Theorem gsumval2 13485

Description: Value of the group sum operation over a finite set of sequential integers. (Contributed by Mario Carneiro, 7-Dec-2014.)

Hypotheses

Ref	Expression
gsumval2.b	⊢ 𝐵 = (Base‘𝐺)
gsumval2.p	⊢ + = (+g‘𝐺)
gsumval2.g	⊢ (𝜑 → 𝐺 ∈ 𝑉)
gsumval2.n	⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
gsumval2.f	⊢ (𝜑 → 𝐹:(𝑀...𝑁)⟶𝐵)

Assertion

Ref	Expression
gsumval2	⊢ (𝜑 → (𝐺 Σg 𝐹) = (seq𝑀( + , 𝐹)‘𝑁))

Proof of Theorem gsumval2

Dummy variables 𝑚 𝑛 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		gsumval2.b	. . 3 ⊢ 𝐵 = (Base‘𝐺)
2		eqid 2231	. . 3 ⊢ (0g‘𝐺) = (0g‘𝐺)
3		gsumval2.p	. . 3 ⊢ + = (+g‘𝐺)
4		gsumval2.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ 𝑉)
5		gsumval2.n	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
6		eluzel2 9760	. . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ)
7	5, 6	syl 14	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ)
8		eluzelz 9765	. . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ)
9	5, 8	syl 14	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℤ)
10	7, 9	fzfigd 10694	. . 3 ⊢ (𝜑 → (𝑀...𝑁) ∈ Fin)
11		gsumval2.f	. . 3 ⊢ (𝜑 → 𝐹:(𝑀...𝑁)⟶𝐵)
12	1, 2, 3, 4, 10, 11	igsumval 13478	. 2 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (℩𝑥(((𝑀...𝑁) = ∅ ∧ 𝑥 = (0g‘𝐺)) ∨ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)))))
13		simprr 533	. . . . . . . 8 ⊢ ((𝑛 ∈ (ℤ≥‘𝑚) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))
14		simprl 531	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ (ℤ≥‘𝑚) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → (𝑀...𝑁) = (𝑚...𝑛))
15		eqcom 2233	. . . . . . . . . . . . . 14 ⊢ ((𝑚...𝑛) = (𝑀...𝑁) ↔ (𝑀...𝑁) = (𝑚...𝑛))
16		fzopth 10296	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (ℤ≥‘𝑚) → ((𝑚...𝑛) = (𝑀...𝑁) ↔ (𝑚 = 𝑀 ∧ 𝑛 = 𝑁)))
17	15, 16	bitr3id 194	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (ℤ≥‘𝑚) → ((𝑀...𝑁) = (𝑚...𝑛) ↔ (𝑚 = 𝑀 ∧ 𝑛 = 𝑁)))
18	17	adantr 276	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ (ℤ≥‘𝑚) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → ((𝑀...𝑁) = (𝑚...𝑛) ↔ (𝑚 = 𝑀 ∧ 𝑛 = 𝑁)))
19	14, 18	mpbid 147	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ (ℤ≥‘𝑚) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → (𝑚 = 𝑀 ∧ 𝑛 = 𝑁))
20	19	simpld 112	. . . . . . . . . 10 ⊢ ((𝑛 ∈ (ℤ≥‘𝑚) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → 𝑚 = 𝑀)
21	20	seqeq1d 10716	. . . . . . . . 9 ⊢ ((𝑛 ∈ (ℤ≥‘𝑚) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → seq𝑚( + , 𝐹) = seq𝑀( + , 𝐹))
22	19	simprd 114	. . . . . . . . 9 ⊢ ((𝑛 ∈ (ℤ≥‘𝑚) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → 𝑛 = 𝑁)
23	21, 22	fveq12d 5646	. . . . . . . 8 ⊢ ((𝑛 ∈ (ℤ≥‘𝑚) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → (seq𝑚( + , 𝐹)‘𝑛) = (seq𝑀( + , 𝐹)‘𝑁))
24	13, 23	eqtrd 2264	. . . . . . 7 ⊢ ((𝑛 ∈ (ℤ≥‘𝑚) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → 𝑥 = (seq𝑀( + , 𝐹)‘𝑁))
25	24	rexlimiva 2645	. . . . . 6 ⊢ (∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)) → 𝑥 = (seq𝑀( + , 𝐹)‘𝑁))
26	25	exlimiv 1646	. . . . 5 ⊢ (∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)) → 𝑥 = (seq𝑀( + , 𝐹)‘𝑁))
27	7	elexd 2816	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ V)
28	27	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑁)) → 𝑀 ∈ V)
29	5	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑁)) → 𝑁 ∈ (ℤ≥‘𝑀))
30		oveq2 6026	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑁 → (𝑀...𝑛) = (𝑀...𝑁))
31	30	eqeq2d 2243	. . . . . . . . . 10 ⊢ (𝑛 = 𝑁 → ((𝑀...𝑁) = (𝑀...𝑛) ↔ (𝑀...𝑁) = (𝑀...𝑁)))
32		fveq2 5639	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑁 → (seq𝑀( + , 𝐹)‘𝑛) = (seq𝑀( + , 𝐹)‘𝑁))
33	32	eqeq2d 2243	. . . . . . . . . 10 ⊢ (𝑛 = 𝑁 → (𝑥 = (seq𝑀( + , 𝐹)‘𝑛) ↔ 𝑥 = (seq𝑀( + , 𝐹)‘𝑁)))
34	31, 33	anbi12d 473	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → (((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑛)) ↔ ((𝑀...𝑁) = (𝑀...𝑁) ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑁))))
35	34	adantl 277	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑁)) ∧ 𝑛 = 𝑁) → (((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑛)) ↔ ((𝑀...𝑁) = (𝑀...𝑁) ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑁))))
36		eqidd 2232	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑁)) → (𝑀...𝑁) = (𝑀...𝑁))
37		simpr 110	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑁)) → 𝑥 = (seq𝑀( + , 𝐹)‘𝑁))
38	36, 37	jca 306	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑁)) → ((𝑀...𝑁) = (𝑀...𝑁) ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑁)))
39	29, 35, 38	rspcedvd 2916	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑁)) → ∃𝑛 ∈ (ℤ≥‘𝑀)((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑛)))
40		fveq2 5639	. . . . . . . 8 ⊢ (𝑚 = 𝑀 → (ℤ≥‘𝑚) = (ℤ≥‘𝑀))
41		oveq1 6025	. . . . . . . . . 10 ⊢ (𝑚 = 𝑀 → (𝑚...𝑛) = (𝑀...𝑛))
42	41	eqeq2d 2243	. . . . . . . . 9 ⊢ (𝑚 = 𝑀 → ((𝑀...𝑁) = (𝑚...𝑛) ↔ (𝑀...𝑁) = (𝑀...𝑛)))
43		seqeq1 10713	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑀 → seq𝑚( + , 𝐹) = seq𝑀( + , 𝐹))
44	43	fveq1d 5641	. . . . . . . . . 10 ⊢ (𝑚 = 𝑀 → (seq𝑚( + , 𝐹)‘𝑛) = (seq𝑀( + , 𝐹)‘𝑛))
45	44	eqeq2d 2243	. . . . . . . . 9 ⊢ (𝑚 = 𝑀 → (𝑥 = (seq𝑚( + , 𝐹)‘𝑛) ↔ 𝑥 = (seq𝑀( + , 𝐹)‘𝑛)))
46	42, 45	anbi12d 473	. . . . . . . 8 ⊢ (𝑚 = 𝑀 → (((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)) ↔ ((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑛))))
47	40, 46	rexeqbidv 2747	. . . . . . 7 ⊢ (𝑚 = 𝑀 → (∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)) ↔ ∃𝑛 ∈ (ℤ≥‘𝑀)((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑛))))
48	28, 39, 47	spcedv 2895	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑁)) → ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)))
49	48	ex 115	. . . . 5 ⊢ (𝜑 → (𝑥 = (seq𝑀( + , 𝐹)‘𝑁) → ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))))
50	26, 49	impbid2 143	. . . 4 ⊢ (𝜑 → (∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)) ↔ 𝑥 = (seq𝑀( + , 𝐹)‘𝑁)))
51		eluzfz2 10267	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁))
52	5, 51	syl 14	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (𝑀...𝑁))
53		n0i 3500	. . . . . . 7 ⊢ (𝑁 ∈ (𝑀...𝑁) → ¬ (𝑀...𝑁) = ∅)
54	52, 53	syl 14	. . . . . 6 ⊢ (𝜑 → ¬ (𝑀...𝑁) = ∅)
55	54	intnanrd 939	. . . . 5 ⊢ (𝜑 → ¬ ((𝑀...𝑁) = ∅ ∧ 𝑥 = (0g‘𝐺)))
56		biorf 751	. . . . 5 ⊢ (¬ ((𝑀...𝑁) = ∅ ∧ 𝑥 = (0g‘𝐺)) → (∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)) ↔ (((𝑀...𝑁) = ∅ ∧ 𝑥 = (0g‘𝐺)) ∨ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)))))
57	55, 56	syl 14	. . . 4 ⊢ (𝜑 → (∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)) ↔ (((𝑀...𝑁) = ∅ ∧ 𝑥 = (0g‘𝐺)) ∨ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)))))
58	50, 57	bitr3d 190	. . 3 ⊢ (𝜑 → (𝑥 = (seq𝑀( + , 𝐹)‘𝑁) ↔ (((𝑀...𝑁) = ∅ ∧ 𝑥 = (0g‘𝐺)) ∨ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)))))
59	58	iotabidv 5309	. 2 ⊢ (𝜑 → (℩𝑥𝑥 = (seq𝑀( + , 𝐹)‘𝑁)) = (℩𝑥(((𝑀...𝑁) = ∅ ∧ 𝑥 = (0g‘𝐺)) ∨ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)))))
60		eqid 2231	. . 3 ⊢ (seq𝑀( + , 𝐹)‘𝑁) = (seq𝑀( + , 𝐹)‘𝑁)
61		seqex 10712	. . . . 5 ⊢ seq𝑀( + , 𝐹) ∈ V
62		fvexg 5658	. . . . 5 ⊢ ((seq𝑀( + , 𝐹) ∈ V ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → (seq𝑀( + , 𝐹)‘𝑁) ∈ V)
63	61, 5, 62	sylancr 414	. . . 4 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) ∈ V)
64		eueq 2977	. . . . 5 ⊢ ((seq𝑀( + , 𝐹)‘𝑁) ∈ V ↔ ∃!𝑥 𝑥 = (seq𝑀( + , 𝐹)‘𝑁))
65	63, 64	sylib 122	. . . 4 ⊢ (𝜑 → ∃!𝑥 𝑥 = (seq𝑀( + , 𝐹)‘𝑁))
66		eqeq1 2238	. . . . 5 ⊢ (𝑥 = (seq𝑀( + , 𝐹)‘𝑁) → (𝑥 = (seq𝑀( + , 𝐹)‘𝑁) ↔ (seq𝑀( + , 𝐹)‘𝑁) = (seq𝑀( + , 𝐹)‘𝑁)))
67	66	iota2 5316	. . . 4 ⊢ (((seq𝑀( + , 𝐹)‘𝑁) ∈ V ∧ ∃!𝑥 𝑥 = (seq𝑀( + , 𝐹)‘𝑁)) → ((seq𝑀( + , 𝐹)‘𝑁) = (seq𝑀( + , 𝐹)‘𝑁) ↔ (℩𝑥𝑥 = (seq𝑀( + , 𝐹)‘𝑁)) = (seq𝑀( + , 𝐹)‘𝑁)))
68	63, 65, 67	syl2anc 411	. . 3 ⊢ (𝜑 → ((seq𝑀( + , 𝐹)‘𝑁) = (seq𝑀( + , 𝐹)‘𝑁) ↔ (℩𝑥𝑥 = (seq𝑀( + , 𝐹)‘𝑁)) = (seq𝑀( + , 𝐹)‘𝑁)))
69	60, 68	mpbii 148	. 2 ⊢ (𝜑 → (℩𝑥𝑥 = (seq𝑀( + , 𝐹)‘𝑁)) = (seq𝑀( + , 𝐹)‘𝑁))
70	12, 59, 69	3eqtr2d 2270	1 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (seq𝑀( + , 𝐹)‘𝑁))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 715   = wceq 1397  ∃wex 1540  ∃!weu 2079   ∈ wcel 2202  ∃wrex 2511  Vcvv 2802  ∅c0 3494  ℩cio 5284  ⟶wf 5322  ‘cfv 5326  (class class class)co 6018  Fincfn 6909  ℤcz 9479  ℤ_≥cuz 9755  ...cfz 10243  seqcseq 10710  Basecbs 13087  +_gcplusg 13165  0_gc0g 13344   Σ_g cgsu 13345

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-addass 8134  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-0id 8140  ax-rnegex 8141  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-frec 6557  df-1o 6582  df-er 6702  df-en 6910  df-fin 6912  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-inn 9144  df-n0 9403  df-z 9480  df-uz 9756  df-fz 10244  df-seqfrec 10711  df-ndx 13090  df-slot 13091  df-base 13093  df-0g 13346  df-igsum 13347

This theorem is referenced by:  gsumsplit1r  13486  gsumprval  13487  gsumwsubmcl  13584  gsumwmhm  13586  mulgnngsum  13719  gsumfzconst  13933  gfsumval  16706  gsumgfsumlem  16709