Theorem gsumval2a 17897

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 2823	. . . 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 2824	. . . 4 ⊢ (𝜑 → (◡𝐹 “ (V ∖ 𝑂)) = (◡𝐹 “ (V ∖ 𝑂)))
6		gsumval2.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝑉)
7		ovexd 7193	. . . 4 ⊢ (𝜑 → (𝑀...𝑁) ∈ V)
8		gsumval2.f	. . . 4 ⊢ (𝜑 → 𝐹:(𝑀...𝑁)⟶𝐵)
9	1, 2, 3, 4, 5, 6, 7, 8	gsumval 17889	. . 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 4480	. . . 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 12899	. . . . . . 7 ⊢ ...:(ℤ × ℤ)⟶𝒫 ℤ
13		ffn 6516	. . . . . . 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 12251	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ)
17	15, 16	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ)
18		eluzelz 12256	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ)
19	15, 18	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℤ)
20		fnovrn 7325	. . . . . 6 ⊢ ((... Fn (ℤ × ℤ) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀...𝑁) ∈ ran ...)
21	14, 17, 19, 20	mp3an2i 1462	. . . . 5 ⊢ (𝜑 → (𝑀...𝑁) ∈ ran ...)
22	21	iftrued 4477	. . . 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 2858	. . 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 2858	. 2 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (℩𝑧∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛))))
25		fvex 6685	. . 3 ⊢ (seq𝑀( + , 𝐹)‘𝑁) ∈ V
26		fzopth 12947	. . . . . . . . . . 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 485	. . . . . . . . . . . . . 14 ⊢ ((𝑀 = 𝑚 ∧ 𝑁 = 𝑛) → 𝑀 = 𝑚)
29	28	seqeq1d 13378	. . . . . . . . . . . . 13 ⊢ ((𝑀 = 𝑚 ∧ 𝑁 = 𝑛) → seq𝑀( + , 𝐹) = seq𝑚( + , 𝐹))
30		simpr 487	. . . . . . . . . . . . 13 ⊢ ((𝑀 = 𝑚 ∧ 𝑁 = 𝑛) → 𝑁 = 𝑛)
31	29, 30	fveq12d 6679	. . . . . . . . . . . 12 ⊢ ((𝑀 = 𝑚 ∧ 𝑁 = 𝑛) → (seq𝑀( + , 𝐹)‘𝑁) = (seq𝑚( + , 𝐹)‘𝑛))
32	31	eqcomd 2829	. . . . . . . . . . 11 ⊢ ((𝑀 = 𝑚 ∧ 𝑁 = 𝑛) → (seq𝑚( + , 𝐹)‘𝑛) = (seq𝑀( + , 𝐹)‘𝑁))
33		eqeq1 2827	. . . . . . . . . . 11 ⊢ (𝑧 = (seq𝑚( + , 𝐹)‘𝑛) → (𝑧 = (seq𝑀( + , 𝐹)‘𝑁) ↔ (seq𝑚( + , 𝐹)‘𝑛) = (seq𝑀( + , 𝐹)‘𝑁)))
34	32, 33	syl5ibrcom 249	. . . . . . . . . 10 ⊢ ((𝑀 = 𝑚 ∧ 𝑁 = 𝑛) → (𝑧 = (seq𝑚( + , 𝐹)‘𝑛) → 𝑧 = (seq𝑀( + , 𝐹)‘𝑁)))
35	27, 34	syl6bi 255	. . . . . . . . 9 ⊢ (𝜑 → ((𝑀...𝑁) = (𝑚...𝑛) → (𝑧 = (seq𝑚( + , 𝐹)‘𝑛) → 𝑧 = (seq𝑀( + , 𝐹)‘𝑁))))
36	35	impd 413	. . . . . . . 8 ⊢ (𝜑 → (((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)) → 𝑧 = (seq𝑀( + , 𝐹)‘𝑁)))
37	36	rexlimdvw 3292	. . . . . . 7 ⊢ (𝜑 → (∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)) → 𝑧 = (seq𝑀( + , 𝐹)‘𝑁)))
38	37	exlimdv 1934	. . . . . 6 ⊢ (𝜑 → (∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)) → 𝑧 = (seq𝑀( + , 𝐹)‘𝑁)))
39	17	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 = (seq𝑀( + , 𝐹)‘𝑁)) → 𝑀 ∈ ℤ)
40		oveq2 7166	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑁 → (𝑀...𝑛) = (𝑀...𝑁))
41	40	eqcomd 2829	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑁 → (𝑀...𝑁) = (𝑀...𝑛))
42	41	biantrurd 535	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑁 → (𝑧 = (seq𝑀( + , 𝐹)‘𝑛) ↔ ((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑧 = (seq𝑀( + , 𝐹)‘𝑛))))
43		fveq2 6672	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑁 → (seq𝑀( + , 𝐹)‘𝑛) = (seq𝑀( + , 𝐹)‘𝑁))
44	43	eqeq2d 2834	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑁 → (𝑧 = (seq𝑀( + , 𝐹)‘𝑛) ↔ 𝑧 = (seq𝑀( + , 𝐹)‘𝑁)))
45	42, 44	bitr3d 283	. . . . . . . . . 10 ⊢ (𝑛 = 𝑁 → (((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑧 = (seq𝑀( + , 𝐹)‘𝑛)) ↔ 𝑧 = (seq𝑀( + , 𝐹)‘𝑁)))
46	45	rspcev 3625	. . . . . . . . 9 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝑧 = (seq𝑀( + , 𝐹)‘𝑁)) → ∃𝑛 ∈ (ℤ≥‘𝑀)((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑧 = (seq𝑀( + , 𝐹)‘𝑛)))
47	15, 46	sylan 582	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 = (seq𝑀( + , 𝐹)‘𝑁)) → ∃𝑛 ∈ (ℤ≥‘𝑀)((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑧 = (seq𝑀( + , 𝐹)‘𝑛)))
48		fveq2 6672	. . . . . . . . . 10 ⊢ (𝑚 = 𝑀 → (ℤ≥‘𝑚) = (ℤ≥‘𝑀))
49		oveq1 7165	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑀 → (𝑚...𝑛) = (𝑀...𝑛))
50	49	eqeq2d 2834	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑀 → ((𝑀...𝑁) = (𝑚...𝑛) ↔ (𝑀...𝑁) = (𝑀...𝑛)))
51		seqeq1 13375	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑀 → seq𝑚( + , 𝐹) = seq𝑀( + , 𝐹))
52	51	fveq1d 6674	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑀 → (seq𝑚( + , 𝐹)‘𝑛) = (seq𝑀( + , 𝐹)‘𝑛))
53	52	eqeq2d 2834	. . . . . . . . . . 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 3404	. . . . . . . . 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 3599	. . . . . . . 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 415	. . . . . 6 ⊢ (𝜑 → (𝑧 = (seq𝑀( + , 𝐹)‘𝑁) → ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛))))
59	38, 58	impbid 214	. . . . 5 ⊢ (𝜑 → (∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)) ↔ 𝑧 = (seq𝑀( + , 𝐹)‘𝑁)))
60	59	adantr 483	. . . 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 6340	. . 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 689	. 2 ⊢ (𝜑 → (℩𝑧∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛))) = (seq𝑀( + , 𝐹)‘𝑁))
63	24, 62	eqtrd 2858	1 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (seq𝑀( + , 𝐹)‘𝑁))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537  ∃wex 1780   ∈ wcel 2114  ∀wral 3140  ∃wrex 3141  {crab 3144  Vcvv 3496   ∖ cdif 3935   ⊆ wss 3938  ifcif 4469  𝒫 cpw 4541   × cxp 5555  ^◡ccnv 5556  ran crn 5558   “ cima 5560   ∘ ccom 5561  ℩cio 6314   Fn wfn 6352  ⟶wf 6353  –1-1-onto→wf1o 6356  ‘cfv 6357  (class class class)co 7158  1c1 10540  ℤcz 11984  ℤ_≥cuz 12246  ...cfz 12895  seqcseq 13372  ♯chash 13693  Basecbs 16485  +_gcplusg 16567  0_gc0g 16715   Σ_g cgsu 16716

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-cnex 10595  ax-resscn 10596  ax-pre-lttri 10613  ax-pre-lttrn 10614

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-nel 3126  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-po 5476  df-so 5477  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-ov 7161  df-oprab 7162  df-mpo 7163  df-1st 7691  df-2nd 7692  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-er 8291  df-en 8512  df-dom 8513  df-sdom 8514  df-pnf 10679  df-mnf 10680  df-xr 10681  df-ltxr 10682  df-le 10683  df-neg 10875  df-z 11985  df-uz 12247  df-fz 12896  df-seq 13373  df-gsum 16718

This theorem is referenced by:  gsumval2  17898