Theorem gsumval2 13099

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 2196	. . 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 9623	. . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ)
7	5, 6	syl 14	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ)
8		eluzelz 9627	. . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ)
9	5, 8	syl 14	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℤ)
10	7, 9	fzfigd 10540	. . 3 ⊢ (𝜑 → (𝑀...𝑁) ∈ Fin)
11		gsumval2.f	. . 3 ⊢ (𝜑 → 𝐹:(𝑀...𝑁)⟶𝐵)
12	1, 2, 3, 4, 10, 11	igsumval 13092	. 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 531	. . . . . . . 8 ⊢ ((𝑛 ∈ (ℤ≥‘𝑚) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))
14		simprl 529	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ (ℤ≥‘𝑚) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → (𝑀...𝑁) = (𝑚...𝑛))
15		eqcom 2198	. . . . . . . . . . . . . 14 ⊢ ((𝑚...𝑛) = (𝑀...𝑁) ↔ (𝑀...𝑁) = (𝑚...𝑛))
16		fzopth 10153	. . . . . . . . . . . . . 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 10562	. . . . . . . . 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 5568	. . . . . . . 8 ⊢ ((𝑛 ∈ (ℤ≥‘𝑚) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → (seq𝑚( + , 𝐹)‘𝑛) = (seq𝑀( + , 𝐹)‘𝑁))
24	13, 23	eqtrd 2229	. . . . . . 7 ⊢ ((𝑛 ∈ (ℤ≥‘𝑚) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → 𝑥 = (seq𝑀( + , 𝐹)‘𝑁))
25	24	rexlimiva 2609	. . . . . 6 ⊢ (∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)) → 𝑥 = (seq𝑀( + , 𝐹)‘𝑁))
26	25	exlimiv 1612	. . . . 5 ⊢ (∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)) → 𝑥 = (seq𝑀( + , 𝐹)‘𝑁))
27	7	elexd 2776	. . . . . . . 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 5933	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑁 → (𝑀...𝑛) = (𝑀...𝑁))
31	30	eqeq2d 2208	. . . . . . . . . 10 ⊢ (𝑛 = 𝑁 → ((𝑀...𝑁) = (𝑀...𝑛) ↔ (𝑀...𝑁) = (𝑀...𝑁)))
32		fveq2 5561	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑁 → (seq𝑀( + , 𝐹)‘𝑛) = (seq𝑀( + , 𝐹)‘𝑁))
33	32	eqeq2d 2208	. . . . . . . . . 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 2197	. . . . . . . . 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 2874	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑁)) → ∃𝑛 ∈ (ℤ≥‘𝑀)((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑛)))
40		fveq2 5561	. . . . . . . 8 ⊢ (𝑚 = 𝑀 → (ℤ≥‘𝑚) = (ℤ≥‘𝑀))
41		oveq1 5932	. . . . . . . . . 10 ⊢ (𝑚 = 𝑀 → (𝑚...𝑛) = (𝑀...𝑛))
42	41	eqeq2d 2208	. . . . . . . . 9 ⊢ (𝑚 = 𝑀 → ((𝑀...𝑁) = (𝑚...𝑛) ↔ (𝑀...𝑁) = (𝑀...𝑛)))
43		seqeq1 10559	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑀 → seq𝑚( + , 𝐹) = seq𝑀( + , 𝐹))
44	43	fveq1d 5563	. . . . . . . . . 10 ⊢ (𝑚 = 𝑀 → (seq𝑚( + , 𝐹)‘𝑛) = (seq𝑀( + , 𝐹)‘𝑛))
45	44	eqeq2d 2208	. . . . . . . . 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 2710	. . . . . . 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 2853	. . . . . 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 10124	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁))
52	5, 51	syl 14	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (𝑀...𝑁))
53		n0i 3457	. . . . . . 7 ⊢ (𝑁 ∈ (𝑀...𝑁) → ¬ (𝑀...𝑁) = ∅)
54	52, 53	syl 14	. . . . . 6 ⊢ (𝜑 → ¬ (𝑀...𝑁) = ∅)
55	54	intnanrd 933	. . . . 5 ⊢ (𝜑 → ¬ ((𝑀...𝑁) = ∅ ∧ 𝑥 = (0g‘𝐺)))
56		biorf 745	. . . . 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 5242	. 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 2196	. . 3 ⊢ (seq𝑀( + , 𝐹)‘𝑁) = (seq𝑀( + , 𝐹)‘𝑁)
61		seqex 10558	. . . . 5 ⊢ seq𝑀( + , 𝐹) ∈ V
62		fvexg 5580	. . . . 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 2935	. . . . 5 ⊢ ((seq𝑀( + , 𝐹)‘𝑁) ∈ V ↔ ∃!𝑥 𝑥 = (seq𝑀( + , 𝐹)‘𝑁))
65	63, 64	sylib 122	. . . 4 ⊢ (𝜑 → ∃!𝑥 𝑥 = (seq𝑀( + , 𝐹)‘𝑁))
66		eqeq1 2203	. . . . 5 ⊢ (𝑥 = (seq𝑀( + , 𝐹)‘𝑁) → (𝑥 = (seq𝑀( + , 𝐹)‘𝑁) ↔ (seq𝑀( + , 𝐹)‘𝑁) = (seq𝑀( + , 𝐹)‘𝑁)))
67	66	iota2 5249	. . . 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 2235	1 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (seq𝑀( + , 𝐹)‘𝑁))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 709   = wceq 1364  ∃wex 1506  ∃!weu 2045   ∈ wcel 2167  ∃wrex 2476  Vcvv 2763  ∅c0 3451  ℩cio 5218  ⟶wf 5255  ‘cfv 5259  (class class class)co 5925  Fincfn 6808  ℤcz 9343  ℤ_≥cuz 9618  ...cfz 10100  seqcseq 10556  Basecbs 12703  +_gcplusg 12780  0_gc0g 12958   Σ_g cgsu 12959

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-addcom 7996  ax-addass 7998  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-0id 8004  ax-rnegex 8005  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-frec 6458  df-1o 6483  df-er 6601  df-en 6809  df-fin 6811  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-inn 9008  df-n0 9267  df-z 9344  df-uz 9619  df-fz 10101  df-seqfrec 10557  df-ndx 12706  df-slot 12707  df-base 12709  df-0g 12960  df-igsum 12961

This theorem is referenced by:  gsumsplit1r  13100  gsumprval  13101  gsumwsubmcl  13198  gsumwmhm  13200  mulgnngsum  13333  gsumfzconst  13547