Theorem gsumfzval 13095

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 10542	. . 3 ⊢ (𝜑 → (𝑀...𝑁) ∈ Fin)
8		gsumfzval.f	. . 3 ⊢ (𝜑 → 𝐹:(𝑀...𝑁)⟶𝐵)
9	1, 2, 3, 4, 7, 8	igsumval 13094	. 2 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (℩𝑥(((𝑀...𝑁) = ∅ ∧ 𝑥 = 0 ) ∨ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)))))
10		fn0g 13079	. . . . . 6 ⊢ 0g Fn V
11	4	elexd 2776	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ V)
12		funfvex 5578	. . . . . . 7 ⊢ ((Fun 0g ∧ 𝐺 ∈ dom 0g) → (0g‘𝐺) ∈ V)
13	12	funfni 5361	. . . . . 6 ⊢ ((0g Fn V ∧ 𝐺 ∈ V) → (0g‘𝐺) ∈ V)
14	10, 11, 13	sylancr 414	. . . . 5 ⊢ (𝜑 → (0g‘𝐺) ∈ V)
15	2, 14	eqeltrid 2283	. . . 4 ⊢ (𝜑 → 0 ∈ V)
16		seqex 10560	. . . . 5 ⊢ seq𝑀( + , 𝐹) ∈ V
17		fvexg 5580	. . . . 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 4520	. . 3 ⊢ (𝜑 → if(𝑁 < 𝑀, 0 , (seq𝑀( + , 𝐹)‘𝑁)) ∈ V)
20		zdclt 9422	. . . . . . . 8 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → DECID 𝑁 < 𝑀)
21	6, 5, 20	syl2anc 411	. . . . . . 7 ⊢ (𝜑 → DECID 𝑁 < 𝑀)
22		eqifdc 3597	. . . . . . 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 10136	. . . . . . . . 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 9467	. . . . . . . . . . . . 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 9467	. . . . . . . . . . . . 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 8166	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (¬ 𝑁 < 𝑀 ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑁))) → 𝑀 ≤ 𝑁)
33		eluz 9633	. . . . . . . . . . . . 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 5933	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑁 → (𝑀...𝑛) = (𝑀...𝑁))
37	36	eqeq2d 2208	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑁 → ((𝑀...𝑁) = (𝑀...𝑛) ↔ (𝑀...𝑁) = (𝑀...𝑁)))
38		fveq2 5561	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑁 → (seq𝑀( + , 𝐹)‘𝑛) = (seq𝑀( + , 𝐹)‘𝑁))
39	38	eqeq2d 2208	. . . . . . . . . . . . 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 2197	. . . . . . . . . . . 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 2874	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (¬ 𝑁 < 𝑀 ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑁))) → ∃𝑛 ∈ (ℤ≥‘𝑀)((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑛)))
46		fveq2 5561	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑀 → (ℤ≥‘𝑚) = (ℤ≥‘𝑀))
47		oveq1 5932	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑀 → (𝑚...𝑛) = (𝑀...𝑛))
48	47	eqeq2d 2208	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑀 → ((𝑀...𝑁) = (𝑚...𝑛) ↔ (𝑀...𝑁) = (𝑀...𝑛)))
49		seqeq1 10561	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 𝑀 → seq𝑚( + , 𝐹) = seq𝑀( + , 𝐹))
50	49	fveq1d 5563	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑀 → (seq𝑚( + , 𝐹)‘𝑛) = (seq𝑀( + , 𝐹)‘𝑛))
51	50	eqeq2d 2208	. . . . . . . . . . . . 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 2710	. . . . . . . . . . 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 2852	. . . . . . . . . 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 9625	. . . . . . . . . . . . . . 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 9467	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑚)) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → 𝑚 ∈ ℝ)
60		eluzelre 9630	. . . . . . . . . . . . . 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 9632	. . . . . . . . . . . . . 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 8167	. . . . . . . . . . . 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 2202	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑚)) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → (𝑚...𝑛) = (𝑀...𝑁))
67		fzopth 10155	. . . . . . . . . . . . . . . 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 4047	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑚)) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → (𝑛 < 𝑚 ↔ 𝑁 < 𝑀))
73	64, 72	mtbid 673	. . . . . . . . . . 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 10564	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑚)) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → seq𝑚( + , 𝐹) = seq𝑀( + , 𝐹))
76	75, 70	fveq12d 5568	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑚)) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → (seq𝑚( + , 𝐹)‘𝑛) = (seq𝑀( + , 𝐹)‘𝑁))
77	74, 76	eqtrd 2229	. . . . . . . . . . 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 2617	. . . . . . . . 9 ⊢ (𝜑 → (∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)) → (¬ 𝑁 < 𝑀 ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑁))))
80	79	exlimdv 1833	. . . . . . . 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 794	. . . . . 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 5241	. . 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 2229	1 ⊢ (𝜑 → (𝐺 Σg 𝐹) = if(𝑁 < 𝑀, 0 , (seq𝑀( + , 𝐹)‘𝑁)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 709  DECID wdc 835   = wceq 1364  ∃wex 1506   ∈ wcel 2167  ∃wrex 2476  Vcvv 2763  ∅c0 3451  ifcif 3562   class class class wbr 4034  ℩cio 5218   Fn wfn 5254  ⟶wf 5255  ‘cfv 5259  (class class class)co 5925  Fincfn 6808  ℝcr 7897   < clt 8080   ≤ cle 8081  ℤcz 9345  ℤ_≥cuz 9620  ...cfz 10102  seqcseq 10558  Basecbs 12705  +_gcplusg 12782  0_gc0g 12960   Σ_g cgsu 12961

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7989  ax-resscn 7990  ax-1cn 7991  ax-1re 7992  ax-icn 7993  ax-addcl 7994  ax-addrcl 7995  ax-mulcl 7996  ax-addcom 7998  ax-addass 8000  ax-distr 8002  ax-i2m1 8003  ax-0lt1 8004  ax-0id 8006  ax-rnegex 8007  ax-cnre 8009  ax-pre-ltirr 8010  ax-pre-ltwlin 8011  ax-pre-lttrn 8012  ax-pre-apti 8013  ax-pre-ltadd 8014

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-frec 6458  df-1o 6483  df-er 6601  df-en 6809  df-fin 6811  df-pnf 8082  df-mnf 8083  df-xr 8084  df-ltxr 8085  df-le 8086  df-sub 8218  df-neg 8219  df-inn 9010  df-n0 9269  df-z 9346  df-uz 9621  df-fz 10103  df-seqfrec 10559  df-ndx 12708  df-slot 12709  df-base 12711  df-0g 12962  df-igsum 12963

This theorem is referenced by:  gsumfzz  13199  gsumfzcl  13203  gsumfzreidx  13545  gsumfzsubmcl  13546  gsumfzmptfidmadd  13547  gsumfzmhm  13551