Theorem gsumval2 18724

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 2740	. . . 4 ⊢ (0g‘𝐺) = (0g‘𝐺)
3		gsumval2.p	. . . 4 ⊢ + = (+g‘𝐺)
4		eqid 2740	. . . 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 480	. . . 4 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → 𝐺 ∈ 𝑉)
7		ovexd 7483	. . . 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 6748	. . . . . 6 ⊢ (𝜑 → 𝐹 Fn (𝑀...𝑁))
10	9	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → 𝐹 Fn (𝑀...𝑁))
11		simpr 484	. . . . 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 6577	. . . . 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 582	. . . 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 18721	. . 3 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → (𝐺 Σg 𝐹) = (0g‘𝐺))
15		simpl 482	. . . . . . . . 9 ⊢ (((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦) → (𝑥 + 𝑦) = 𝑦)
16	15	ralimi 3089	. . . . . . . 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 4100	. . . . . 6 ⊢ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = 𝑦}
19		fvex 6933	. . . . . . . 8 ⊢ (0g‘𝐺) ∈ V
20	19	snid 4684	. . . . . . 7 ⊢ (0g‘𝐺) ∈ {(0g‘𝐺)}
21	8	fdmd 6757	. . . . . . . . . . . . 13 ⊢ (𝜑 → dom 𝐹 = (𝑀...𝑁))
22		gsumval2.n	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
23		eluzfz1 13591	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁))
24		ne0i 4364	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ (𝑀...𝑁) → (𝑀...𝑁) ≠ ∅)
25	22, 23, 24	3syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑀...𝑁) ≠ ∅)
26	21, 25	eqnetrd 3014	. . . . . . . . . . . 12 ⊢ (𝜑 → dom 𝐹 ≠ ∅)
27		dm0rn0 5949	. . . . . . . . . . . . 13 ⊢ (dom 𝐹 = ∅ ↔ ran 𝐹 = ∅)
28	27	necon3bii 2999	. . . . . . . . . . . 12 ⊢ (dom 𝐹 ≠ ∅ ↔ ran 𝐹 ≠ ∅)
29	26, 28	sylib 218	. . . . . . . . . . 11 ⊢ (𝜑 → ran 𝐹 ≠ ∅)
30	29	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → ran 𝐹 ≠ ∅)
31		ssn0 4427	. . . . . . . . . 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 583	. . . . . . . . 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 2951	. . . . . . . 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 18710	. . . . . . . . . . 11 ⊢ (𝐺 ∈ 𝑉 → {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} ⊆ {(0g‘𝐺)})
35	5, 34	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} ⊆ {(0g‘𝐺)})
36		sssn 4851	. . . . . . . . . 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 218	. . . . . . . . 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 1003	. . . . . . . 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 590	. . . . . . 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 2851	. . . . . 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 4006	. . . . 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 7455	. . . . . . . . 9 ⊢ (𝑥 = (0g‘𝐺) → (𝑥 + 𝑦) = ((0g‘𝐺) + 𝑦))
43	42	eqeq1d 2742	. . . . . . . 8 ⊢ (𝑥 = (0g‘𝐺) → ((𝑥 + 𝑦) = 𝑦 ↔ ((0g‘𝐺) + 𝑦) = 𝑦))
44	43	ralbidv 3184	. . . . . . 7 ⊢ (𝑥 = (0g‘𝐺) → (∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = 𝑦 ↔ ∀𝑦 ∈ 𝐵 ((0g‘𝐺) + 𝑦) = 𝑦))
45	44	elrab 3708	. . . . . 6 ⊢ ((0g‘𝐺) ∈ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = 𝑦} ↔ ((0g‘𝐺) ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ((0g‘𝐺) + 𝑦) = 𝑦))
46		oveq2 7456	. . . . . . . 8 ⊢ (𝑦 = (0g‘𝐺) → ((0g‘𝐺) + 𝑦) = ((0g‘𝐺) + (0g‘𝐺)))
47		id 22	. . . . . . . 8 ⊢ (𝑦 = (0g‘𝐺) → 𝑦 = (0g‘𝐺))
48	46, 47	eqeq12d 2756	. . . . . . 7 ⊢ (𝑦 = (0g‘𝐺) → (((0g‘𝐺) + 𝑦) = 𝑦 ↔ ((0g‘𝐺) + (0g‘𝐺)) = (0g‘𝐺)))
49	48	rspcva 3633	. . . . . 6 ⊢ (((0g‘𝐺) ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ((0g‘𝐺) + 𝑦) = 𝑦) → ((0g‘𝐺) + (0g‘𝐺)) = (0g‘𝐺))
50	45, 49	sylbi 217	. . . . 5 ⊢ ((0g‘𝐺) ∈ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = 𝑦} → ((0g‘𝐺) + (0g‘𝐺)) = (0g‘𝐺))
51	41, 50	syl 17	. . . 4 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → ((0g‘𝐺) + (0g‘𝐺)) = (0g‘𝐺))
52	22	adantr 480	. . . 4 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → 𝑁 ∈ (ℤ≥‘𝑀))
53	35	ad2antrr 725	. . . . . 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 7118	. . . . . 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 4009	. . . . 5 ⊢ (((𝜑 ∧ ran 𝐹 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) ∧ 𝑧 ∈ (𝑀...𝑁)) → (𝐹‘𝑧) ∈ {(0g‘𝐺)})
56		elsni 4665	. . . . 5 ⊢ ((𝐹‘𝑧) ∈ {(0g‘𝐺)} → (𝐹‘𝑧) = (0g‘𝐺))
57	55, 56	syl 17	. . . 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 14097	. . 3 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → (seq𝑀( + , 𝐹)‘𝑁) = (0g‘𝐺))
59	14, 58	eqtr4d 2783	. 2 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → (𝐺 Σg 𝐹) = (seq𝑀( + , 𝐹)‘𝑁))
60	5	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → 𝐺 ∈ 𝑉)
61	22	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → 𝑁 ∈ (ℤ≥‘𝑀))
62	8	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → 𝐹:(𝑀...𝑁)⟶𝐵)
63		simpr 484	. . 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 18723	. 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 812	1 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (seq𝑀( + , 𝐹)‘𝑁))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 846   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  ∀wral 3067  {crab 3443  Vcvv 3488   ⊆ wss 3976  ∅c0 4352  {csn 4648  dom cdm 5700  ran crn 5701   Fn wfn 6568  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448  ℤ_≥cuz 12903  ...cfz 13567  seqcseq 14052  Basecbs 17258  +_gcplusg 17311  0_gc0g 17499   Σ_g cgsu 17500

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-er 8763  df-en 9004  df-dom 9005  df-sdom 9006  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-n0 12554  df-z 12640  df-uz 12904  df-fz 13568  df-seq 14053  df-0g 17501  df-gsum 17502

This theorem is referenced by:  gsumsplit1r  18725  gsumprval  18726  gsumwsubmcl  18872  gsumws1  18873  gsumsgrpccat  18875  gsumwmhm  18880  mulgnngsum  19119  gsumval3  19949  gsummptfzcl  20011  gsumncl  34517  gsumnunsn  34518