Theorem gsumval2 18654

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 2736	. . . 4 ⊢ (0g‘𝐺) = (0g‘𝐺)
3		gsumval2.p	. . . 4 ⊢ + = (+g‘𝐺)
4		eqid 2736	. . . 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 7402	. . . 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 6669	. . . . . 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 6502	. . . . 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 584	. . . 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 18651	. . 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 3074	. . . . . . . 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 4016	. . . . . 6 ⊢ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = 𝑦}
19		fvex 6853	. . . . . . . 8 ⊢ (0g‘𝐺) ∈ V
20	19	snid 4606	. . . . . . 7 ⊢ (0g‘𝐺) ∈ {(0g‘𝐺)}
21	8	fdmd 6678	. . . . . . . . . . . . 13 ⊢ (𝜑 → dom 𝐹 = (𝑀...𝑁))
22		gsumval2.n	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
23		eluzfz1 13485	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁))
24		ne0i 4281	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ (𝑀...𝑁) → (𝑀...𝑁) ≠ ∅)
25	22, 23, 24	3syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑀...𝑁) ≠ ∅)
26	21, 25	eqnetrd 2999	. . . . . . . . . . . 12 ⊢ (𝜑 → dom 𝐹 ≠ ∅)
27		dm0rn0 5879	. . . . . . . . . . . . 13 ⊢ (dom 𝐹 = ∅ ↔ ran 𝐹 = ∅)
28	27	necon3bii 2984	. . . . . . . . . . . 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 4344	. . . . . . . . . 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 585	. . . . . . . . 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 2937	. . . . . . . 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 18640	. . . . . . . . . . 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 4769	. . . . . . . . . 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 1005	. . . . . . . 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 592	. . . . . . 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 2843	. . . . . 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 3919	. . . . 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 7374	. . . . . . . . 9 ⊢ (𝑥 = (0g‘𝐺) → (𝑥 + 𝑦) = ((0g‘𝐺) + 𝑦))
43	42	eqeq1d 2738	. . . . . . . 8 ⊢ (𝑥 = (0g‘𝐺) → ((𝑥 + 𝑦) = 𝑦 ↔ ((0g‘𝐺) + 𝑦) = 𝑦))
44	43	ralbidv 3160	. . . . . . 7 ⊢ (𝑥 = (0g‘𝐺) → (∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = 𝑦 ↔ ∀𝑦 ∈ 𝐵 ((0g‘𝐺) + 𝑦) = 𝑦))
45	44	elrab 3634	. . . . . 6 ⊢ ((0g‘𝐺) ∈ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = 𝑦} ↔ ((0g‘𝐺) ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ((0g‘𝐺) + 𝑦) = 𝑦))
46		oveq2 7375	. . . . . . . 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 2752	. . . . . . 7 ⊢ (𝑦 = (0g‘𝐺) → (((0g‘𝐺) + 𝑦) = 𝑦 ↔ ((0g‘𝐺) + (0g‘𝐺)) = (0g‘𝐺)))
49	48	rspcva 3562	. . . . . 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 727	. . . . . 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 7036	. . . . . 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 3922	. . . . 5 ⊢ (((𝜑 ∧ ran 𝐹 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) ∧ 𝑧 ∈ (𝑀...𝑁)) → (𝐹‘𝑧) ∈ {(0g‘𝐺)})
56		elsni 4584	. . . . 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 14008	. . 3 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → (seq𝑀( + , 𝐹)‘𝑁) = (0g‘𝐺))
59	14, 58	eqtr4d 2774	. 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 18653	. 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 813	1 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (seq𝑀( + , 𝐹)‘𝑁))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 848   = wceq 1542   ∈ wcel 2114   ≠ wne 2932  ∀wral 3051  {crab 3389  Vcvv 3429   ⊆ wss 3889  ∅c0 4273  {csn 4567  dom cdm 5631  ran crn 5632   Fn wfn 6493  ⟶wf 6494  ‘cfv 6498  (class class class)co 7367  ℤ_≥cuz 12788  ...cfz 13461  seqcseq 13963  Basecbs 17179  +_gcplusg 17220  0_gc0g 17402   Σ_g cgsu 17403

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-er 8643  df-en 8894  df-dom 8895  df-sdom 8896  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-n0 12438  df-z 12525  df-uz 12789  df-fz 13462  df-seq 13964  df-0g 17404  df-gsum 17405

This theorem is referenced by:  gsumsplit1r  18655  gsumprval  18656  gsumwsubmcl  18805  gsumws1  18806  gsumsgrpccat  18808  gsumwmhm  18813  mulgnngsum  19055  gsumval3  19882  gsummptfzcl  19944  gsumncl  34684  gsumnunsn  34685