Theorem gsumfzval 13654

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 10817	. . 3 ⊢ (𝜑 → (𝑀...𝑁) ∈ Fin)
8		gsumfzval.f	. . 3 ⊢ (𝜑 → 𝐹:(𝑀...𝑁)⟶𝐵)
9	1, 2, 3, 4, 7, 8	igsumval 13653	. 2 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (℩𝑥(((𝑀...𝑁) = ∅ ∧ 𝑥 = 0 ) ∨ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)))))
10		fn0g 13638	. . . . . 6 ⊢ 0g Fn V
11	4	elexd 2829	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ V)
12		funfvex 5692	. . . . . . 7 ⊢ ((Fun 0g ∧ 𝐺 ∈ dom 0g) → (0g‘𝐺) ∈ V)
13	12	funfni 5463	. . . . . 6 ⊢ ((0g Fn V ∧ 𝐺 ∈ V) → (0g‘𝐺) ∈ V)
14	10, 11, 13	sylancr 414	. . . . 5 ⊢ (𝜑 → (0g‘𝐺) ∈ V)
15	2, 14	eqeltrid 2321	. . . 4 ⊢ (𝜑 → 0 ∈ V)
16		seqex 10835	. . . . 5 ⊢ seq𝑀( + , 𝐹) ∈ V
17		fvexg 5694	. . . . 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 4610	. . 3 ⊢ (𝜑 → if(𝑁 < 𝑀, 0 , (seq𝑀( + , 𝐹)‘𝑁)) ∈ V)
20		zdclt 9672	. . . . . . . 8 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → DECID 𝑁 < 𝑀)
21	6, 5, 20	syl2anc 411	. . . . . . 7 ⊢ (𝜑 → DECID 𝑁 < 𝑀)
22		eqifdc 3663	. . . . . . 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 10396	. . . . . . . . 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 9718	. . . . . . . . . . . . 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 9718	. . . . . . . . . . . . 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 8410	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (¬ 𝑁 < 𝑀 ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑁))) → 𝑀 ≤ 𝑁)
33		eluz 9885	. . . . . . . . . . . . 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 6066	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑁 → (𝑀...𝑛) = (𝑀...𝑁))
37	36	eqeq2d 2246	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑁 → ((𝑀...𝑁) = (𝑀...𝑛) ↔ (𝑀...𝑁) = (𝑀...𝑁)))
38		fveq2 5675	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑁 → (seq𝑀( + , 𝐹)‘𝑛) = (seq𝑀( + , 𝐹)‘𝑁))
39	38	eqeq2d 2246	. . . . . . . . . . . . 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 2235	. . . . . . . . . . . 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 2929	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (¬ 𝑁 < 𝑀 ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑁))) → ∃𝑛 ∈ (ℤ≥‘𝑀)((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑛)))
46		fveq2 5675	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑀 → (ℤ≥‘𝑚) = (ℤ≥‘𝑀))
47		oveq1 6065	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑀 → (𝑚...𝑛) = (𝑀...𝑛))
48	47	eqeq2d 2246	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑀 → ((𝑀...𝑁) = (𝑚...𝑛) ↔ (𝑀...𝑁) = (𝑀...𝑛)))
49		seqeq1 10836	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 𝑀 → seq𝑚( + , 𝐹) = seq𝑀( + , 𝐹))
50	49	fveq1d 5677	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑀 → (seq𝑚( + , 𝐹)‘𝑛) = (seq𝑀( + , 𝐹)‘𝑛))
51	50	eqeq2d 2246	. . . . . . . . . . . . 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 2760	. . . . . . . . . . 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 2907	. . . . . . . . . 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 9876	. . . . . . . . . . . . . . 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 9718	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑚)) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → 𝑚 ∈ ℝ)
60		eluzelre 9882	. . . . . . . . . . . . . 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 9884	. . . . . . . . . . . . . 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 8411	. . . . . . . . . . . 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 2240	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑚)) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → (𝑚...𝑛) = (𝑀...𝑁))
67		fzopth 10416	. . . . . . . . . . . . . . . 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 4127	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑚)) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → (𝑛 < 𝑚 ↔ 𝑁 < 𝑀))
73	64, 72	mtbid 679	. . . . . . . . . . 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 10839	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑚)) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → seq𝑚( + , 𝐹) = seq𝑀( + , 𝐹))
76	75, 70	fveq12d 5682	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑚)) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → (seq𝑚( + , 𝐹)‘𝑛) = (seq𝑀( + , 𝐹)‘𝑁))
77	74, 76	eqtrd 2267	. . . . . . . . . . 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 2665	. . . . . . . . 9 ⊢ (𝜑 → (∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)) → (¬ 𝑁 < 𝑀 ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑁))))
80	79	exlimdv 1868	. . . . . . . 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 801	. . . . . 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 5339	. . 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 2267	1 ⊢ (𝜑 → (𝐺 Σg 𝐹) = if(𝑁 < 𝑀, 0 , (seq𝑀( + , 𝐹)‘𝑁)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 716  DECID wdc 842   = wceq 1398  ∃wex 1541   ∈ wcel 2205  ∃wrex 2523  Vcvv 2815  ∅c0 3512  ifcif 3624   class class class wbr 4114  ℩cio 5315   Fn wfn 5352  ⟶wf 5353  ‘cfv 5357  (class class class)co 6058  Fincfn 6988  ℝcr 8142   < clt 8324   ≤ cle 8325  ℤcz 9594  ℤ_≥cuz 9871  ...cfz 10361  seqcseq 10833  Basecbs 13296  +_gcplusg 13374  0_gc0g 13553   Σ_g cgsu 13554

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-1o 6660  df-er 6780  df-en 6989  df-fin 6991  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-inn 9255  df-n0 9514  df-z 9595  df-uz 9872  df-fz 10362  df-seqfrec 10834  df-ndx 13299  df-slot 13300  df-base 13302  df-0g 13555  df-igsum 13556

This theorem is referenced by:  gsumfzz  13750  gsumfzcl  13754  gsumfzreidx  14090  gsumfzsubmcl  14091  gsumfzmptfidmadd  14092  gsumfzmhm  14096