Theorem gsumval2 18925

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 2763	. . . 4 ⊢ (0g‘𝐺) = (0g‘𝐺)
3		gsumval2.p	. . . 4 ⊢ + = (+g‘𝐺)
4		eqid 2763	. . . 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 7486	. . . 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 6740	. . . . . 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 6570	. . . . 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 18922	. . 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 3103	. . . . . . . 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 4004	. . . . . 6 ⊢ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = 𝑦}
19		fvex 6929	. . . . . . . 8 ⊢ (0g‘𝐺) ∈ V
20	19	snid 4624	. . . . . . 7 ⊢ (0g‘𝐺) ∈ {(0g‘𝐺)}
21	8	fdmd 6750	. . . . . . . . . . . . 13 ⊢ (𝜑 → dom 𝐹 = (𝑀...𝑁))
22		gsumval2.n	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
23		eluzfz1 13704	. . . . . . . . . . . . . 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 3026	. . . . . . . . . . . 12 ⊢ (𝜑 → dom 𝐹 ≠ ∅)
27		dm0rn0 5918	. . . . . . . . . . . . 13 ⊢ (dom 𝐹 = ∅ ↔ ran 𝐹 = ∅)
28	27	necon3bii 3011	. . . . . . . . . . . 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 2964	. . . . . . . 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 18903	. . . . . . . . . . 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 4787	. . . . . . . . . 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 2871	. . . . . 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 3911	. . . . 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 7458	. . . . . . . . 9 ⊢ (𝑥 = (0g‘𝐺) → (𝑥 + 𝑦) = ((0g‘𝐺) + 𝑦))
43	42	eqeq1d 2765	. . . . . . . 8 ⊢ (𝑥 = (0g‘𝐺) → ((𝑥 + 𝑦) = 𝑦 ↔ ((0g‘𝐺) + 𝑦) = 𝑦))
44	43	ralbidv 3189	. . . . . . 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 7459	. . . . . . . 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 2779	. . . . . . 7 ⊢ (𝑦 = (0g‘𝐺) → (((0g‘𝐺) + 𝑦) = 𝑦 ↔ ((0g‘𝐺) + (0g‘𝐺)) = (0g‘𝐺)))
49	48	rspcva 3578	. . . . . 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 7116	. . . . . 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 3914	. . . . 5 ⊢ (((𝜑 ∧ ran 𝐹 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) ∧ 𝑧 ∈ (𝑀...𝑁)) → (𝐹‘𝑧) ∈ {(0g‘𝐺)})
56		elsni 4601	. . . . 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 14229	. . 3 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → (seq𝑀( + , 𝐹)‘𝑁) = (0g‘𝐺))
59	14, 58	eqtr4d 2801	. 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 18924	. 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 2147   ≠ wne 2959  ∀wral 3080  {crab 3416  Vcvv 3454   ⊆ wss 3900  ∅c0 4281  {csn 4588  dom cdm 5658  ran crn 5659   Fn wfn 6561  ⟶wf 6562  ‘cfv 6566  (class class class)co 7451  ℤ_≥cuz 13005  ...cfz 13679  seqcseq 14184  Basecbs 17431  +_gcplusg 17472  0_gc0g 17660   Σ_g cgsu 17661

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-cnex 11295  ax-resscn 11296  ax-1cn 11297  ax-icn 11298  ax-addcl 11299  ax-addrcl 11300  ax-mulcl 11301  ax-mulrcl 11302  ax-mulcom 11303  ax-addass 11304  ax-mulass 11305  ax-distr 11306  ax-i2m1 11307  ax-1ne0 11308  ax-1rid 11309  ax-rnegex 11310  ax-rrecex 11311  ax-cnre 11312  ax-pre-lttri 11313  ax-pre-lttrn 11314  ax-pre-ltadd 11315  ax-pre-mulgt0 11316

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-1st 8035  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-er 8755  df-en 9012  df-dom 9013  df-sdom 9014  df-pnf 11384  df-mnf 11385  df-xr 11386  df-ltxr 11387  df-le 11388  df-sub 11582  df-neg 11583  df-nn 12376  df-n0 12647  df-z 12734  df-uz 13006  df-fz 13680  df-seq 14185  df-0g 17662  df-gsum 17663

This theorem is used by:  gsumsplit1r  18926  gsumprval  18927  gsumwsubmcl  19083  gsumws1  19084  gsumsgrpccat  19086  gsumwmhm  19091  mulgnngsum  19339  gsumval3  20172  gsummptfzcl  20234  gsumncl  35283  gsumnunsn  35284  veroquadgsumlem  51411