Theorem gsumfzval 13497

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 10699	. . 3 ⊢ (𝜑 → (𝑀...𝑁) ∈ Fin)
8		gsumfzval.f	. . 3 ⊢ (𝜑 → 𝐹:(𝑀...𝑁)⟶𝐵)
9	1, 2, 3, 4, 7, 8	igsumval 13496	. 2 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (℩𝑥(((𝑀...𝑁) = ∅ ∧ 𝑥 = 0 ) ∨ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)))))
10		fn0g 13481	. . . . . 6 ⊢ 0g Fn V
11	4	elexd 2815	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ V)
12		funfvex 5659	. . . . . . 7 ⊢ ((Fun 0g ∧ 𝐺 ∈ dom 0g) → (0g‘𝐺) ∈ V)
13	12	funfni 5434	. . . . . 6 ⊢ ((0g Fn V ∧ 𝐺 ∈ V) → (0g‘𝐺) ∈ V)
14	10, 11, 13	sylancr 414	. . . . 5 ⊢ (𝜑 → (0g‘𝐺) ∈ V)
15	2, 14	eqeltrid 2317	. . . 4 ⊢ (𝜑 → 0 ∈ V)
16		seqex 10717	. . . . 5 ⊢ seq𝑀( + , 𝐹) ∈ V
17		fvexg 5661	. . . . 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 4583	. . 3 ⊢ (𝜑 → if(𝑁 < 𝑀, 0 , (seq𝑀( + , 𝐹)‘𝑁)) ∈ V)
20		zdclt 9562	. . . . . . . 8 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → DECID 𝑁 < 𝑀)
21	6, 5, 20	syl2anc 411	. . . . . . 7 ⊢ (𝜑 → DECID 𝑁 < 𝑀)
22		eqifdc 3643	. . . . . . 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 10282	. . . . . . . . 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 9607	. . . . . . . . . . . . 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 9607	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (¬ 𝑁 < 𝑀 ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑁))) → 𝑁 ∈ ℝ)
31		simprl 531	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (¬ 𝑁 < 𝑀 ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑁))) → ¬ 𝑁 < 𝑀)
32	28, 30, 31	nltled 8305	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (¬ 𝑁 < 𝑀 ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑁))) → 𝑀 ≤ 𝑁)
33		eluz 9774	. . . . . . . . . . . . 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 6031	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑁 → (𝑀...𝑛) = (𝑀...𝑁))
37	36	eqeq2d 2242	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑁 → ((𝑀...𝑁) = (𝑀...𝑛) ↔ (𝑀...𝑁) = (𝑀...𝑁)))
38		fveq2 5642	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑁 → (seq𝑀( + , 𝐹)‘𝑛) = (seq𝑀( + , 𝐹)‘𝑁))
39	38	eqeq2d 2242	. . . . . . . . . . . . 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 2231	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (¬ 𝑁 < 𝑀 ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑁))) → (𝑀...𝑁) = (𝑀...𝑁))
43		simprr 533	. . . . . . . . . . . 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 2915	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (¬ 𝑁 < 𝑀 ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑁))) → ∃𝑛 ∈ (ℤ≥‘𝑀)((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑛)))
46		fveq2 5642	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑀 → (ℤ≥‘𝑚) = (ℤ≥‘𝑀))
47		oveq1 6030	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑀 → (𝑚...𝑛) = (𝑀...𝑛))
48	47	eqeq2d 2242	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑀 → ((𝑀...𝑁) = (𝑚...𝑛) ↔ (𝑀...𝑁) = (𝑀...𝑛)))
49		seqeq1 10718	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 𝑀 → seq𝑚( + , 𝐹) = seq𝑀( + , 𝐹))
50	49	fveq1d 5644	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑀 → (seq𝑚( + , 𝐹)‘𝑛) = (seq𝑀( + , 𝐹)‘𝑛))
51	50	eqeq2d 2242	. . . . . . . . . . . . 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 2746	. . . . . . . . . . 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 2893	. . . . . . . . . 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 9765	. . . . . . . . . . . . . . 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 9607	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑚)) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → 𝑚 ∈ ℝ)
60		eluzelre 9771	. . . . . . . . . . . . . 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 9773	. . . . . . . . . . . . . 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 8306	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑚)) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → ¬ 𝑛 < 𝑚)
65		simprl 531	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑚)) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → (𝑀...𝑁) = (𝑚...𝑛))
66	65	eqcomd 2236	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑚)) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → (𝑚...𝑛) = (𝑀...𝑁))
67		fzopth 10301	. . . . . . . . . . . . . . . 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 4102	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑚)) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → (𝑛 < 𝑚 ↔ 𝑁 < 𝑀))
73	64, 72	mtbid 678	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑚)) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → ¬ 𝑁 < 𝑀)
74		simprr 533	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑚)) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))
75	71	seqeq1d 10721	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑚)) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → seq𝑚( + , 𝐹) = seq𝑀( + , 𝐹))
76	75, 70	fveq12d 5649	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑚)) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → (seq𝑚( + , 𝐹)‘𝑛) = (seq𝑀( + , 𝐹)‘𝑁))
77	74, 76	eqtrd 2263	. . . . . . . . . . 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 2652	. . . . . . . . 9 ⊢ (𝜑 → (∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)) → (¬ 𝑁 < 𝑀 ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑁))))
80	79	exlimdv 1866	. . . . . . . 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 800	. . . . . 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 5310	. . 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 2263	1 ⊢ (𝜑 → (𝐺 Σg 𝐹) = if(𝑁 < 𝑀, 0 , (seq𝑀( + , 𝐹)‘𝑁)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 715  DECID wdc 841   = wceq 1397  ∃wex 1540   ∈ wcel 2201  ∃wrex 2510  Vcvv 2801  ∅c0 3493  ifcif 3604   class class class wbr 4089  ℩cio 5286   Fn wfn 5323  ⟶wf 5324  ‘cfv 5328  (class class class)co 6023  Fincfn 6914  ℝcr 8036   < clt 8219   ≤ cle 8220  ℤcz 9484  ℤ_≥cuz 9760  ...cfz 10248  seqcseq 10715  Basecbs 13105  +_gcplusg 13183  0_gc0g 13362   Σ_g cgsu 13363

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-coll 4205  ax-sep 4208  ax-nul 4216  ax-pow 4266  ax-pr 4301  ax-un 4532  ax-setind 4637  ax-iinf 4688  ax-cnex 8128  ax-resscn 8129  ax-1cn 8130  ax-1re 8131  ax-icn 8132  ax-addcl 8133  ax-addrcl 8134  ax-mulcl 8135  ax-addcom 8137  ax-addass 8139  ax-distr 8141  ax-i2m1 8142  ax-0lt1 8143  ax-0id 8145  ax-rnegex 8146  ax-cnre 8148  ax-pre-ltirr 8149  ax-pre-ltwlin 8150  ax-pre-lttrn 8151  ax-pre-apti 8152  ax-pre-ltadd 8153

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-nel 2497  df-ral 2514  df-rex 2515  df-reu 2516  df-rab 2518  df-v 2803  df-sbc 3031  df-csb 3127  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-nul 3494  df-if 3605  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-int 3930  df-iun 3973  df-br 4090  df-opab 4152  df-mpt 4153  df-tr 4189  df-id 4392  df-iord 4465  df-on 4467  df-ilim 4468  df-suc 4470  df-iom 4691  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-res 4739  df-ima 4740  df-iota 5288  df-fun 5330  df-fn 5331  df-f 5332  df-f1 5333  df-fo 5334  df-f1o 5335  df-fv 5336  df-riota 5976  df-ov 6026  df-oprab 6027  df-mpo 6028  df-1st 6308  df-2nd 6309  df-recs 6476  df-frec 6562  df-1o 6587  df-er 6707  df-en 6915  df-fin 6917  df-pnf 8221  df-mnf 8222  df-xr 8223  df-ltxr 8224  df-le 8225  df-sub 8357  df-neg 8358  df-inn 9149  df-n0 9408  df-z 9485  df-uz 9761  df-fz 10249  df-seqfrec 10716  df-ndx 13108  df-slot 13109  df-base 13111  df-0g 13364  df-igsum 13365

This theorem is referenced by:  gsumfzz  13601  gsumfzcl  13605  gsumfzreidx  13947  gsumfzsubmcl  13948  gsumfzmptfidmadd  13949  gsumfzmhm  13953