Theorem gzsumval2 13691

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

Hypotheses

Ref	Expression
gzsumval2.b	⊢ 𝐵 = (Base‘𝐺)
gzsumval2.p	⊢ + = (+g‘𝐺)
gzsumval2.g	⊢ (𝜑 → 𝐺 ∈ 𝑉)
gzsumval2.n	⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
gzsumval2.f	⊢ (𝜑 → 𝐹:(𝑀...𝑁)⟶𝐵)

Assertion

Ref	Expression
gzsumval2	⊢ (𝜑 → (𝐺 Σgz 𝐹) = (seq𝑀( + , 𝐹)‘𝑁))

Proof of Theorem gzsumval2

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

Step	Hyp	Ref	Expression
1		gzsumval2.b	. . 3 ⊢ 𝐵 = (Base‘𝐺)
2		eqid 2238	. . 3 ⊢ (0g‘𝐺) = (0g‘𝐺)
3		gzsumval2.p	. . 3 ⊢ + = (+g‘𝐺)
4		gzsumval2.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ 𝑉)
5		gzsumval2.n	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
6		eluzel2 9905	. . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ)
7	5, 6	syl 14	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ)
8		eluzelz 9910	. . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ)
9	5, 8	syl 14	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℤ)
10	7, 9	fzfigd 10846	. . 3 ⊢ (𝜑 → (𝑀...𝑁) ∈ Fin)
11		gzsumval2.f	. . 3 ⊢ (𝜑 → 𝐹:(𝑀...𝑁)⟶𝐵)
12	1, 2, 3, 4, 10, 11	gzsumval 13687	. 2 ⊢ (𝜑 → (𝐺 Σgz 𝐹) = (℩𝑥(((𝑀...𝑁) = ∅ ∧ 𝑥 = (0g‘𝐺)) ∨ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)))))
13		simprr 537	. . . . . . . 8 ⊢ ((𝑛 ∈ (ℤ≥‘𝑚) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))
14		simprl 535	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ (ℤ≥‘𝑚) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → (𝑀...𝑁) = (𝑚...𝑛))
15		eqcom 2240	. . . . . . . . . . . . . 14 ⊢ ((𝑚...𝑛) = (𝑀...𝑁) ↔ (𝑀...𝑁) = (𝑚...𝑛))
16		fzopth 10445	. . . . . . . . . . . . . 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 10868	. . . . . . . . 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 5697	. . . . . . . 8 ⊢ ((𝑛 ∈ (ℤ≥‘𝑚) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → (seq𝑚( + , 𝐹)‘𝑛) = (seq𝑀( + , 𝐹)‘𝑁))
24	13, 23	eqtrd 2271	. . . . . . 7 ⊢ ((𝑛 ∈ (ℤ≥‘𝑚) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → 𝑥 = (seq𝑀( + , 𝐹)‘𝑁))
25	24	rexlimiva 2663	. . . . . 6 ⊢ (∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)) → 𝑥 = (seq𝑀( + , 𝐹)‘𝑁))
26	25	exlimiv 1651	. . . . 5 ⊢ (∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)) → 𝑥 = (seq𝑀( + , 𝐹)‘𝑁))
27	7	elexd 2835	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ V)
28	27	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑁)) → 𝑀 ∈ V)
29		oveq2 6083	. . . . . . . . . 10 ⊢ (𝑛 = 𝑁 → (𝑀...𝑛) = (𝑀...𝑁))
30	29	eqeq2d 2250	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → ((𝑀...𝑁) = (𝑀...𝑛) ↔ (𝑀...𝑁) = (𝑀...𝑁)))
31		fveq2 5690	. . . . . . . . . 10 ⊢ (𝑛 = 𝑁 → (seq𝑀( + , 𝐹)‘𝑛) = (seq𝑀( + , 𝐹)‘𝑁))
32	31	eqeq2d 2250	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → (𝑥 = (seq𝑀( + , 𝐹)‘𝑛) ↔ 𝑥 = (seq𝑀( + , 𝐹)‘𝑁)))
33	30, 32	anbi12d 477	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑛)) ↔ ((𝑀...𝑁) = (𝑀...𝑁) ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑁))))
34	5	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑁)) → 𝑁 ∈ (ℤ≥‘𝑀))
35		eqidd 2239	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑁)) → (𝑀...𝑁) = (𝑀...𝑁))
36		simpr 110	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑁)) → 𝑥 = (seq𝑀( + , 𝐹)‘𝑁))
37	35, 36	jca 306	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑁)) → ((𝑀...𝑁) = (𝑀...𝑁) ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑁)))
38	33, 34, 37	rspcedvdw 2936	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑁)) → ∃𝑛 ∈ (ℤ≥‘𝑀)((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑛)))
39		fveq2 5690	. . . . . . . 8 ⊢ (𝑚 = 𝑀 → (ℤ≥‘𝑚) = (ℤ≥‘𝑀))
40		oveq1 6082	. . . . . . . . . 10 ⊢ (𝑚 = 𝑀 → (𝑚...𝑛) = (𝑀...𝑛))
41	40	eqeq2d 2250	. . . . . . . . 9 ⊢ (𝑚 = 𝑀 → ((𝑀...𝑁) = (𝑚...𝑛) ↔ (𝑀...𝑁) = (𝑀...𝑛)))
42		seqeq1 10865	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑀 → seq𝑚( + , 𝐹) = seq𝑀( + , 𝐹))
43	42	fveq1d 5692	. . . . . . . . . 10 ⊢ (𝑚 = 𝑀 → (seq𝑚( + , 𝐹)‘𝑛) = (seq𝑀( + , 𝐹)‘𝑛))
44	43	eqeq2d 2250	. . . . . . . . 9 ⊢ (𝑚 = 𝑀 → (𝑥 = (seq𝑚( + , 𝐹)‘𝑛) ↔ 𝑥 = (seq𝑀( + , 𝐹)‘𝑛)))
45	41, 44	anbi12d 477	. . . . . . . 8 ⊢ (𝑚 = 𝑀 → (((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)) ↔ ((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑛))))
46	39, 45	rexeqbidv 2766	. . . . . . 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))))
47	28, 38, 46	spcedv 2914	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑁)) → ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)))
48	47	ex 115	. . . . 5 ⊢ (𝜑 → (𝑥 = (seq𝑀( + , 𝐹)‘𝑁) → ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))))
49	26, 48	impbid2 143	. . . 4 ⊢ (𝜑 → (∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)) ↔ 𝑥 = (seq𝑀( + , 𝐹)‘𝑁)))
50		eluzfz2 10415	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁))
51		n0i 3527	. . . . . . 7 ⊢ (𝑁 ∈ (𝑀...𝑁) → ¬ (𝑀...𝑁) = ∅)
52	5, 50, 51	3syl 17	. . . . . 6 ⊢ (𝜑 → ¬ (𝑀...𝑁) = ∅)
53	52	intnanrd 944	. . . . 5 ⊢ (𝜑 → ¬ ((𝑀...𝑁) = ∅ ∧ 𝑥 = (0g‘𝐺)))
54		biorf 756	. . . . 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)))))
55	53, 54	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)))))
56	49, 55	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)))))
57	56	iotabidv 5355	. 2 ⊢ (𝜑 → (℩𝑥𝑥 = (seq𝑀( + , 𝐹)‘𝑁)) = (℩𝑥(((𝑀...𝑁) = ∅ ∧ 𝑥 = (0g‘𝐺)) ∨ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)))))
58		eqid 2238	. . 3 ⊢ (seq𝑀( + , 𝐹)‘𝑁) = (seq𝑀( + , 𝐹)‘𝑁)
59		seqex 10864	. . . . 5 ⊢ seq𝑀( + , 𝐹) ∈ V
60		fvexg 5709	. . . . 5 ⊢ ((seq𝑀( + , 𝐹) ∈ V ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → (seq𝑀( + , 𝐹)‘𝑁) ∈ V)
61	59, 5, 60	sylancr 418	. . . 4 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) ∈ V)
62		eueq 2997	. . . . 5 ⊢ ((seq𝑀( + , 𝐹)‘𝑁) ∈ V ↔ ∃!𝑥 𝑥 = (seq𝑀( + , 𝐹)‘𝑁))
63	61, 62	sylib 122	. . . 4 ⊢ (𝜑 → ∃!𝑥 𝑥 = (seq𝑀( + , 𝐹)‘𝑁))
64		eqeq1 2245	. . . . 5 ⊢ (𝑥 = (seq𝑀( + , 𝐹)‘𝑁) → (𝑥 = (seq𝑀( + , 𝐹)‘𝑁) ↔ (seq𝑀( + , 𝐹)‘𝑁) = (seq𝑀( + , 𝐹)‘𝑁)))
65	64	iota2 5362	. . . 4 ⊢ (((seq𝑀( + , 𝐹)‘𝑁) ∈ V ∧ ∃!𝑥 𝑥 = (seq𝑀( + , 𝐹)‘𝑁)) → ((seq𝑀( + , 𝐹)‘𝑁) = (seq𝑀( + , 𝐹)‘𝑁) ↔ (℩𝑥𝑥 = (seq𝑀( + , 𝐹)‘𝑁)) = (seq𝑀( + , 𝐹)‘𝑁)))
66	61, 63, 65	syl2anc 415	. . 3 ⊢ (𝜑 → ((seq𝑀( + , 𝐹)‘𝑁) = (seq𝑀( + , 𝐹)‘𝑁) ↔ (℩𝑥𝑥 = (seq𝑀( + , 𝐹)‘𝑁)) = (seq𝑀( + , 𝐹)‘𝑁)))
67	58, 66	mpbii 148	. 2 ⊢ (𝜑 → (℩𝑥𝑥 = (seq𝑀( + , 𝐹)‘𝑁)) = (seq𝑀( + , 𝐹)‘𝑁))
68	12, 57, 67	3eqtr2d 2277	1 ⊢ (𝜑 → (𝐺 Σgz 𝐹) = (seq𝑀( + , 𝐹)‘𝑁))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 720   = wceq 1402  ∃wex 1545  ∃!weu 2086   ∈ wcel 2209  ∃wrex 2529  Vcvv 2821  ∅c0 3520  ℩cio 5330  ⟶wf 5368  ‘cfv 5372  (class class class)co 6075  Fincfn 7012  ℤcz 9623  ℤ_≥cuz 9900  ...cfz 10390  seqcseq 10862  Basecbs 13330  +_gcplusg 13408  0_gc0g 13587   Σ_gz cgzsu 13588

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-frec 6652  df-1o 6677  df-er 6797  df-en 7013  df-fin 7015  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-inn 9284  df-n0 9543  df-z 9624  df-uz 9901  df-fz 10391  df-seqfrec 10863  df-ndx 13333  df-slot 13334  df-base 13336  df-0g 13589  df-gzsum 13590

This theorem is referenced by:  gzsumsplit1r  13692  gzsumwsubmcl  13778  gzsumwmhm  13780  mulgnngzsum  13907  gzsumconst  14120  gzsumshift  14126  gsumvalfi  14129