Theorem gsumval2 18790

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)

Assertion

Ref	Expression
gsumval2	⊢ (𝜑 → (𝐺 Σg 𝐹) = (seq𝑀( + , 𝐹)‘𝑁))

Proof of Theorem gsumval2

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		gsumval2.b	. . . 4 ⊢ 𝐵 = (Base‘𝐺)
2		eqid 2762	. . . 4 ⊢ (0g‘𝐺) = (0g‘𝐺)
3		gsumval2.p	. . . 4 ⊢ + = (+g‘𝐺)
4		eqid 2762	. . . 4 ⊢ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}
5		gsumval2.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ 𝑉)
6	5	adantr 486	. . . 4 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → 𝐺 ∈ 𝑉)
7		ovexd 7451	. . . 4 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → (𝑀...𝑁) ∈ V)
8		gsumval2.f	. . . . . . 7 ⊢ (𝜑 → 𝐹:(𝑀...𝑁)⟶𝐵)
9	8	ffnd 6707	. . . . . 6 ⊢ (𝜑 → 𝐹 Fn (𝑀...𝑁))
10	9	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → 𝐹 Fn (𝑀...𝑁))
11		simpr 490	. . . . 5 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → ran 𝐹 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)})
12		df-f 6541	. . . . 5 ⊢ (𝐹:(𝑀...𝑁)⟶{𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} ↔ (𝐹 Fn (𝑀...𝑁) ∧ ran 𝐹 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}))
13	10, 11, 12	sylanbrc 595	. . . 4 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → 𝐹:(𝑀...𝑁)⟶{𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)})
14	1, 2, 3, 4, 6, 7, 13	gsumval1 18787	. . 3 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → (𝐺 Σg 𝐹) = (0g‘𝐺))
15		simpl 488	. . . . . . . . 9 ⊢ (((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦) → (𝑥 + 𝑦) = 𝑦)
16	15	ralimi 3101	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦) → ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = 𝑦)
17	16	a1i 11	. . . . . . 7 ⊢ (𝑥 ∈ 𝐵 → (∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦) → ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = 𝑦))
18	17	ss2rabi 4027	. . . . . 6 ⊢ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = 𝑦}
19		fvex 6895	. . . . . . . 8 ⊢ (0g‘𝐺) ∈ V
20	19	snid 4626	. . . . . . 7 ⊢ (0g‘𝐺) ∈ {(0g‘𝐺)}
21	8	fdmd 6717	. . . . . . . . . . . . 13 ⊢ (𝜑 → dom 𝐹 = (𝑀...𝑁))
22		gsumval2.n	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
23		eluzfz1 13587	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁))
24		ne0i 4290	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ (𝑀...𝑁) → (𝑀...𝑁) ≠ ∅)
25	22, 23, 24	3syl 19	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑀...𝑁) ≠ ∅)
26	21, 25	eqnetrd 3024	. . . . . . . . . . . 12 ⊢ (𝜑 → dom 𝐹 ≠ ∅)
27		dm0rn0 5912	. . . . . . . . . . . . 13 ⊢ (dom 𝐹 = ∅ ↔ ran 𝐹 = ∅)
28	27	necon3bii 3009	. . . . . . . . . . . 12 ⊢ (dom 𝐹 ≠ ∅ ↔ ran 𝐹 ≠ ∅)
29	26, 28	sylib 221	. . . . . . . . . . 11 ⊢ (𝜑 → ran 𝐹 ≠ ∅)
30	29	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → ran 𝐹 ≠ ∅)
31		ssn0 4358	. . . . . . . . . 10 ⊢ ((ran 𝐹 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} ∧ ran 𝐹 ≠ ∅) → {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} ≠ ∅)
32	11, 30, 31	syl2anc 596	. . . . . . . . 9 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} ≠ ∅)
33	32	neneqd 2962	. . . . . . . 8 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → ¬ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} = ∅)
34	1, 2, 3, 4	mgmidsssn0 18770	. . . . . . . . . . 11 ⊢ (𝐺 ∈ 𝑉 → {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} ⊆ {(0g‘𝐺)})
35	5, 34	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} ⊆ {(0g‘𝐺)})
36		sssn 4790	. . . . . . . . . 10 ⊢ ({𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} ⊆ {(0g‘𝐺)} ↔ ({𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} = ∅ ∨ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} = {(0g‘𝐺)}))
37	35, 36	sylib 221	. . . . . . . . 9 ⊢ (𝜑 → ({𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} = ∅ ∨ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} = {(0g‘𝐺)}))
38	37	orcanai 1018	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} = ∅) → {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} = {(0g‘𝐺)})
39	33, 38	syldan 603	. . . . . . 7 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} = {(0g‘𝐺)})
40	20, 39	eleqtrrid 2869	. . . . . 6 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → (0g‘𝐺) ∈ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)})
41	18, 40	sselid 3932	. . . . 5 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → (0g‘𝐺) ∈ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = 𝑦})
42		oveq1 7423	. . . . . . . . 9 ⊢ (𝑥 = (0g‘𝐺) → (𝑥 + 𝑦) = ((0g‘𝐺) + 𝑦))
43	42	eqeq1d 2764	. . . . . . . 8 ⊢ (𝑥 = (0g‘𝐺) → ((𝑥 + 𝑦) = 𝑦 ↔ ((0g‘𝐺) + 𝑦) = 𝑦))
44	43	ralbidv 3187	. . . . . . 7 ⊢ (𝑥 = (0g‘𝐺) → (∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = 𝑦 ↔ ∀𝑦 ∈ 𝐵 ((0g‘𝐺) + 𝑦) = 𝑦))
45	44	elrab 3648	. . . . . 6 ⊢ ((0g‘𝐺) ∈ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = 𝑦} ↔ ((0g‘𝐺) ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ((0g‘𝐺) + 𝑦) = 𝑦))
46		oveq2 7424	. . . . . . . 8 ⊢ (𝑦 = (0g‘𝐺) → ((0g‘𝐺) + 𝑦) = ((0g‘𝐺) + (0g‘𝐺)))
47		id 23	. . . . . . . 8 ⊢ (𝑦 = (0g‘𝐺) → 𝑦 = (0g‘𝐺))
48	46, 47	eqeq12d 2778	. . . . . . 7 ⊢ (𝑦 = (0g‘𝐺) → (((0g‘𝐺) + 𝑦) = 𝑦 ↔ ((0g‘𝐺) + (0g‘𝐺)) = (0g‘𝐺)))
49	48	rspcva 3577	. . . . . 6 ⊢ (((0g‘𝐺) ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ((0g‘𝐺) + 𝑦) = 𝑦) → ((0g‘𝐺) + (0g‘𝐺)) = (0g‘𝐺))
50	45, 49	sylbi 220	. . . . 5 ⊢ ((0g‘𝐺) ∈ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = 𝑦} → ((0g‘𝐺) + (0g‘𝐺)) = (0g‘𝐺))
51	41, 50	syl 18	. . . 4 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → ((0g‘𝐺) + (0g‘𝐺)) = (0g‘𝐺))
52	22	adantr 486	. . . 4 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → 𝑁 ∈ (ℤ≥‘𝑀))
53	35	ad2antrr 739	. . . . . 6 ⊢ (((𝜑 ∧ ran 𝐹 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) ∧ 𝑧 ∈ (𝑀...𝑁)) → {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} ⊆ {(0g‘𝐺)})
54	13	ffvelcdmda 7080	. . . . . 6 ⊢ (((𝜑 ∧ ran 𝐹 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) ∧ 𝑧 ∈ (𝑀...𝑁)) → (𝐹‘𝑧) ∈ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)})
55	53, 54	sseldd 3935	. . . . 5 ⊢ (((𝜑 ∧ ran 𝐹 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) ∧ 𝑧 ∈ (𝑀...𝑁)) → (𝐹‘𝑧) ∈ {(0g‘𝐺)})
56		elsni 4604	. . . . 5 ⊢ ((𝐹‘𝑧) ∈ {(0g‘𝐺)} → (𝐹‘𝑧) = (0g‘𝐺))
57	55, 56	syl 18	. . . 4 ⊢ (((𝜑 ∧ ran 𝐹 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) ∧ 𝑧 ∈ (𝑀...𝑁)) → (𝐹‘𝑧) = (0g‘𝐺))
58	51, 52, 57	seqid3 14112	. . 3 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → (seq𝑀( + , 𝐹)‘𝑁) = (0g‘𝐺))
59	14, 58	eqtr4d 2800	. 2 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → (𝐺 Σg 𝐹) = (seq𝑀( + , 𝐹)‘𝑁))
60	5	adantr 486	. . 3 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → 𝐺 ∈ 𝑉)
61	22	adantr 486	. . 3 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → 𝑁 ∈ (ℤ≥‘𝑀))
62	8	adantr 486	. . 3 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → 𝐹:(𝑀...𝑁)⟶𝐵)
63		simpr 490	. . 3 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → ¬ ran 𝐹 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)})
64	1, 3, 60, 61, 62, 4, 63	gsumval2a 18789	. 2 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → (𝐺 Σg 𝐹) = (seq𝑀( + , 𝐹)‘𝑁))
65	59, 64	pm2.61dan 825	1 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (seq𝑀( + , 𝐹)‘𝑁))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2145   ≠ wne 2957  ∀wral 3078  {crab 3414  Vcvv 3453   ⊆ wss 3902  ∅c0 4282  {csn 4587  dom cdm 5659  ran crn 5660   Fn wfn 6532  ⟶wf 6533  ‘cfv 6537  (class class class)co 7416  ℤ_≥cuz 12890  ...cfz 13563  seqcseq 14067  Basecbs 17305  +_gcplusg 17346  0_gc0g 17528   Σ_g cgsu 17529

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739  ax-cnex 11183  ax-resscn 11184  ax-1cn 11185  ax-icn 11186  ax-addcl 11187  ax-addrcl 11188  ax-mulcl 11189  ax-mulrcl 11190  ax-mulcom 11191  ax-addass 11192  ax-mulass 11193  ax-distr 11194  ax-i2m1 11195  ax-1ne0 11196  ax-1rid 11197  ax-rnegex 11198  ax-rrecex 11199  ax-cnre 11200  ax-pre-lttri 11201  ax-pre-lttrn 11202  ax-pre-ltadd 11203  ax-pre-mulgt0 11204

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7866  df-1st 7989  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-er 8699  df-en 8956  df-dom 8957  df-sdom 8958  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11470  df-neg 11471  df-nn 12261  df-n0 12532  df-z 12619  df-uz 12891  df-fz 13564  df-seq 14068  df-0g 17530  df-gsum 17531

This theorem is used by:  gsumsplit1r  18791  gsumprval  18792  gsumwsubmcl  18947  gsumws1  18948  gsumsgrpccat  18950  gsumwmhm  18955  mulgnngsum  19203  gsumval3  20035  gsummptfzcl  20097  gsumncl  35038  gsumnunsn  35039  veroquadgsumlem  50803