Theorem sumss 14302

Description: Change the index set to a subset in an upper integer sum. (Contributed by Mario Carneiro, 21-Apr-2014.)

Hypotheses

Ref	Expression
sumss.1	⊢ (𝜑 → 𝐴 ⊆ 𝐵)
sumss.2	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ)
sumss.3	⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ 𝐴)) → 𝐶 = 0)
sumss.4	⊢ (𝜑 → 𝐵 ⊆ (ℤ≥‘𝑀))

Assertion

Ref	Expression
sumss	⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐶)

Distinct variable groups:   𝐴,𝑘   𝐵,𝑘   𝜑,𝑘   𝑘,𝑀

Allowed substitution hint:   𝐶(𝑘)

Proof of Theorem sumss

Dummy variable 𝑚 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2610	. . . . 5 ⊢ (ℤ≥‘𝑀) = (ℤ≥‘𝑀)
2		simpr 476	. . . . 5 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → 𝑀 ∈ ℤ)
3		sumss.1	. . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
4		sumss.4	. . . . . . 7 ⊢ (𝜑 → 𝐵 ⊆ (ℤ≥‘𝑀))
5	3, 4	sstrd 3578	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ (ℤ≥‘𝑀))
6	5	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → 𝐴 ⊆ (ℤ≥‘𝑀))
7		nfcv 2751	. . . . . . 7 ⊢ Ⅎ𝑘𝑚
8		nffvmpt1 6111	. . . . . . . 8 ⊢ Ⅎ𝑘((𝑘 ∈ (ℤ≥‘𝑀) ↦ if(𝑘 ∈ 𝐴, 𝐶, 0))‘𝑚)
9		nfv 1830	. . . . . . . . 9 ⊢ Ⅎ𝑘 𝑚 ∈ 𝐴
10		nffvmpt1 6111	. . . . . . . . 9 ⊢ Ⅎ𝑘((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚)
11		nfcv 2751	. . . . . . . . 9 ⊢ Ⅎ𝑘0
12	9, 10, 11	nfif 4065	. . . . . . . 8 ⊢ Ⅎ𝑘if(𝑚 ∈ 𝐴, ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚), 0)
13	8, 12	nfeq 2762	. . . . . . 7 ⊢ Ⅎ𝑘((𝑘 ∈ (ℤ≥‘𝑀) ↦ if(𝑘 ∈ 𝐴, 𝐶, 0))‘𝑚) = if(𝑚 ∈ 𝐴, ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚), 0)
14		fveq2 6103	. . . . . . . 8 ⊢ (𝑘 = 𝑚 → ((𝑘 ∈ (ℤ≥‘𝑀) ↦ if(𝑘 ∈ 𝐴, 𝐶, 0))‘𝑘) = ((𝑘 ∈ (ℤ≥‘𝑀) ↦ if(𝑘 ∈ 𝐴, 𝐶, 0))‘𝑚))
15		eleq1 2676	. . . . . . . . 9 ⊢ (𝑘 = 𝑚 → (𝑘 ∈ 𝐴 ↔ 𝑚 ∈ 𝐴))
16		fveq2 6103	. . . . . . . . 9 ⊢ (𝑘 = 𝑚 → ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑘) = ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚))
17	15, 16	ifbieq1d 4059	. . . . . . . 8 ⊢ (𝑘 = 𝑚 → if(𝑘 ∈ 𝐴, ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑘), 0) = if(𝑚 ∈ 𝐴, ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚), 0))
18	14, 17	eqeq12d 2625	. . . . . . 7 ⊢ (𝑘 = 𝑚 → (((𝑘 ∈ (ℤ≥‘𝑀) ↦ if(𝑘 ∈ 𝐴, 𝐶, 0))‘𝑘) = if(𝑘 ∈ 𝐴, ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑘), 0) ↔ ((𝑘 ∈ (ℤ≥‘𝑀) ↦ if(𝑘 ∈ 𝐴, 𝐶, 0))‘𝑚) = if(𝑚 ∈ 𝐴, ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚), 0)))
19		eqid 2610	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) ↦ if(𝑘 ∈ 𝐴, 𝐶, 0)) = (𝑘 ∈ (ℤ≥‘𝑀) ↦ if(𝑘 ∈ 𝐴, 𝐶, 0))
20	19	fvmpt2i 6199	. . . . . . . . . 10 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → ((𝑘 ∈ (ℤ≥‘𝑀) ↦ if(𝑘 ∈ 𝐴, 𝐶, 0))‘𝑘) = ( I ‘if(𝑘 ∈ 𝐴, 𝐶, 0)))
21		iftrue 4042	. . . . . . . . . . 11 ⊢ (𝑘 ∈ 𝐴 → if(𝑘 ∈ 𝐴, 𝐶, 0) = 𝐶)
22	21	fveq2d 6107	. . . . . . . . . 10 ⊢ (𝑘 ∈ 𝐴 → ( I ‘if(𝑘 ∈ 𝐴, 𝐶, 0)) = ( I ‘𝐶))
23	20, 22	sylan9eq 2664	. . . . . . . . 9 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ 𝑘 ∈ 𝐴) → ((𝑘 ∈ (ℤ≥‘𝑀) ↦ if(𝑘 ∈ 𝐴, 𝐶, 0))‘𝑘) = ( I ‘𝐶))
24		iftrue 4042	. . . . . . . . . . 11 ⊢ (𝑘 ∈ 𝐴 → if(𝑘 ∈ 𝐴, ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑘), 0) = ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑘))
25		eqid 2610	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐶) = (𝑘 ∈ 𝐴 ↦ 𝐶)
26	25	fvmpt2i 6199	. . . . . . . . . . 11 ⊢ (𝑘 ∈ 𝐴 → ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑘) = ( I ‘𝐶))
27	24, 26	eqtrd 2644	. . . . . . . . . 10 ⊢ (𝑘 ∈ 𝐴 → if(𝑘 ∈ 𝐴, ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑘), 0) = ( I ‘𝐶))
28	27	adantl 481	. . . . . . . . 9 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ 𝑘 ∈ 𝐴) → if(𝑘 ∈ 𝐴, ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑘), 0) = ( I ‘𝐶))
29	23, 28	eqtr4d 2647	. . . . . . . 8 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ 𝑘 ∈ 𝐴) → ((𝑘 ∈ (ℤ≥‘𝑀) ↦ if(𝑘 ∈ 𝐴, 𝐶, 0))‘𝑘) = if(𝑘 ∈ 𝐴, ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑘), 0))
30		iffalse 4045	. . . . . . . . . . . 12 ⊢ (¬ 𝑘 ∈ 𝐴 → if(𝑘 ∈ 𝐴, 𝐶, 0) = 0)
31	30	fveq2d 6107	. . . . . . . . . . 11 ⊢ (¬ 𝑘 ∈ 𝐴 → ( I ‘if(𝑘 ∈ 𝐴, 𝐶, 0)) = ( I ‘0))
32		0z 11265	. . . . . . . . . . . 12 ⊢ 0 ∈ ℤ
33		fvi 6165	. . . . . . . . . . . 12 ⊢ (0 ∈ ℤ → ( I ‘0) = 0)
34	32, 33	ax-mp 5	. . . . . . . . . . 11 ⊢ ( I ‘0) = 0
35	31, 34	syl6eq 2660	. . . . . . . . . 10 ⊢ (¬ 𝑘 ∈ 𝐴 → ( I ‘if(𝑘 ∈ 𝐴, 𝐶, 0)) = 0)
36	20, 35	sylan9eq 2664	. . . . . . . . 9 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ¬ 𝑘 ∈ 𝐴) → ((𝑘 ∈ (ℤ≥‘𝑀) ↦ if(𝑘 ∈ 𝐴, 𝐶, 0))‘𝑘) = 0)
37		iffalse 4045	. . . . . . . . . 10 ⊢ (¬ 𝑘 ∈ 𝐴 → if(𝑘 ∈ 𝐴, ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑘), 0) = 0)
38	37	adantl 481	. . . . . . . . 9 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ¬ 𝑘 ∈ 𝐴) → if(𝑘 ∈ 𝐴, ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑘), 0) = 0)
39	36, 38	eqtr4d 2647	. . . . . . . 8 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ¬ 𝑘 ∈ 𝐴) → ((𝑘 ∈ (ℤ≥‘𝑀) ↦ if(𝑘 ∈ 𝐴, 𝐶, 0))‘𝑘) = if(𝑘 ∈ 𝐴, ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑘), 0))
40	29, 39	pm2.61dan 828	. . . . . . 7 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → ((𝑘 ∈ (ℤ≥‘𝑀) ↦ if(𝑘 ∈ 𝐴, 𝐶, 0))‘𝑘) = if(𝑘 ∈ 𝐴, ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑘), 0))
41	7, 13, 18, 40	vtoclgaf 3244	. . . . . 6 ⊢ (𝑚 ∈ (ℤ≥‘𝑀) → ((𝑘 ∈ (ℤ≥‘𝑀) ↦ if(𝑘 ∈ 𝐴, 𝐶, 0))‘𝑚) = if(𝑚 ∈ 𝐴, ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚), 0))
42	41	adantl 481	. . . . 5 ⊢ (((𝜑 ∧ 𝑀 ∈ ℤ) ∧ 𝑚 ∈ (ℤ≥‘𝑀)) → ((𝑘 ∈ (ℤ≥‘𝑀) ↦ if(𝑘 ∈ 𝐴, 𝐶, 0))‘𝑚) = if(𝑚 ∈ 𝐴, ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚), 0))
43		sumss.2	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ)
44	43, 25	fmptd 6292	. . . . . . 7 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐶):𝐴⟶ℂ)
45	44	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → (𝑘 ∈ 𝐴 ↦ 𝐶):𝐴⟶ℂ)
46	45	ffvelrnda 6267	. . . . 5 ⊢ (((𝜑 ∧ 𝑀 ∈ ℤ) ∧ 𝑚 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚) ∈ ℂ)
47	1, 2, 6, 42, 46	zsum 14296	. . . 4 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚) = ( ⇝ ‘seq𝑀( + , (𝑘 ∈ (ℤ≥‘𝑀) ↦ if(𝑘 ∈ 𝐴, 𝐶, 0)))))
48	4	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → 𝐵 ⊆ (ℤ≥‘𝑀))
49		nfv 1830	. . . . . . . . 9 ⊢ Ⅎ𝑘𝜑
50		nfv 1830	. . . . . . . . . . 11 ⊢ Ⅎ𝑘 𝑚 ∈ 𝐵
51		nffvmpt1 6111	. . . . . . . . . . 11 ⊢ Ⅎ𝑘((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚)
52	50, 51, 11	nfif 4065	. . . . . . . . . 10 ⊢ Ⅎ𝑘if(𝑚 ∈ 𝐵, ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚), 0)
53	8, 52	nfeq 2762	. . . . . . . . 9 ⊢ Ⅎ𝑘((𝑘 ∈ (ℤ≥‘𝑀) ↦ if(𝑘 ∈ 𝐴, 𝐶, 0))‘𝑚) = if(𝑚 ∈ 𝐵, ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚), 0)
54	49, 53	nfim 1813	. . . . . . . 8 ⊢ Ⅎ𝑘(𝜑 → ((𝑘 ∈ (ℤ≥‘𝑀) ↦ if(𝑘 ∈ 𝐴, 𝐶, 0))‘𝑚) = if(𝑚 ∈ 𝐵, ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚), 0))
55		eleq1 2676	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑚 → (𝑘 ∈ 𝐵 ↔ 𝑚 ∈ 𝐵))
56		fveq2 6103	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑚 → ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑘) = ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚))
57	55, 56	ifbieq1d 4059	. . . . . . . . . 10 ⊢ (𝑘 = 𝑚 → if(𝑘 ∈ 𝐵, ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑘), 0) = if(𝑚 ∈ 𝐵, ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚), 0))
58	14, 57	eqeq12d 2625	. . . . . . . . 9 ⊢ (𝑘 = 𝑚 → (((𝑘 ∈ (ℤ≥‘𝑀) ↦ if(𝑘 ∈ 𝐴, 𝐶, 0))‘𝑘) = if(𝑘 ∈ 𝐵, ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑘), 0) ↔ ((𝑘 ∈ (ℤ≥‘𝑀) ↦ if(𝑘 ∈ 𝐴, 𝐶, 0))‘𝑚) = if(𝑚 ∈ 𝐵, ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚), 0)))
59	58	imbi2d 329	. . . . . . . 8 ⊢ (𝑘 = 𝑚 → ((𝜑 → ((𝑘 ∈ (ℤ≥‘𝑀) ↦ if(𝑘 ∈ 𝐴, 𝐶, 0))‘𝑘) = if(𝑘 ∈ 𝐵, ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑘), 0)) ↔ (𝜑 → ((𝑘 ∈ (ℤ≥‘𝑀) ↦ if(𝑘 ∈ 𝐴, 𝐶, 0))‘𝑚) = if(𝑚 ∈ 𝐵, ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚), 0))))
60	23	adantll 746	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) ∧ 𝑘 ∈ 𝐴) → ((𝑘 ∈ (ℤ≥‘𝑀) ↦ if(𝑘 ∈ 𝐴, 𝐶, 0))‘𝑘) = ( I ‘𝐶))
61	3	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → 𝐴 ⊆ 𝐵)
62	61	sselda 3568	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ 𝐵)
63		iftrue 4042	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ 𝐵 → if(𝑘 ∈ 𝐵, ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑘), 0) = ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑘))
64		eqid 2610	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ 𝐵 ↦ 𝐶) = (𝑘 ∈ 𝐵 ↦ 𝐶)
65	64	fvmpt2i 6199	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ 𝐵 → ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑘) = ( I ‘𝐶))
66	63, 65	eqtrd 2644	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ 𝐵 → if(𝑘 ∈ 𝐵, ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑘), 0) = ( I ‘𝐶))
67	62, 66	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) ∧ 𝑘 ∈ 𝐴) → if(𝑘 ∈ 𝐵, ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑘), 0) = ( I ‘𝐶))
68	60, 67	eqtr4d 2647	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) ∧ 𝑘 ∈ 𝐴) → ((𝑘 ∈ (ℤ≥‘𝑀) ↦ if(𝑘 ∈ 𝐴, 𝐶, 0))‘𝑘) = if(𝑘 ∈ 𝐵, ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑘), 0))
69	36	adantll 746	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) ∧ ¬ 𝑘 ∈ 𝐴) → ((𝑘 ∈ (ℤ≥‘𝑀) ↦ if(𝑘 ∈ 𝐴, 𝐶, 0))‘𝑘) = 0)
70	66	ad2antrl 760	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐵 ∧ ¬ 𝑘 ∈ 𝐴)) → if(𝑘 ∈ 𝐵, ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑘), 0) = ( I ‘𝐶))
71		eldif 3550	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (𝐵 ∖ 𝐴) ↔ (𝑘 ∈ 𝐵 ∧ ¬ 𝑘 ∈ 𝐴))
72		sumss.3	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ 𝐴)) → 𝐶 = 0)
73	72	fveq2d 6107	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ 𝐴)) → ( I ‘𝐶) = ( I ‘0))
74		0cn 9911	. . . . . . . . . . . . . . . . . . 19 ⊢ 0 ∈ ℂ
75		fvi 6165	. . . . . . . . . . . . . . . . . . 19 ⊢ (0 ∈ ℂ → ( I ‘0) = 0)
76	74, 75	ax-mp 5	. . . . . . . . . . . . . . . . . 18 ⊢ ( I ‘0) = 0
77	73, 76	syl6eq 2660	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ 𝐴)) → ( I ‘𝐶) = 0)
78	71, 77	sylan2br 492	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐵 ∧ ¬ 𝑘 ∈ 𝐴)) → ( I ‘𝐶) = 0)
79	70, 78	eqtrd 2644	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐵 ∧ ¬ 𝑘 ∈ 𝐴)) → if(𝑘 ∈ 𝐵, ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑘), 0) = 0)
80	79	expr 641	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (¬ 𝑘 ∈ 𝐴 → if(𝑘 ∈ 𝐵, ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑘), 0) = 0))
81		iffalse 4045	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝑘 ∈ 𝐵 → if(𝑘 ∈ 𝐵, ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑘), 0) = 0)
82	81	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ¬ 𝑘 ∈ 𝐵) → if(𝑘 ∈ 𝐵, ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑘), 0) = 0)
83	82	a1d 25	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ¬ 𝑘 ∈ 𝐵) → (¬ 𝑘 ∈ 𝐴 → if(𝑘 ∈ 𝐵, ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑘), 0) = 0))
84	80, 83	pm2.61dan 828	. . . . . . . . . . . . 13 ⊢ (𝜑 → (¬ 𝑘 ∈ 𝐴 → if(𝑘 ∈ 𝐵, ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑘), 0) = 0))
85	84	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (¬ 𝑘 ∈ 𝐴 → if(𝑘 ∈ 𝐵, ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑘), 0) = 0))
86	85	imp 444	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) ∧ ¬ 𝑘 ∈ 𝐴) → if(𝑘 ∈ 𝐵, ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑘), 0) = 0)
87	69, 86	eqtr4d 2647	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) ∧ ¬ 𝑘 ∈ 𝐴) → ((𝑘 ∈ (ℤ≥‘𝑀) ↦ if(𝑘 ∈ 𝐴, 𝐶, 0))‘𝑘) = if(𝑘 ∈ 𝐵, ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑘), 0))
88	68, 87	pm2.61dan 828	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → ((𝑘 ∈ (ℤ≥‘𝑀) ↦ if(𝑘 ∈ 𝐴, 𝐶, 0))‘𝑘) = if(𝑘 ∈ 𝐵, ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑘), 0))
89	88	expcom 450	. . . . . . . 8 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → (𝜑 → ((𝑘 ∈ (ℤ≥‘𝑀) ↦ if(𝑘 ∈ 𝐴, 𝐶, 0))‘𝑘) = if(𝑘 ∈ 𝐵, ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑘), 0)))
90	7, 54, 59, 89	vtoclgaf 3244	. . . . . . 7 ⊢ (𝑚 ∈ (ℤ≥‘𝑀) → (𝜑 → ((𝑘 ∈ (ℤ≥‘𝑀) ↦ if(𝑘 ∈ 𝐴, 𝐶, 0))‘𝑚) = if(𝑚 ∈ 𝐵, ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚), 0)))
91	90	impcom 445	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘𝑀)) → ((𝑘 ∈ (ℤ≥‘𝑀) ↦ if(𝑘 ∈ 𝐴, 𝐶, 0))‘𝑚) = if(𝑚 ∈ 𝐵, ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚), 0))
92	91	adantlr 747	. . . . 5 ⊢ (((𝜑 ∧ 𝑀 ∈ ℤ) ∧ 𝑚 ∈ (ℤ≥‘𝑀)) → ((𝑘 ∈ (ℤ≥‘𝑀) ↦ if(𝑘 ∈ 𝐴, 𝐶, 0))‘𝑚) = if(𝑚 ∈ 𝐵, ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚), 0))
93	43	ex 449	. . . . . . . . . 10 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → 𝐶 ∈ ℂ))
94	93	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑘 ∈ 𝐴 → 𝐶 ∈ ℂ))
95	72, 74	syl6eqel 2696	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ 𝐴)) → 𝐶 ∈ ℂ)
96	71, 95	sylan2br 492	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐵 ∧ ¬ 𝑘 ∈ 𝐴)) → 𝐶 ∈ ℂ)
97	96	expr 641	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (¬ 𝑘 ∈ 𝐴 → 𝐶 ∈ ℂ))
98	94, 97	pm2.61d 169	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℂ)
99	98, 64	fmptd 6292	. . . . . . 7 ⊢ (𝜑 → (𝑘 ∈ 𝐵 ↦ 𝐶):𝐵⟶ℂ)
100	99	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → (𝑘 ∈ 𝐵 ↦ 𝐶):𝐵⟶ℂ)
101	100	ffvelrnda 6267	. . . . 5 ⊢ (((𝜑 ∧ 𝑀 ∈ ℤ) ∧ 𝑚 ∈ 𝐵) → ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚) ∈ ℂ)
102	1, 2, 48, 92, 101	zsum 14296	. . . 4 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → Σ𝑚 ∈ 𝐵 ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚) = ( ⇝ ‘seq𝑀( + , (𝑘 ∈ (ℤ≥‘𝑀) ↦ if(𝑘 ∈ 𝐴, 𝐶, 0)))))
103	47, 102	eqtr4d 2647	. . 3 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚) = Σ𝑚 ∈ 𝐵 ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚))
104		sumfc 14287	. . 3 ⊢ Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚) = Σ𝑘 ∈ 𝐴 𝐶
105		sumfc 14287	. . 3 ⊢ Σ𝑚 ∈ 𝐵 ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚) = Σ𝑘 ∈ 𝐵 𝐶
106	103, 104, 105	3eqtr3g 2667	. 2 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐶)
107	3	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑀 ∈ ℤ) → 𝐴 ⊆ 𝐵)
108		uzf 11566	. . . . . . . . . . . 12 ⊢ ℤ≥:ℤ⟶𝒫 ℤ
109	108	fdmi 5965	. . . . . . . . . . 11 ⊢ dom ℤ≥ = ℤ
110	109	eleq2i 2680	. . . . . . . . . 10 ⊢ (𝑀 ∈ dom ℤ≥ ↔ 𝑀 ∈ ℤ)
111		ndmfv 6128	. . . . . . . . . 10 ⊢ (¬ 𝑀 ∈ dom ℤ≥ → (ℤ≥‘𝑀) = ∅)
112	110, 111	sylnbir 320	. . . . . . . . 9 ⊢ (¬ 𝑀 ∈ ℤ → (ℤ≥‘𝑀) = ∅)
113	112	sseq2d 3596	. . . . . . . 8 ⊢ (¬ 𝑀 ∈ ℤ → (𝐵 ⊆ (ℤ≥‘𝑀) ↔ 𝐵 ⊆ ∅))
114	4, 113	syl5ib 233	. . . . . . 7 ⊢ (¬ 𝑀 ∈ ℤ → (𝜑 → 𝐵 ⊆ ∅))
115	114	impcom 445	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑀 ∈ ℤ) → 𝐵 ⊆ ∅)
116	107, 115	sstrd 3578	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑀 ∈ ℤ) → 𝐴 ⊆ ∅)
117		ss0 3926	. . . . 5 ⊢ (𝐴 ⊆ ∅ → 𝐴 = ∅)
118	116, 117	syl 17	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑀 ∈ ℤ) → 𝐴 = ∅)
119		ss0 3926	. . . . 5 ⊢ (𝐵 ⊆ ∅ → 𝐵 = ∅)
120	115, 119	syl 17	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑀 ∈ ℤ) → 𝐵 = ∅)
121	118, 120	eqtr4d 2647	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑀 ∈ ℤ) → 𝐴 = 𝐵)
122	121	sumeq1d 14279	. 2 ⊢ ((𝜑 ∧ ¬ 𝑀 ∈ ℤ) → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐶)
123	106, 122	pm2.61dan 828	1 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ∖ cdif 3537   ⊆ wss 3540  ∅c0 3874  ifcif 4036  𝒫 cpw 4108   ↦ cmpt 4643   I cid 4948  dom cdm 5038  ⟶wf 5800  ‘cfv 5804  ℂcc 9813  0cc0 9815   + caddc 9818  ℤcz 11254  ℤ_≥cuz 11563  seqcseq 12663   ⇝ cli 14063  Σcsu 14264

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-fzo 12335  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-sum 14265

This theorem is referenced by:  fsumss  14303  sumss2  14304  binomlem  14400  eulerpartlemsv2  29747  eulerpartlemsv3  29750  eulerpartlemv  29753  eulerpartlemb  29757