Theorem gsumfzval 13419

Description: An expression for Σ_g when summing over a finite set of sequential integers. (Contributed by Jim Kingdon, 14-Aug-2025.)

Hypotheses

Ref	Expression
gsumval.b	⊢ 𝐵 = (Base‘𝐺)
gsumval.z	⊢ 0 = (0g‘𝐺)
gsumval.p	⊢ + = (+g‘𝐺)
gsumval.g	⊢ (𝜑 → 𝐺 ∈ 𝑉)
gsumfzval.m	⊢ (𝜑 → 𝑀 ∈ ℤ)
gsumfzval.n	⊢ (𝜑 → 𝑁 ∈ ℤ)
gsumfzval.f	⊢ (𝜑 → 𝐹:(𝑀...𝑁)⟶𝐵)

Assertion

Ref	Expression
gsumfzval	⊢ (𝜑 → (𝐺 Σg 𝐹) = if(𝑁 < 𝑀, 0 , (seq𝑀( + , 𝐹)‘𝑁)))

Proof of Theorem gsumfzval

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

Step	Hyp	Ref	Expression
1		gsumval.b	. . 3 ⊢ 𝐵 = (Base‘𝐺)
2		gsumval.z	. . 3 ⊢ 0 = (0g‘𝐺)
3		gsumval.p	. . 3 ⊢ + = (+g‘𝐺)
4		gsumval.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ 𝑉)
5		gsumfzval.m	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ)
6		gsumfzval.n	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℤ)
7	5, 6	fzfigd 10648	. . 3 ⊢ (𝜑 → (𝑀...𝑁) ∈ Fin)
8		gsumfzval.f	. . 3 ⊢ (𝜑 → 𝐹:(𝑀...𝑁)⟶𝐵)
9	1, 2, 3, 4, 7, 8	igsumval 13418	. 2 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (℩𝑥(((𝑀...𝑁) = ∅ ∧ 𝑥 = 0 ) ∨ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)))))
10		fn0g 13403	. . . . . 6 ⊢ 0g Fn V
11	4	elexd 2813	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ V)
12		funfvex 5643	. . . . . . 7 ⊢ ((Fun 0g ∧ 𝐺 ∈ dom 0g) → (0g‘𝐺) ∈ V)
13	12	funfni 5422	. . . . . 6 ⊢ ((0g Fn V ∧ 𝐺 ∈ V) → (0g‘𝐺) ∈ V)
14	10, 11, 13	sylancr 414	. . . . 5 ⊢ (𝜑 → (0g‘𝐺) ∈ V)
15	2, 14	eqeltrid 2316	. . . 4 ⊢ (𝜑 → 0 ∈ V)
16		seqex 10666	. . . . 5 ⊢ seq𝑀( + , 𝐹) ∈ V
17		fvexg 5645	. . . . 5 ⊢ ((seq𝑀( + , 𝐹) ∈ V ∧ 𝑁 ∈ ℤ) → (seq𝑀( + , 𝐹)‘𝑁) ∈ V)
18	16, 6, 17	sylancr 414	. . . 4 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) ∈ V)
19	15, 18	ifexd 4574	. . 3 ⊢ (𝜑 → if(𝑁 < 𝑀, 0 , (seq𝑀( + , 𝐹)‘𝑁)) ∈ V)
20		zdclt 9520	. . . . . . . 8 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → DECID 𝑁 < 𝑀)
21	6, 5, 20	syl2anc 411	. . . . . . 7 ⊢ (𝜑 → DECID 𝑁 < 𝑀)
22		eqifdc 3639	. . . . . . 7 ⊢ (DECID 𝑁 < 𝑀 → (𝑥 = if(𝑁 < 𝑀, 0 , (seq𝑀( + , 𝐹)‘𝑁)) ↔ ((𝑁 < 𝑀 ∧ 𝑥 = 0 ) ∨ (¬ 𝑁 < 𝑀 ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑁)))))
23	21, 22	syl 14	. . . . . 6 ⊢ (𝜑 → (𝑥 = if(𝑁 < 𝑀, 0 , (seq𝑀( + , 𝐹)‘𝑁)) ↔ ((𝑁 < 𝑀 ∧ 𝑥 = 0 ) ∨ (¬ 𝑁 < 𝑀 ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑁)))))
24		fzn 10234	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 < 𝑀 ↔ (𝑀...𝑁) = ∅))
25	5, 6, 24	syl2anc 411	. . . . . . . 8 ⊢ (𝜑 → (𝑁 < 𝑀 ↔ (𝑀...𝑁) = ∅))
26	25	anbi1d 465	. . . . . . 7 ⊢ (𝜑 → ((𝑁 < 𝑀 ∧ 𝑥 = 0 ) ↔ ((𝑀...𝑁) = ∅ ∧ 𝑥 = 0 )))
27	5	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (¬ 𝑁 < 𝑀 ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑁))) → 𝑀 ∈ ℤ)
28	27	zred 9565	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (¬ 𝑁 < 𝑀 ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑁))) → 𝑀 ∈ ℝ)
29	6	adantr 276	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (¬ 𝑁 < 𝑀 ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑁))) → 𝑁 ∈ ℤ)
30	29	zred 9565	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (¬ 𝑁 < 𝑀 ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑁))) → 𝑁 ∈ ℝ)
31		simprl 529	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (¬ 𝑁 < 𝑀 ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑁))) → ¬ 𝑁 < 𝑀)
32	28, 30, 31	nltled 8263	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (¬ 𝑁 < 𝑀 ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑁))) → 𝑀 ≤ 𝑁)
33		eluz 9731	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ≥‘𝑀) ↔ 𝑀 ≤ 𝑁))
34	27, 29, 33	syl2anc 411	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (¬ 𝑁 < 𝑀 ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑁))) → (𝑁 ∈ (ℤ≥‘𝑀) ↔ 𝑀 ≤ 𝑁))
35	32, 34	mpbird 167	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (¬ 𝑁 < 𝑀 ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑁))) → 𝑁 ∈ (ℤ≥‘𝑀))
36		oveq2 6008	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑁 → (𝑀...𝑛) = (𝑀...𝑁))
37	36	eqeq2d 2241	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑁 → ((𝑀...𝑁) = (𝑀...𝑛) ↔ (𝑀...𝑁) = (𝑀...𝑁)))
38		fveq2 5626	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑁 → (seq𝑀( + , 𝐹)‘𝑛) = (seq𝑀( + , 𝐹)‘𝑁))
39	38	eqeq2d 2241	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑁 → (𝑥 = (seq𝑀( + , 𝐹)‘𝑛) ↔ 𝑥 = (seq𝑀( + , 𝐹)‘𝑁)))
40	37, 39	anbi12d 473	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑁 → (((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑛)) ↔ ((𝑀...𝑁) = (𝑀...𝑁) ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑁))))
41	40	adantl 277	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (¬ 𝑁 < 𝑀 ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑁))) ∧ 𝑛 = 𝑁) → (((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑛)) ↔ ((𝑀...𝑁) = (𝑀...𝑁) ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑁))))
42		eqidd 2230	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (¬ 𝑁 < 𝑀 ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑁))) → (𝑀...𝑁) = (𝑀...𝑁))
43		simprr 531	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (¬ 𝑁 < 𝑀 ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑁))) → 𝑥 = (seq𝑀( + , 𝐹)‘𝑁))
44	42, 43	jca 306	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (¬ 𝑁 < 𝑀 ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑁))) → ((𝑀...𝑁) = (𝑀...𝑁) ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑁)))
45	35, 41, 44	rspcedvd 2913	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (¬ 𝑁 < 𝑀 ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑁))) → ∃𝑛 ∈ (ℤ≥‘𝑀)((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑛)))
46		fveq2 5626	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑀 → (ℤ≥‘𝑚) = (ℤ≥‘𝑀))
47		oveq1 6007	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑀 → (𝑚...𝑛) = (𝑀...𝑛))
48	47	eqeq2d 2241	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑀 → ((𝑀...𝑁) = (𝑚...𝑛) ↔ (𝑀...𝑁) = (𝑀...𝑛)))
49		seqeq1 10667	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 𝑀 → seq𝑚( + , 𝐹) = seq𝑀( + , 𝐹))
50	49	fveq1d 5628	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑀 → (seq𝑚( + , 𝐹)‘𝑛) = (seq𝑀( + , 𝐹)‘𝑛))
51	50	eqeq2d 2241	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑀 → (𝑥 = (seq𝑚( + , 𝐹)‘𝑛) ↔ 𝑥 = (seq𝑀( + , 𝐹)‘𝑛)))
52	48, 51	anbi12d 473	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑀 → (((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)) ↔ ((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑛))))
53	46, 52	rexeqbidv 2745	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑀 → (∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)) ↔ ∃𝑛 ∈ (ℤ≥‘𝑀)((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑛))))
54	53	spcegv 2891	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℤ → (∃𝑛 ∈ (ℤ≥‘𝑀)((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑛)) → ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))))
55	27, 45, 54	sylc 62	. . . . . . . . 9 ⊢ ((𝜑 ∧ (¬ 𝑁 < 𝑀 ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑁))) → ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)))
56	55	ex 115	. . . . . . . 8 ⊢ (𝜑 → ((¬ 𝑁 < 𝑀 ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑁)) → ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))))
57		eluzel2 9723	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ (ℤ≥‘𝑚) → 𝑚 ∈ ℤ)
58	57	ad2antlr 489	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑚)) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → 𝑚 ∈ ℤ)
59	58	zred 9565	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑚)) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → 𝑚 ∈ ℝ)
60		eluzelre 9728	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (ℤ≥‘𝑚) → 𝑛 ∈ ℝ)
61	60	ad2antlr 489	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑚)) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → 𝑛 ∈ ℝ)
62		eluzle 9730	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (ℤ≥‘𝑚) → 𝑚 ≤ 𝑛)
63	62	ad2antlr 489	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑚)) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → 𝑚 ≤ 𝑛)
64	59, 61, 63	lensymd 8264	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑚)) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → ¬ 𝑛 < 𝑚)
65		simprl 529	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑚)) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → (𝑀...𝑁) = (𝑚...𝑛))
66	65	eqcomd 2235	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑚)) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → (𝑚...𝑛) = (𝑀...𝑁))
67		fzopth 10253	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ (ℤ≥‘𝑚) → ((𝑚...𝑛) = (𝑀...𝑁) ↔ (𝑚 = 𝑀 ∧ 𝑛 = 𝑁)))
68	67	ad2antlr 489	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑚)) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → ((𝑚...𝑛) = (𝑀...𝑁) ↔ (𝑚 = 𝑀 ∧ 𝑛 = 𝑁)))
69	66, 68	mpbid 147	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑚)) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → (𝑚 = 𝑀 ∧ 𝑛 = 𝑁))
70	69	simprd 114	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑚)) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → 𝑛 = 𝑁)
71	69	simpld 112	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑚)) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → 𝑚 = 𝑀)
72	70, 71	breq12d 4095	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑚)) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → (𝑛 < 𝑚 ↔ 𝑁 < 𝑀))
73	64, 72	mtbid 676	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑚)) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → ¬ 𝑁 < 𝑀)
74		simprr 531	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑚)) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))
75	71	seqeq1d 10670	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑚)) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → seq𝑚( + , 𝐹) = seq𝑀( + , 𝐹))
76	75, 70	fveq12d 5633	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑚)) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → (seq𝑚( + , 𝐹)‘𝑛) = (seq𝑀( + , 𝐹)‘𝑁))
77	74, 76	eqtrd 2262	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑚)) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → 𝑥 = (seq𝑀( + , 𝐹)‘𝑁))
78	73, 77	jca 306	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑚)) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → (¬ 𝑁 < 𝑀 ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑁)))
79	78	rexlimdva2 2651	. . . . . . . . 9 ⊢ (𝜑 → (∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)) → (¬ 𝑁 < 𝑀 ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑁))))
80	79	exlimdv 1865	. . . . . . . 8 ⊢ (𝜑 → (∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)) → (¬ 𝑁 < 𝑀 ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑁))))
81	56, 80	impbid 129	. . . . . . 7 ⊢ (𝜑 → ((¬ 𝑁 < 𝑀 ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑁)) ↔ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))))
82	26, 81	orbi12d 798	. . . . . 6 ⊢ (𝜑 → (((𝑁 < 𝑀 ∧ 𝑥 = 0 ) ∨ (¬ 𝑁 < 𝑀 ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑁))) ↔ (((𝑀...𝑁) = ∅ ∧ 𝑥 = 0 ) ∨ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)))))
83	23, 82	bitr2d 189	. . . . 5 ⊢ (𝜑 → ((((𝑀...𝑁) = ∅ ∧ 𝑥 = 0 ) ∨ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) ↔ 𝑥 = if(𝑁 < 𝑀, 0 , (seq𝑀( + , 𝐹)‘𝑁))))
84	83	adantr 276	. . . 4 ⊢ ((𝜑 ∧ if(𝑁 < 𝑀, 0 , (seq𝑀( + , 𝐹)‘𝑁)) ∈ V) → ((((𝑀...𝑁) = ∅ ∧ 𝑥 = 0 ) ∨ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) ↔ 𝑥 = if(𝑁 < 𝑀, 0 , (seq𝑀( + , 𝐹)‘𝑁))))
85	84	iota5 5299	. . 3 ⊢ ((𝜑 ∧ if(𝑁 < 𝑀, 0 , (seq𝑀( + , 𝐹)‘𝑁)) ∈ V) → (℩𝑥(((𝑀...𝑁) = ∅ ∧ 𝑥 = 0 ) ∨ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)))) = if(𝑁 < 𝑀, 0 , (seq𝑀( + , 𝐹)‘𝑁)))
86	19, 85	mpdan 421	. 2 ⊢ (𝜑 → (℩𝑥(((𝑀...𝑁) = ∅ ∧ 𝑥 = 0 ) ∨ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)))) = if(𝑁 < 𝑀, 0 , (seq𝑀( + , 𝐹)‘𝑁)))
87	9, 86	eqtrd 2262	1 ⊢ (𝜑 → (𝐺 Σg 𝐹) = if(𝑁 < 𝑀, 0 , (seq𝑀( + , 𝐹)‘𝑁)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 713  DECID wdc 839   = wceq 1395  ∃wex 1538   ∈ wcel 2200  ∃wrex 2509  Vcvv 2799  ∅c0 3491  ifcif 3602   class class class wbr 4082  ℩cio 5275   Fn wfn 5312  ⟶wf 5313  ‘cfv 5317  (class class class)co 6000  Fincfn 6885  ℝcr 7994   < clt 8177   ≤ cle 8178  ℤcz 9442  ℤ_≥cuz 9718  ...cfz 10200  seqcseq 10664  Basecbs 13027  +_gcplusg 13105  0_gc0g 13284   Σ_g cgsu 13285

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-addcom 8095  ax-addass 8097  ax-distr 8099  ax-i2m1 8100  ax-0lt1 8101  ax-0id 8103  ax-rnegex 8104  ax-cnre 8106  ax-pre-ltirr 8107  ax-pre-ltwlin 8108  ax-pre-lttrn 8109  ax-pre-apti 8110  ax-pre-ltadd 8111

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4383  df-iord 4456  df-on 4458  df-ilim 4459  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-recs 6449  df-frec 6535  df-1o 6560  df-er 6678  df-en 6886  df-fin 6888  df-pnf 8179  df-mnf 8180  df-xr 8181  df-ltxr 8182  df-le 8183  df-sub 8315  df-neg 8316  df-inn 9107  df-n0 9366  df-z 9443  df-uz 9719  df-fz 10201  df-seqfrec 10665  df-ndx 13030  df-slot 13031  df-base 13033  df-0g 13286  df-igsum 13287

This theorem is referenced by:  gsumfzz  13523  gsumfzcl  13527  gsumfzreidx  13869  gsumfzsubmcl  13870  gsumfzmptfidmadd  13871  gsumfzmhm  13875