Theorem gsumval2a 18612

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 2729	. . . 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 2730	. . . 4 ⊢ (𝜑 → (◡𝐹 “ (V ∖ 𝑂)) = (◡𝐹 “ (V ∖ 𝑂)))
6		gsumval2.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝑉)
7		ovexd 7422	. . . 4 ⊢ (𝜑 → (𝑀...𝑁) ∈ V)
8		gsumval2.f	. . . 4 ⊢ (𝜑 → 𝐹:(𝑀...𝑁)⟶𝐵)
9	1, 2, 3, 4, 5, 6, 7, 8	gsumval 18604	. . 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 4499	. . . 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 13472	. . . . . . 7 ⊢ ...:(ℤ × ℤ)⟶𝒫 ℤ
13		ffn 6688	. . . . . . 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 12798	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ)
17	15, 16	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ)
18		eluzelz 12803	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ)
19	15, 18	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℤ)
20		fnovrn 7564	. . . . . 6 ⊢ ((... Fn (ℤ × ℤ) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀...𝑁) ∈ ran ...)
21	14, 17, 19, 20	mp3an2i 1468	. . . . 5 ⊢ (𝜑 → (𝑀...𝑁) ∈ ran ...)
22	21	iftrued 4496	. . . 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 2764	. . 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 2764	. 2 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (℩𝑧∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛))))
25		fvex 6871	. . 3 ⊢ (seq𝑀( + , 𝐹)‘𝑁) ∈ V
26		fzopth 13522	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → ((𝑀...𝑁) = (𝑚...𝑛) ↔ (𝑀 = 𝑚 ∧ 𝑁 = 𝑛)))
27	15, 26	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑀...𝑁) = (𝑚...𝑛) ↔ (𝑀 = 𝑚 ∧ 𝑁 = 𝑛)))
28		simpl 482	. . . . . . . . . . . . . 14 ⊢ ((𝑀 = 𝑚 ∧ 𝑁 = 𝑛) → 𝑀 = 𝑚)
29	28	seqeq1d 13972	. . . . . . . . . . . . 13 ⊢ ((𝑀 = 𝑚 ∧ 𝑁 = 𝑛) → seq𝑀( + , 𝐹) = seq𝑚( + , 𝐹))
30		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝑀 = 𝑚 ∧ 𝑁 = 𝑛) → 𝑁 = 𝑛)
31	29, 30	fveq12d 6865	. . . . . . . . . . . 12 ⊢ ((𝑀 = 𝑚 ∧ 𝑁 = 𝑛) → (seq𝑀( + , 𝐹)‘𝑁) = (seq𝑚( + , 𝐹)‘𝑛))
32	31	eqcomd 2735	. . . . . . . . . . 11 ⊢ ((𝑀 = 𝑚 ∧ 𝑁 = 𝑛) → (seq𝑚( + , 𝐹)‘𝑛) = (seq𝑀( + , 𝐹)‘𝑁))
33		eqeq1 2733	. . . . . . . . . . 11 ⊢ (𝑧 = (seq𝑚( + , 𝐹)‘𝑛) → (𝑧 = (seq𝑀( + , 𝐹)‘𝑁) ↔ (seq𝑚( + , 𝐹)‘𝑛) = (seq𝑀( + , 𝐹)‘𝑁)))
34	32, 33	syl5ibrcom 247	. . . . . . . . . 10 ⊢ ((𝑀 = 𝑚 ∧ 𝑁 = 𝑛) → (𝑧 = (seq𝑚( + , 𝐹)‘𝑛) → 𝑧 = (seq𝑀( + , 𝐹)‘𝑁)))
35	27, 34	biimtrdi 253	. . . . . . . . 9 ⊢ (𝜑 → ((𝑀...𝑁) = (𝑚...𝑛) → (𝑧 = (seq𝑚( + , 𝐹)‘𝑛) → 𝑧 = (seq𝑀( + , 𝐹)‘𝑁))))
36	35	impd 410	. . . . . . . 8 ⊢ (𝜑 → (((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)) → 𝑧 = (seq𝑀( + , 𝐹)‘𝑁)))
37	36	rexlimdvw 3139	. . . . . . 7 ⊢ (𝜑 → (∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)) → 𝑧 = (seq𝑀( + , 𝐹)‘𝑁)))
38	37	exlimdv 1933	. . . . . 6 ⊢ (𝜑 → (∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)) → 𝑧 = (seq𝑀( + , 𝐹)‘𝑁)))
39	17	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 = (seq𝑀( + , 𝐹)‘𝑁)) → 𝑀 ∈ ℤ)
40		oveq2 7395	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑁 → (𝑀...𝑛) = (𝑀...𝑁))
41	40	eqcomd 2735	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑁 → (𝑀...𝑁) = (𝑀...𝑛))
42	41	biantrurd 532	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑁 → (𝑧 = (seq𝑀( + , 𝐹)‘𝑛) ↔ ((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑧 = (seq𝑀( + , 𝐹)‘𝑛))))
43		fveq2 6858	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑁 → (seq𝑀( + , 𝐹)‘𝑛) = (seq𝑀( + , 𝐹)‘𝑁))
44	43	eqeq2d 2740	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑁 → (𝑧 = (seq𝑀( + , 𝐹)‘𝑛) ↔ 𝑧 = (seq𝑀( + , 𝐹)‘𝑁)))
45	42, 44	bitr3d 281	. . . . . . . . . 10 ⊢ (𝑛 = 𝑁 → (((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑧 = (seq𝑀( + , 𝐹)‘𝑛)) ↔ 𝑧 = (seq𝑀( + , 𝐹)‘𝑁)))
46	45	rspcev 3588	. . . . . . . . 9 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝑧 = (seq𝑀( + , 𝐹)‘𝑁)) → ∃𝑛 ∈ (ℤ≥‘𝑀)((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑧 = (seq𝑀( + , 𝐹)‘𝑛)))
47	15, 46	sylan 580	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 = (seq𝑀( + , 𝐹)‘𝑁)) → ∃𝑛 ∈ (ℤ≥‘𝑀)((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑧 = (seq𝑀( + , 𝐹)‘𝑛)))
48		fveq2 6858	. . . . . . . . . 10 ⊢ (𝑚 = 𝑀 → (ℤ≥‘𝑚) = (ℤ≥‘𝑀))
49		oveq1 7394	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑀 → (𝑚...𝑛) = (𝑀...𝑛))
50	49	eqeq2d 2740	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑀 → ((𝑀...𝑁) = (𝑚...𝑛) ↔ (𝑀...𝑁) = (𝑀...𝑛)))
51		seqeq1 13969	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑀 → seq𝑚( + , 𝐹) = seq𝑀( + , 𝐹))
52	51	fveq1d 6860	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑀 → (seq𝑚( + , 𝐹)‘𝑛) = (seq𝑀( + , 𝐹)‘𝑛))
53	52	eqeq2d 2740	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑀 → (𝑧 = (seq𝑚( + , 𝐹)‘𝑛) ↔ 𝑧 = (seq𝑀( + , 𝐹)‘𝑛)))
54	50, 53	anbi12d 632	. . . . . . . . . 10 ⊢ (𝑚 = 𝑀 → (((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)) ↔ ((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑧 = (seq𝑀( + , 𝐹)‘𝑛))))
55	48, 54	rexeqbidv 3320	. . . . . . . . 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 3563	. . . . . . . 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 65	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 = (seq𝑀( + , 𝐹)‘𝑁)) → ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)))
58	57	ex 412	. . . . . 6 ⊢ (𝜑 → (𝑧 = (seq𝑀( + , 𝐹)‘𝑁) → ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛))))
59	38, 58	impbid 212	. . . . 5 ⊢ (𝜑 → (∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)) ↔ 𝑧 = (seq𝑀( + , 𝐹)‘𝑁)))
60	59	adantr 480	. . . 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 6494	. . 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 691	. 2 ⊢ (𝜑 → (℩𝑧∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛))) = (seq𝑀( + , 𝐹)‘𝑁))
63	24, 62	eqtrd 2764	1 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (seq𝑀( + , 𝐹)‘𝑁))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053  {crab 3405  Vcvv 3447   ∖ cdif 3911   ⊆ wss 3914  ifcif 4488  𝒫 cpw 4563   × cxp 5636  ^◡ccnv 5637  ran crn 5639   “ cima 5641   ∘ ccom 5642  ℩cio 6462   Fn wfn 6506  ⟶wf 6507  –1-1-onto→wf1o 6510  ‘cfv 6511  (class class class)co 7387  1c1 11069  ℤcz 12529  ℤ_≥cuz 12793  ...cfz 13468  seqcseq 13966  ♯chash 14295  Basecbs 17179  +_gcplusg 17220  0_gc0g 17402   Σ_g cgsu 17403

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-pre-lttri 11142  ax-pre-lttrn 11143

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-po 5546  df-so 5547  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-er 8671  df-en 8919  df-dom 8920  df-sdom 8921  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-neg 11408  df-z 12530  df-uz 12794  df-fz 13469  df-seq 13967  df-gsum 17405

This theorem is referenced by:  gsumval2  18613