df-imas 16610 df-qus 16611 df-xps 16612 df-mre 16686 df-mrc 16687 df-acs 16689 df-mgm 17681 df-sgrp 17723 df-mnd 17734 df-mhm 17774 df-submnd 17775 df-grp 17864 df-minusg 17865 df-sbg 17866 df-mulg 17982 df-subg 18030 df-nsg 18031 df-eqg 18032 df-ghm 18097 df-gim 18140 df-ga 18161 df-cntz 18188 df-oppg 18215 df-od 18387 df-gex 18388 df-pgp 18389 df-lsm 18491 df-pj1 18492 df-cmn 18635 df-abl 18636 df-cyg 18720 df-dprd 18834 df-dpj 18835 df-mgp 18930 df-ur 18942 df-ring 18989 df-cring 18990 df-oppr 19063 df-dvdsr 19081 df-unit 19082 df-invr 19112 df-rnghom 19157 df-subrg 19223 df-lmod 19326 df-lss 19394 df-lsp 19434 df-sra 19634 df-rgmod 19635 df-lidl 19636 df-rsp 19637 df-2idl 19694 df-psmet 20219 df-xmet 20220 df-met 20221 df-bl 20222 df-mopn 20223 df-fbas 20224 df-fg 20225 df-cnfld 20228 df-zring 20300 df-zrh 20333 df-zn 20336 df-top 21186 df-topon 21203 df-topsp 21225 df-bases 21238 df-cld 21311 df-ntr 21312 df-cls 21313 df-nei 21390 df-lp 21428 df-perf 21429 df-cn 21519 df-cnp 21520 df-haus 21607 df-tx 21854 df-hmeo 22047 df-fil 22138 df-fm 22230 df-flim 22231 df-flf 22232 df-xms 22613 df-ms 22614 df-tms 22615 df-cncf 23169 df-0p 23954 df-limc 24147 df-dv 24148 df-ply 24461 df-idp 24462 df-coe 24463 df-dgr 24464 df-quot 24563 df-log 24821 df-cxp 24822 df-dchr 25491

This theorem is referenced by: dchrhash 25529 sumdchr 25530

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