Theorem gsumval2a 18731

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)
gsumval2a.o	⊢ 𝑂 = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}
gsumval2a.f	⊢ (𝜑 → ¬ ran 𝐹 ⊆ 𝑂)

Assertion

Ref	Expression
gsumval2a	⊢ (𝜑 → (𝐺 Σg 𝐹) = (seq𝑀( + , 𝐹)‘𝑁))

Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐺,𝑦   𝑥,𝑉   𝑥, + ,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝑀(𝑥,𝑦)   𝑁(𝑥,𝑦)   𝑂(𝑥,𝑦)   𝑉(𝑦)

Proof of Theorem gsumval2a

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

Step	Hyp	Ref	Expression
1		gsumval2.b	. . . 4 ⊢ 𝐵 = (Base‘𝐺)
2		eqid 2765	. . . 4 ⊢ (0g‘𝐺) = (0g‘𝐺)
3		gsumval2.p	. . . 4 ⊢ + = (+g‘𝐺)
4		gsumval2a.o	. . . 4 ⊢ 𝑂 = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}
5		eqidd 2766	. . . 4 ⊢ (𝜑 → (◡𝐹 “ (V ∖ 𝑂)) = (◡𝐹 “ (V ∖ 𝑂)))
6		gsumval2.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝑉)
7		ovexd 7435	. . . 4 ⊢ (𝜑 → (𝑀...𝑁) ∈ V)
8		gsumval2.f	. . . 4 ⊢ (𝜑 → 𝐹:(𝑀...𝑁)⟶𝐵)
9	1, 2, 3, 4, 5, 6, 7, 8	gsumval 18723	. . 3 ⊢ (𝜑 → (𝐺 Σg 𝐹) = if(ran 𝐹 ⊆ 𝑂, (0g‘𝐺), if((𝑀...𝑁) ∈ ran ..., (℩𝑧∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛))), (℩𝑧∃𝑓(𝑓:(1...(♯‘(◡𝐹 “ (V ∖ 𝑂))))–1-1-onto→(◡𝐹 “ (V ∖ 𝑂)) ∧ 𝑧 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘(◡𝐹 “ (V ∖ 𝑂)))))))))
10		gsumval2a.f	. . . . 5 ⊢ (𝜑 → ¬ ran 𝐹 ⊆ 𝑂)
11	10	iffalsed 4494	. . . 4 ⊢ (𝜑 → if(ran 𝐹 ⊆ 𝑂, (0g‘𝐺), if((𝑀...𝑁) ∈ ran ..., (℩𝑧∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛))), (℩𝑧∃𝑓(𝑓:(1...(♯‘(◡𝐹 “ (V ∖ 𝑂))))–1-1-onto→(◡𝐹 “ (V ∖ 𝑂)) ∧ 𝑧 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘(◡𝐹 “ (V ∖ 𝑂)))))))) = if((𝑀...𝑁) ∈ ran ..., (℩𝑧∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛))), (℩𝑧∃𝑓(𝑓:(1...(♯‘(◡𝐹 “ (V ∖ 𝑂))))–1-1-onto→(◡𝐹 “ (V ∖ 𝑂)) ∧ 𝑧 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘(◡𝐹 “ (V ∖ 𝑂))))))))
12		fzf 13527	. . . . . . 7 ⊢ ...:(ℤ × ℤ)⟶𝒫 ℤ
13		ffn 6695	. . . . . . 7 ⊢ (...:(ℤ × ℤ)⟶𝒫 ℤ → ... Fn (ℤ × ℤ))
14	12, 13	ax-mp 5	. . . . . 6 ⊢ ... Fn (ℤ × ℤ)
15		gsumval2.n	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
16		eluzel2 12855	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ)
17	15, 16	syl 18	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ)
18		eluzelz 12860	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ)
19	15, 18	syl 18	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℤ)
20		fnovrn 7575	. . . . . 6 ⊢ ((... Fn (ℤ × ℤ) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀...𝑁) ∈ ran ...)
21	14, 17, 19, 20	mp3an2i 1490	. . . . 5 ⊢ (𝜑 → (𝑀...𝑁) ∈ ran ...)
22	21	iftrued 4491	. . . 4 ⊢ (𝜑 → if((𝑀...𝑁) ∈ ran ..., (℩𝑧∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛))), (℩𝑧∃𝑓(𝑓:(1...(♯‘(◡𝐹 “ (V ∖ 𝑂))))–1-1-onto→(◡𝐹 “ (V ∖ 𝑂)) ∧ 𝑧 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘(◡𝐹 “ (V ∖ 𝑂))))))) = (℩𝑧∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛))))
23	11, 22	eqtrd 2800	. . 3 ⊢ (𝜑 → if(ran 𝐹 ⊆ 𝑂, (0g‘𝐺), if((𝑀...𝑁) ∈ ran ..., (℩𝑧∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛))), (℩𝑧∃𝑓(𝑓:(1...(♯‘(◡𝐹 “ (V ∖ 𝑂))))–1-1-onto→(◡𝐹 “ (V ∖ 𝑂)) ∧ 𝑧 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘(◡𝐹 “ (V ∖ 𝑂)))))))) = (℩𝑧∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛))))
24	9, 23	eqtrd 2800	. 2 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (℩𝑧∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛))))
25		fvex 6884	. . 3 ⊢ (seq𝑀( + , 𝐹)‘𝑁) ∈ V
26		fzopth 13577	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → ((𝑀...𝑁) = (𝑚...𝑛) ↔ (𝑀 = 𝑚 ∧ 𝑁 = 𝑛)))
27	15, 26	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑀...𝑁) = (𝑚...𝑛) ↔ (𝑀 = 𝑚 ∧ 𝑁 = 𝑛)))
28		simpl 487	. . . . . . . . . . . . . 14 ⊢ ((𝑀 = 𝑚 ∧ 𝑁 = 𝑛) → 𝑀 = 𝑚)
29	28	seqeq1d 14031	. . . . . . . . . . . . 13 ⊢ ((𝑀 = 𝑚 ∧ 𝑁 = 𝑛) → seq𝑀( + , 𝐹) = seq𝑚( + , 𝐹))
30		simpr 489	. . . . . . . . . . . . 13 ⊢ ((𝑀 = 𝑚 ∧ 𝑁 = 𝑛) → 𝑁 = 𝑛)
31	29, 30	fveq12d 6878	. . . . . . . . . . . 12 ⊢ ((𝑀 = 𝑚 ∧ 𝑁 = 𝑛) → (seq𝑀( + , 𝐹)‘𝑁) = (seq𝑚( + , 𝐹)‘𝑛))
32	31	eqcomd 2771	. . . . . . . . . . 11 ⊢ ((𝑀 = 𝑚 ∧ 𝑁 = 𝑛) → (seq𝑚( + , 𝐹)‘𝑛) = (seq𝑀( + , 𝐹)‘𝑁))
33		eqeq1 2769	. . . . . . . . . . 11 ⊢ (𝑧 = (seq𝑚( + , 𝐹)‘𝑛) → (𝑧 = (seq𝑀( + , 𝐹)‘𝑁) ↔ (seq𝑚( + , 𝐹)‘𝑛) = (seq𝑀( + , 𝐹)‘𝑁)))
34	32, 33	syl5ibrcom 250	. . . . . . . . . 10 ⊢ ((𝑀 = 𝑚 ∧ 𝑁 = 𝑛) → (𝑧 = (seq𝑚( + , 𝐹)‘𝑛) → 𝑧 = (seq𝑀( + , 𝐹)‘𝑁)))
35	27, 34	biimtrdi 256	. . . . . . . . 9 ⊢ (𝜑 → ((𝑀...𝑁) = (𝑚...𝑛) → (𝑧 = (seq𝑚( + , 𝐹)‘𝑛) → 𝑧 = (seq𝑀( + , 𝐹)‘𝑁))))
36	35	impd 415	. . . . . . . 8 ⊢ (𝜑 → (((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)) → 𝑧 = (seq𝑀( + , 𝐹)‘𝑁)))
37	36	rexlimdvw 3171	. . . . . . 7 ⊢ (𝜑 → (∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)) → 𝑧 = (seq𝑀( + , 𝐹)‘𝑁)))
38	37	exlimdv 1956	. . . . . 6 ⊢ (𝜑 → (∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)) → 𝑧 = (seq𝑀( + , 𝐹)‘𝑁)))
39	17	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 = (seq𝑀( + , 𝐹)‘𝑁)) → 𝑀 ∈ ℤ)
40		oveq2 7408	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑁 → (𝑀...𝑛) = (𝑀...𝑁))
41	40	eqcomd 2771	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑁 → (𝑀...𝑁) = (𝑀...𝑛))
42	41	biantrurd 541	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑁 → (𝑧 = (seq𝑀( + , 𝐹)‘𝑛) ↔ ((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑧 = (seq𝑀( + , 𝐹)‘𝑛))))
43		fveq2 6871	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑁 → (seq𝑀( + , 𝐹)‘𝑛) = (seq𝑀( + , 𝐹)‘𝑁))
44	43	eqeq2d 2776	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑁 → (𝑧 = (seq𝑀( + , 𝐹)‘𝑛) ↔ 𝑧 = (seq𝑀( + , 𝐹)‘𝑁)))
45	42, 44	bitr3d 284	. . . . . . . . . 10 ⊢ (𝑛 = 𝑁 → (((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑧 = (seq𝑀( + , 𝐹)‘𝑛)) ↔ 𝑧 = (seq𝑀( + , 𝐹)‘𝑁)))
46	45	rspcev 3584	. . . . . . . . 9 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝑧 = (seq𝑀( + , 𝐹)‘𝑁)) → ∃𝑛 ∈ (ℤ≥‘𝑀)((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑧 = (seq𝑀( + , 𝐹)‘𝑛)))
47	15, 46	sylan 591	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 = (seq𝑀( + , 𝐹)‘𝑁)) → ∃𝑛 ∈ (ℤ≥‘𝑀)((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑧 = (seq𝑀( + , 𝐹)‘𝑛)))
48		fveq2 6871	. . . . . . . . . 10 ⊢ (𝑚 = 𝑀 → (ℤ≥‘𝑚) = (ℤ≥‘𝑀))
49		oveq1 7407	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑀 → (𝑚...𝑛) = (𝑀...𝑛))
50	49	eqeq2d 2776	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑀 → ((𝑀...𝑁) = (𝑚...𝑛) ↔ (𝑀...𝑁) = (𝑀...𝑛)))
51		seqeq1 14028	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑀 → seq𝑚( + , 𝐹) = seq𝑀( + , 𝐹))
52	51	fveq1d 6873	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑀 → (seq𝑚( + , 𝐹)‘𝑛) = (seq𝑀( + , 𝐹)‘𝑛))
53	52	eqeq2d 2776	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑀 → (𝑧 = (seq𝑚( + , 𝐹)‘𝑛) ↔ 𝑧 = (seq𝑀( + , 𝐹)‘𝑛)))
54	50, 53	anbi12d 643	. . . . . . . . . 10 ⊢ (𝑚 = 𝑀 → (((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)) ↔ ((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑧 = (seq𝑀( + , 𝐹)‘𝑛))))
55	48, 54	rexeqbidv 3340	. . . . . . . . 9 ⊢ (𝑚 = 𝑀 → (∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)) ↔ ∃𝑛 ∈ (ℤ≥‘𝑀)((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑧 = (seq𝑀( + , 𝐹)‘𝑛))))
56	55	spcegv 3559	. . . . . . . 8 ⊢ (𝑀 ∈ ℤ → (∃𝑛 ∈ (ℤ≥‘𝑀)((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑧 = (seq𝑀( + , 𝐹)‘𝑛)) → ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛))))
57	39, 47, 56	sylc 66	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 = (seq𝑀( + , 𝐹)‘𝑁)) → ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)))
58	57	ex 417	. . . . . 6 ⊢ (𝜑 → (𝑧 = (seq𝑀( + , 𝐹)‘𝑁) → ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛))))
59	38, 58	impbid 215	. . . . 5 ⊢ (𝜑 → (∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)) ↔ 𝑧 = (seq𝑀( + , 𝐹)‘𝑁)))
60	59	adantr 485	. . . 4 ⊢ ((𝜑 ∧ (seq𝑀( + , 𝐹)‘𝑁) ∈ V) → (∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)) ↔ 𝑧 = (seq𝑀( + , 𝐹)‘𝑁)))
61	60	iota5 6508	. . 3 ⊢ ((𝜑 ∧ (seq𝑀( + , 𝐹)‘𝑁) ∈ V) → (℩𝑧∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛))) = (seq𝑀( + , 𝐹)‘𝑁))
62	25, 61	mpan2 703	. 2 ⊢ (𝜑 → (℩𝑧∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛))) = (seq𝑀( + , 𝐹)‘𝑁))
63	24, 62	eqtrd 2800	1 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (seq𝑀( + , 𝐹)‘𝑁))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1563  ∃wex 1802   ∈ wcel 2145  ∀wral 3079  ∃wrex 3089  {crab 3417  Vcvv 3457   ∖ cdif 3904   ⊆ wss 3907  ifcif 4483  𝒫 cpw 4558   × cxp 5649  ^◡ccnv 5650  ran crn 5652   “ cima 5654   ∘ ccom 5655  ℩cio 6479   Fn wfn 6520  ⟶wf 6521  –1-1-onto→wf1o 6524  ‘cfv 6525  (class class class)co 7400  1c1 11089  ℤcz 12579  ℤ_≥cuz 12850  ...cfz 13523  seqcseq 14025  ♯chash 14354  Basecbs 17257  +_gcplusg 17298  0_gc0g 17480   Σ_g cgsu 17481

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-sep 5250  ax-nul 5260  ax-pow 5326  ax-pr 5394  ax-un 7722  ax-cnex 11144  ax-resscn 11145  ax-pre-lttri 11162  ax-pre-lttrn 11163

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-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5105  df-opab 5167  df-mpt 5186  df-id 5546  df-po 5559  df-so 5560  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6291  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-ov 7403  df-oprab 7404  df-mpo 7405  df-1st 7974  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-er 8682  df-en 8932  df-dom 8933  df-sdom 8934  df-pnf 11233  df-mnf 11234  df-xr 11235  df-ltxr 11236  df-le 11237  df-neg 11432  df-z 12580  df-uz 12851  df-fz 13524  df-seq 14026  df-gsum 17483

This theorem is referenced by:  gsumval2  18732