Theorem sumss 15765

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 2765	. . . . 5 ⊢ (ℤ≥‘𝑀) = (ℤ≥‘𝑀)
2		simpr 489	. . . . 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 3949	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ (ℤ≥‘𝑀))
6	5	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → 𝐴 ⊆ (ℤ≥‘𝑀))
7		nfcv 2927	. . . . . . 7 ⊢ Ⅎ𝑘𝑚
8		nffvmpt1 6882	. . . . . . . 8 ⊢ Ⅎ𝑘((𝑘 ∈ (ℤ≥‘𝑀) ↦ if(𝑘 ∈ 𝐴, 𝐶, 0))‘𝑚)
9		nfv 1937	. . . . . . . . 9 ⊢ Ⅎ𝑘 𝑚 ∈ 𝐴
10		nffvmpt1 6882	. . . . . . . . 9 ⊢ Ⅎ𝑘((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚)
11		nfcv 2927	. . . . . . . . 9 ⊢ Ⅎ𝑘0
12	9, 10, 11	nfif 4514	. . . . . . . 8 ⊢ Ⅎ𝑘if(𝑚 ∈ 𝐴, ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚), 0)
13	8, 12	nfeq 2940	. . . . . . 7 ⊢ Ⅎ𝑘((𝑘 ∈ (ℤ≥‘𝑀) ↦ if(𝑘 ∈ 𝐴, 𝐶, 0))‘𝑚) = if(𝑚 ∈ 𝐴, ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚), 0)
14		fveq2 6871	. . . . . . . 8 ⊢ (𝑘 = 𝑚 → ((𝑘 ∈ (ℤ≥‘𝑀) ↦ if(𝑘 ∈ 𝐴, 𝐶, 0))‘𝑘) = ((𝑘 ∈ (ℤ≥‘𝑀) ↦ if(𝑘 ∈ 𝐴, 𝐶, 0))‘𝑚))
15		eleq1w 2848	. . . . . . . . 9 ⊢ (𝑘 = 𝑚 → (𝑘 ∈ 𝐴 ↔ 𝑚 ∈ 𝐴))
16		fveq2 6871	. . . . . . . . 9 ⊢ (𝑘 = 𝑚 → ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑘) = ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚))
17	15, 16	ifbieq1d 4508	. . . . . . . 8 ⊢ (𝑘 = 𝑚 → if(𝑘 ∈ 𝐴, ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑘), 0) = if(𝑚 ∈ 𝐴, ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚), 0))
18	14, 17	eqeq12d 2781	. . . . . . 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 2765	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) ↦ if(𝑘 ∈ 𝐴, 𝐶, 0)) = (𝑘 ∈ (ℤ≥‘𝑀) ↦ if(𝑘 ∈ 𝐴, 𝐶, 0))
20	19	fvmpt2i 6990	. . . . . . . . . 10 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → ((𝑘 ∈ (ℤ≥‘𝑀) ↦ if(𝑘 ∈ 𝐴, 𝐶, 0))‘𝑘) = ( I ‘if(𝑘 ∈ 𝐴, 𝐶, 0)))
21		iftrue 4489	. . . . . . . . . . 11 ⊢ (𝑘 ∈ 𝐴 → if(𝑘 ∈ 𝐴, 𝐶, 0) = 𝐶)
22	21	fveq2d 6875	. . . . . . . . . 10 ⊢ (𝑘 ∈ 𝐴 → ( I ‘if(𝑘 ∈ 𝐴, 𝐶, 0)) = ( I ‘𝐶))
23	20, 22	sylan9eq 2820	. . . . . . . . 9 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ 𝑘 ∈ 𝐴) → ((𝑘 ∈ (ℤ≥‘𝑀) ↦ if(𝑘 ∈ 𝐴, 𝐶, 0))‘𝑘) = ( I ‘𝐶))
24		iftrue 4489	. . . . . . . . . . 11 ⊢ (𝑘 ∈ 𝐴 → if(𝑘 ∈ 𝐴, ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑘), 0) = ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑘))
25		eqid 2765	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐶) = (𝑘 ∈ 𝐴 ↦ 𝐶)
26	25	fvmpt2i 6990	. . . . . . . . . . 11 ⊢ (𝑘 ∈ 𝐴 → ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑘) = ( I ‘𝐶))
27	24, 26	eqtrd 2800	. . . . . . . . . 10 ⊢ (𝑘 ∈ 𝐴 → if(𝑘 ∈ 𝐴, ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑘), 0) = ( I ‘𝐶))
28	27	adantl 486	. . . . . . . . 9 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ 𝑘 ∈ 𝐴) → if(𝑘 ∈ 𝐴, ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑘), 0) = ( I ‘𝐶))
29	23, 28	eqtr4d 2803	. . . . . . . 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 4492	. . . . . . . . . . . 12 ⊢ (¬ 𝑘 ∈ 𝐴 → if(𝑘 ∈ 𝐴, 𝐶, 0) = 0)
31	30	fveq2d 6875	. . . . . . . . . . 11 ⊢ (¬ 𝑘 ∈ 𝐴 → ( I ‘if(𝑘 ∈ 𝐴, 𝐶, 0)) = ( I ‘0))
32		0z 12593	. . . . . . . . . . . 12 ⊢ 0 ∈ ℤ
33		fvi 6947	. . . . . . . . . . . 12 ⊢ (0 ∈ ℤ → ( I ‘0) = 0)
34	32, 33	ax-mp 5	. . . . . . . . . . 11 ⊢ ( I ‘0) = 0
35	31, 34	eqtrdi 2816	. . . . . . . . . 10 ⊢ (¬ 𝑘 ∈ 𝐴 → ( I ‘if(𝑘 ∈ 𝐴, 𝐶, 0)) = 0)
36	20, 35	sylan9eq 2820	. . . . . . . . 9 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ¬ 𝑘 ∈ 𝐴) → ((𝑘 ∈ (ℤ≥‘𝑀) ↦ if(𝑘 ∈ 𝐴, 𝐶, 0))‘𝑘) = 0)
37		iffalse 4492	. . . . . . . . . 10 ⊢ (¬ 𝑘 ∈ 𝐴 → if(𝑘 ∈ 𝐴, ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑘), 0) = 0)
38	37	adantl 486	. . . . . . . . 9 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ¬ 𝑘 ∈ 𝐴) → if(𝑘 ∈ 𝐴, ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑘), 0) = 0)
39	36, 38	eqtr4d 2803	. . . . . . . 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 824	. . . . . . 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 3543	. . . . . 6 ⊢ (𝑚 ∈ (ℤ≥‘𝑀) → ((𝑘 ∈ (ℤ≥‘𝑀) ↦ if(𝑘 ∈ 𝐴, 𝐶, 0))‘𝑚) = if(𝑚 ∈ 𝐴, ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚), 0))
42	41	adantl 486	. . . . 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	fmpttd 7100	. . . . . . 7 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐶):𝐴⟶ℂ)
45	44	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → (𝑘 ∈ 𝐴 ↦ 𝐶):𝐴⟶ℂ)
46	45	ffvelcdmda 7069	. . . . 5 ⊢ (((𝜑 ∧ 𝑀 ∈ ℤ) ∧ 𝑚 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚) ∈ ℂ)
47	1, 2, 6, 42, 46	zsum 15759	. . . 4 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚) = ( ⇝ ‘seq𝑀( + , (𝑘 ∈ (ℤ≥‘𝑀) ↦ if(𝑘 ∈ 𝐴, 𝐶, 0)))))
48	4	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → 𝐵 ⊆ (ℤ≥‘𝑀))
49		nfv 1937	. . . . . . . . 9 ⊢ Ⅎ𝑘𝜑
50		nfv 1937	. . . . . . . . . . 11 ⊢ Ⅎ𝑘 𝑚 ∈ 𝐵
51		nffvmpt1 6882	. . . . . . . . . . 11 ⊢ Ⅎ𝑘((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚)
52	50, 51, 11	nfif 4514	. . . . . . . . . 10 ⊢ Ⅎ𝑘if(𝑚 ∈ 𝐵, ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚), 0)
53	8, 52	nfeq 2940	. . . . . . . . 9 ⊢ Ⅎ𝑘((𝑘 ∈ (ℤ≥‘𝑀) ↦ if(𝑘 ∈ 𝐴, 𝐶, 0))‘𝑚) = if(𝑚 ∈ 𝐵, ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚), 0)
54	49, 53	nfim 1919	. . . . . . . 8 ⊢ Ⅎ𝑘(𝜑 → ((𝑘 ∈ (ℤ≥‘𝑀) ↦ if(𝑘 ∈ 𝐴, 𝐶, 0))‘𝑚) = if(𝑚 ∈ 𝐵, ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚), 0))
55		eleq1w 2848	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑚 → (𝑘 ∈ 𝐵 ↔ 𝑚 ∈ 𝐵))
56		fveq2 6871	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑚 → ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑘) = ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚))
57	55, 56	ifbieq1d 4508	. . . . . . . . . 10 ⊢ (𝑘 = 𝑚 → if(𝑘 ∈ 𝐵, ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑘), 0) = if(𝑚 ∈ 𝐵, ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚), 0))
58	14, 57	eqeq12d 2781	. . . . . . . . 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 343	. . . . . . . 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 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) ∧ 𝑘 ∈ 𝐴) → ((𝑘 ∈ (ℤ≥‘𝑀) ↦ if(𝑘 ∈ 𝐴, 𝐶, 0))‘𝑘) = ( I ‘𝐶))
61	3	adantr 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → 𝐴 ⊆ 𝐵)
62	61	sselda 3939	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ 𝐵)
63		iftrue 4489	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ 𝐵 → if(𝑘 ∈ 𝐵, ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑘), 0) = ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑘))
64		eqid 2765	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ 𝐵 ↦ 𝐶) = (𝑘 ∈ 𝐵 ↦ 𝐶)
65	64	fvmpt2i 6990	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ 𝐵 → ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑘) = ( I ‘𝐶))
66	63, 65	eqtrd 2800	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ 𝐵 → if(𝑘 ∈ 𝐵, ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑘), 0) = ( I ‘𝐶))
67	62, 66	syl 18	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) ∧ 𝑘 ∈ 𝐴) → if(𝑘 ∈ 𝐵, ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑘), 0) = ( I ‘𝐶))
68	60, 67	eqtr4d 2803	. . . . . . . . . 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 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) ∧ ¬ 𝑘 ∈ 𝐴) → ((𝑘 ∈ (ℤ≥‘𝑀) ↦ if(𝑘 ∈ 𝐴, 𝐶, 0))‘𝑘) = 0)
70	66	ad2antrl 740	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐵 ∧ ¬ 𝑘 ∈ 𝐴)) → if(𝑘 ∈ 𝐵, ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑘), 0) = ( I ‘𝐶))
71		eldif 3917	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (𝐵 ∖ 𝐴) ↔ (𝑘 ∈ 𝐵 ∧ ¬ 𝑘 ∈ 𝐴))
72		sumss.3	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ 𝐴)) → 𝐶 = 0)
73	72	fveq2d 6875	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ 𝐴)) → ( I ‘𝐶) = ( I ‘0))
74		0cn 11186	. . . . . . . . . . . . . . . . . . 19 ⊢ 0 ∈ ℂ
75		fvi 6947	. . . . . . . . . . . . . . . . . . 19 ⊢ (0 ∈ ℂ → ( I ‘0) = 0)
76	74, 75	ax-mp 5	. . . . . . . . . . . . . . . . . 18 ⊢ ( I ‘0) = 0
77	73, 76	eqtrdi 2816	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ 𝐴)) → ( I ‘𝐶) = 0)
78	71, 77	sylan2br 606	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐵 ∧ ¬ 𝑘 ∈ 𝐴)) → ( I ‘𝐶) = 0)
79	70, 78	eqtrd 2800	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐵 ∧ ¬ 𝑘 ∈ 𝐴)) → if(𝑘 ∈ 𝐵, ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑘), 0) = 0)
80	79	expr 461	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (¬ 𝑘 ∈ 𝐴 → if(𝑘 ∈ 𝐵, ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑘), 0) = 0))
81		iffalse 4492	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝑘 ∈ 𝐵 → if(𝑘 ∈ 𝐵, ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑘), 0) = 0)
82	81	adantl 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ¬ 𝑘 ∈ 𝐵) → if(𝑘 ∈ 𝐵, ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑘), 0) = 0)
83	82	a1d 26	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ¬ 𝑘 ∈ 𝐵) → (¬ 𝑘 ∈ 𝐴 → if(𝑘 ∈ 𝐵, ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑘), 0) = 0))
84	80, 83	pm2.61dan 824	. . . . . . . . . . . . 13 ⊢ (𝜑 → (¬ 𝑘 ∈ 𝐴 → if(𝑘 ∈ 𝐵, ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑘), 0) = 0))
85	84	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (¬ 𝑘 ∈ 𝐴 → if(𝑘 ∈ 𝐵, ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑘), 0) = 0))
86	85	imp 411	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) ∧ ¬ 𝑘 ∈ 𝐴) → if(𝑘 ∈ 𝐵, ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑘), 0) = 0)
87	69, 86	eqtr4d 2803	. . . . . . . . . 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 824	. . . . . . . . 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 418	. . . . . . . 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 3543	. . . . . . 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 412	. . . . . 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 727	. . . . 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 417	. . . . . . . . . 10 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → 𝐶 ∈ ℂ))
94	93	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑘 ∈ 𝐴 → 𝐶 ∈ ℂ))
95	72, 74	eqeltrdi 2873	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ 𝐴)) → 𝐶 ∈ ℂ)
96	71, 95	sylan2br 606	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐵 ∧ ¬ 𝑘 ∈ 𝐴)) → 𝐶 ∈ ℂ)
97	96	expr 461	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (¬ 𝑘 ∈ 𝐴 → 𝐶 ∈ ℂ))
98	94, 97	pm2.61d 181	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℂ)
99	98	fmpttd 7100	. . . . . . 7 ⊢ (𝜑 → (𝑘 ∈ 𝐵 ↦ 𝐶):𝐵⟶ℂ)
100	99	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → (𝑘 ∈ 𝐵 ↦ 𝐶):𝐵⟶ℂ)
101	100	ffvelcdmda 7069	. . . . 5 ⊢ (((𝜑 ∧ 𝑀 ∈ ℤ) ∧ 𝑚 ∈ 𝐵) → ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚) ∈ ℂ)
102	1, 2, 48, 92, 101	zsum 15759	. . . 4 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → Σ𝑚 ∈ 𝐵 ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚) = ( ⇝ ‘seq𝑀( + , (𝑘 ∈ (ℤ≥‘𝑀) ↦ if(𝑘 ∈ 𝐴, 𝐶, 0)))))
103	47, 102	eqtr4d 2803	. . 3 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚) = Σ𝑚 ∈ 𝐵 ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚))
104		sumfc 15750	. . 3 ⊢ Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚) = Σ𝑘 ∈ 𝐴 𝐶
105		sumfc 15750	. . 3 ⊢ Σ𝑚 ∈ 𝐵 ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚) = Σ𝑘 ∈ 𝐵 𝐶
106	103, 104, 105	3eqtr3g 2823	. 2 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐶)
107	3	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑀 ∈ ℤ) → 𝐴 ⊆ 𝐵)
108		uzf 12856	. . . . . . . . . . . 12 ⊢ ℤ≥:ℤ⟶𝒫 ℤ
109	108	fdmi 6707	. . . . . . . . . . 11 ⊢ dom ℤ≥ = ℤ
110	109	eleq2i 2857	. . . . . . . . . 10 ⊢ (𝑀 ∈ dom ℤ≥ ↔ 𝑀 ∈ ℤ)
111		ndmfv 6903	. . . . . . . . . 10 ⊢ (¬ 𝑀 ∈ dom ℤ≥ → (ℤ≥‘𝑀) = ∅)
112	110, 111	sylnbir 334	. . . . . . . . 9 ⊢ (¬ 𝑀 ∈ ℤ → (ℤ≥‘𝑀) = ∅)
113	112	sseq2d 3971	. . . . . . . 8 ⊢ (¬ 𝑀 ∈ ℤ → (𝐵 ⊆ (ℤ≥‘𝑀) ↔ 𝐵 ⊆ ∅))
114	4, 113	imbitrid 247	. . . . . . 7 ⊢ (¬ 𝑀 ∈ ℤ → (𝜑 → 𝐵 ⊆ ∅))
115	114	impcom 412	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑀 ∈ ℤ) → 𝐵 ⊆ ∅)
116	107, 115	sstrd 3949	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑀 ∈ ℤ) → 𝐴 ⊆ ∅)
117		ss0 4359	. . . . 5 ⊢ (𝐴 ⊆ ∅ → 𝐴 = ∅)
118	116, 117	syl 18	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑀 ∈ ℤ) → 𝐴 = ∅)
119		ss0 4359	. . . . 5 ⊢ (𝐵 ⊆ ∅ → 𝐵 = ∅)
120	115, 119	syl 18	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑀 ∈ ℤ) → 𝐵 = ∅)
121	118, 120	eqtr4d 2803	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑀 ∈ ℤ) → 𝐴 = 𝐵)
122	121	sumeq1d 15741	. 2 ⊢ ((𝜑 ∧ ¬ 𝑀 ∈ ℤ) → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐶)
123	106, 122	pm2.61dan 824	1 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1563   ∈ wcel 2145   ∖ cdif 3904   ⊆ wss 3907  ∅c0 4288  ifcif 4483  𝒫 cpw 4558   ↦ cmpt 5186   I cid 5546  dom cdm 5652  ⟶wf 6521  ‘cfv 6525  ℂcc 11086  0cc0 11088   + caddc 11091  ℤcz 12582  ℤ_≥cuz 12853  seqcseq 14028   ⇝ cli 15525  Σcsu 15727

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722  ax-inf2 9598  ax-cnex 11144  ax-resscn 11145  ax-1cn 11146  ax-icn 11147  ax-addcl 11148  ax-addrcl 11149  ax-mulcl 11150  ax-mulrcl 11151  ax-mulcom 11152  ax-addass 11153  ax-mulass 11154  ax-distr 11155  ax-i2m1 11156  ax-1ne0 11157  ax-1rid 11158  ax-rnegex 11159  ax-rrecex 11160  ax-cnre 11161  ax-pre-lttri 11162  ax-pre-lttrn 11163  ax-pre-ltadd 11164  ax-pre-mulgt0 11165

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-int 4909  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-se 5606  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-isom 6534  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-om 7851  df-1st 7974  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-1o 8441  df-er 8682  df-en 8932  df-dom 8933  df-sdom 8934  df-fin 8935  df-oi 9460  df-card 9913  df-pnf 11233  df-mnf 11234  df-xr 11235  df-ltxr 11236  df-le 11237  df-sub 11431  df-neg 11432  df-div 11860  df-nn 12225  df-2 12294  df-n0 12496  df-z 12583  df-uz 12854  df-rp 13008  df-fz 13527  df-fzo 13674  df-seq 14029  df-exp 14089  df-hash 14358  df-cj 15140  df-re 15141  df-im 15142  df-sqrt 15276  df-abs 15277  df-clim 15529  df-sum 15728

This theorem is referenced by:  fsumss  15766  sumss2  15767  binomlem  15873  eulerpartlemsv2  34665  eulerpartlemsv3  34668  eulerpartlemv  34671  eulerpartlemb  34675