Theorem addsrpr 6673

Description: Addition of signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)

Assertion

Ref	Expression
addsrpr	⊢ (((A ∈ P ∧ B ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) → ([⟨A, B⟩] ~R +R [⟨𝐶, 𝐷⟩] ~R ) = [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R )

Proof of Theorem addsrpr

Dummy variables x y z w v u 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		opelxpi 4319	. . . 4 ⊢ ((A ∈ P ∧ B ∈ P) → ⟨A, B⟩ ∈ (P × P))
2		enrex 6665	. . . . 5 ⊢ ~R ∈ V
3	2	ecelqsi 6096	. . . 4 ⊢ (⟨A, B⟩ ∈ (P × P) → [⟨A, B⟩] ~R ∈ ((P × P) / ~R ))
4	1, 3	syl 14	. . 3 ⊢ ((A ∈ P ∧ B ∈ P) → [⟨A, B⟩] ~R ∈ ((P × P) / ~R ))
5		opelxpi 4319	. . . 4 ⊢ ((𝐶 ∈ P ∧ 𝐷 ∈ P) → ⟨𝐶, 𝐷⟩ ∈ (P × P))
6	2	ecelqsi 6096	. . . 4 ⊢ (⟨𝐶, 𝐷⟩ ∈ (P × P) → [⟨𝐶, 𝐷⟩] ~R ∈ ((P × P) / ~R ))
7	5, 6	syl 14	. . 3 ⊢ ((𝐶 ∈ P ∧ 𝐷 ∈ P) → [⟨𝐶, 𝐷⟩] ~R ∈ ((P × P) / ~R ))
8	4, 7	anim12i 321	. 2 ⊢ (((A ∈ P ∧ B ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) → ([⟨A, B⟩] ~R ∈ ((P × P) / ~R ) ∧ [⟨𝐶, 𝐷⟩] ~R ∈ ((P × P) / ~R )))
9		eqid 2037	. . . 4 ⊢ [⟨A, B⟩] ~R = [⟨A, B⟩] ~R
10		eqid 2037	. . . 4 ⊢ [⟨𝐶, 𝐷⟩] ~R = [⟨𝐶, 𝐷⟩] ~R
11	9, 10	pm3.2i 257	. . 3 ⊢ ([⟨A, B⟩] ~R = [⟨A, B⟩] ~R ∧ [⟨𝐶, 𝐷⟩] ~R = [⟨𝐶, 𝐷⟩] ~R )
12		eqid 2037	. . 3 ⊢ [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R = [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R
13		opeq12 3542	. . . . . . . . 9 ⊢ ((w = A ∧ v = B) → ⟨w, v⟩ = ⟨A, B⟩)
14	13	eceq1d 6078	. . . . . . . 8 ⊢ ((w = A ∧ v = B) → [⟨w, v⟩] ~R = [⟨A, B⟩] ~R )
15	14	eqeq2d 2048	. . . . . . 7 ⊢ ((w = A ∧ v = B) → ([⟨A, B⟩] ~R = [⟨w, v⟩] ~R ↔ [⟨A, B⟩] ~R = [⟨A, B⟩] ~R ))
16	15	anbi1d 438	. . . . . 6 ⊢ ((w = A ∧ v = B) → (([⟨A, B⟩] ~R = [⟨w, v⟩] ~R ∧ [⟨𝐶, 𝐷⟩] ~R = [⟨𝐶, 𝐷⟩] ~R ) ↔ ([⟨A, B⟩] ~R = [⟨A, B⟩] ~R ∧ [⟨𝐶, 𝐷⟩] ~R = [⟨𝐶, 𝐷⟩] ~R )))
17		simpl 102	. . . . . . . . . 10 ⊢ ((w = A ∧ v = B) → w = A)
18	17	oveq1d 5470	. . . . . . . . 9 ⊢ ((w = A ∧ v = B) → (w +P 𝐶) = (A +P 𝐶))
19		simpr 103	. . . . . . . . . 10 ⊢ ((w = A ∧ v = B) → v = B)
20	19	oveq1d 5470	. . . . . . . . 9 ⊢ ((w = A ∧ v = B) → (v +P 𝐷) = (B +P 𝐷))
21	18, 20	opeq12d 3548	. . . . . . . 8 ⊢ ((w = A ∧ v = B) → ⟨(w +P 𝐶), (v +P 𝐷)⟩ = ⟨(A +P 𝐶), (B +P 𝐷)⟩)
22	21	eceq1d 6078	. . . . . . 7 ⊢ ((w = A ∧ v = B) → [⟨(w +P 𝐶), (v +P 𝐷)⟩] ~R = [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R )
23	22	eqeq2d 2048	. . . . . 6 ⊢ ((w = A ∧ v = B) → ([⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R = [⟨(w +P 𝐶), (v +P 𝐷)⟩] ~R ↔ [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R = [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R ))
24	16, 23	anbi12d 442	. . . . 5 ⊢ ((w = A ∧ v = B) → ((([⟨A, B⟩] ~R = [⟨w, v⟩] ~R ∧ [⟨𝐶, 𝐷⟩] ~R = [⟨𝐶, 𝐷⟩] ~R ) ∧ [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R = [⟨(w +P 𝐶), (v +P 𝐷)⟩] ~R ) ↔ (([⟨A, B⟩] ~R = [⟨A, B⟩] ~R ∧ [⟨𝐶, 𝐷⟩] ~R = [⟨𝐶, 𝐷⟩] ~R ) ∧ [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R = [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R )))
25	24	spc2egv 2636	. . . 4 ⊢ ((A ∈ P ∧ B ∈ P) → ((([⟨A, B⟩] ~R = [⟨A, B⟩] ~R ∧ [⟨𝐶, 𝐷⟩] ~R = [⟨𝐶, 𝐷⟩] ~R ) ∧ [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R = [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R ) → ∃w∃v(([⟨A, B⟩] ~R = [⟨w, v⟩] ~R ∧ [⟨𝐶, 𝐷⟩] ~R = [⟨𝐶, 𝐷⟩] ~R ) ∧ [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R = [⟨(w +P 𝐶), (v +P 𝐷)⟩] ~R )))
26		opeq12 3542	. . . . . . . . . 10 ⊢ ((u = 𝐶 ∧ 𝑡 = 𝐷) → ⟨u, 𝑡⟩ = ⟨𝐶, 𝐷⟩)
27	26	eceq1d 6078	. . . . . . . . 9 ⊢ ((u = 𝐶 ∧ 𝑡 = 𝐷) → [⟨u, 𝑡⟩] ~R = [⟨𝐶, 𝐷⟩] ~R )
28	27	eqeq2d 2048	. . . . . . . 8 ⊢ ((u = 𝐶 ∧ 𝑡 = 𝐷) → ([⟨𝐶, 𝐷⟩] ~R = [⟨u, 𝑡⟩] ~R ↔ [⟨𝐶, 𝐷⟩] ~R = [⟨𝐶, 𝐷⟩] ~R ))
29	28	anbi2d 437	. . . . . . 7 ⊢ ((u = 𝐶 ∧ 𝑡 = 𝐷) → (([⟨A, B⟩] ~R = [⟨w, v⟩] ~R ∧ [⟨𝐶, 𝐷⟩] ~R = [⟨u, 𝑡⟩] ~R ) ↔ ([⟨A, B⟩] ~R = [⟨w, v⟩] ~R ∧ [⟨𝐶, 𝐷⟩] ~R = [⟨𝐶, 𝐷⟩] ~R )))
30		simpl 102	. . . . . . . . . . 11 ⊢ ((u = 𝐶 ∧ 𝑡 = 𝐷) → u = 𝐶)
31	30	oveq2d 5471	. . . . . . . . . 10 ⊢ ((u = 𝐶 ∧ 𝑡 = 𝐷) → (w +P u) = (w +P 𝐶))
32		simpr 103	. . . . . . . . . . 11 ⊢ ((u = 𝐶 ∧ 𝑡 = 𝐷) → 𝑡 = 𝐷)
33	32	oveq2d 5471	. . . . . . . . . 10 ⊢ ((u = 𝐶 ∧ 𝑡 = 𝐷) → (v +P 𝑡) = (v +P 𝐷))
34	31, 33	opeq12d 3548	. . . . . . . . 9 ⊢ ((u = 𝐶 ∧ 𝑡 = 𝐷) → ⟨(w +P u), (v +P 𝑡)⟩ = ⟨(w +P 𝐶), (v +P 𝐷)⟩)
35	34	eceq1d 6078	. . . . . . . 8 ⊢ ((u = 𝐶 ∧ 𝑡 = 𝐷) → [⟨(w +P u), (v +P 𝑡)⟩] ~R = [⟨(w +P 𝐶), (v +P 𝐷)⟩] ~R )
36	35	eqeq2d 2048	. . . . . . 7 ⊢ ((u = 𝐶 ∧ 𝑡 = 𝐷) → ([⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R = [⟨(w +P u), (v +P 𝑡)⟩] ~R ↔ [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R = [⟨(w +P 𝐶), (v +P 𝐷)⟩] ~R ))
37	29, 36	anbi12d 442	. . . . . 6 ⊢ ((u = 𝐶 ∧ 𝑡 = 𝐷) → ((([⟨A, B⟩] ~R = [⟨w, v⟩] ~R ∧ [⟨𝐶, 𝐷⟩] ~R = [⟨u, 𝑡⟩] ~R ) ∧ [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R = [⟨(w +P u), (v +P 𝑡)⟩] ~R ) ↔ (([⟨A, B⟩] ~R = [⟨w, v⟩] ~R ∧ [⟨𝐶, 𝐷⟩] ~R = [⟨𝐶, 𝐷⟩] ~R ) ∧ [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R = [⟨(w +P 𝐶), (v +P 𝐷)⟩] ~R )))
38	37	spc2egv 2636	. . . . 5 ⊢ ((𝐶 ∈ P ∧ 𝐷 ∈ P) → ((([⟨A, B⟩] ~R = [⟨w, v⟩] ~R ∧ [⟨𝐶, 𝐷⟩] ~R = [⟨𝐶, 𝐷⟩] ~R ) ∧ [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R = [⟨(w +P 𝐶), (v +P 𝐷)⟩] ~R ) → ∃u∃𝑡(([⟨A, B⟩] ~R = [⟨w, v⟩] ~R ∧ [⟨𝐶, 𝐷⟩] ~R = [⟨u, 𝑡⟩] ~R ) ∧ [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R = [⟨(w +P u), (v +P 𝑡)⟩] ~R )))
39	38	2eximdv 1759	. . . 4 ⊢ ((𝐶 ∈ P ∧ 𝐷 ∈ P) → (∃w∃v(([⟨A, B⟩] ~R = [⟨w, v⟩] ~R ∧ [⟨𝐶, 𝐷⟩] ~R = [⟨𝐶, 𝐷⟩] ~R ) ∧ [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R = [⟨(w +P 𝐶), (v +P 𝐷)⟩] ~R ) → ∃w∃v∃u∃𝑡(([⟨A, B⟩] ~R = [⟨w, v⟩] ~R ∧ [⟨𝐶, 𝐷⟩] ~R = [⟨u, 𝑡⟩] ~R ) ∧ [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R = [⟨(w +P u), (v +P 𝑡)⟩] ~R )))
40	25, 39	sylan9 389	. . 3 ⊢ (((A ∈ P ∧ B ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) → ((([⟨A, B⟩] ~R = [⟨A, B⟩] ~R ∧ [⟨𝐶, 𝐷⟩] ~R = [⟨𝐶, 𝐷⟩] ~R ) ∧ [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R = [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R ) → ∃w∃v∃u∃𝑡(([⟨A, B⟩] ~R = [⟨w, v⟩] ~R ∧ [⟨𝐶, 𝐷⟩] ~R = [⟨u, 𝑡⟩] ~R ) ∧ [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R = [⟨(w +P u), (v +P 𝑡)⟩] ~R )))
41	11, 12, 40	mp2ani 408	. 2 ⊢ (((A ∈ P ∧ B ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) → ∃w∃v∃u∃𝑡(([⟨A, B⟩] ~R = [⟨w, v⟩] ~R ∧ [⟨𝐶, 𝐷⟩] ~R = [⟨u, 𝑡⟩] ~R ) ∧ [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R = [⟨(w +P u), (v +P 𝑡)⟩] ~R ))
42		ecexg 6046	. . . 4 ⊢ ( ~R ∈ V → [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R ∈ V)
43	2, 42	ax-mp 7	. . 3 ⊢ [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R ∈ V
44		simp1 903	. . . . . . . 8 ⊢ ((x = [⟨A, B⟩] ~R ∧ y = [⟨𝐶, 𝐷⟩] ~R ∧ z = [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R ) → x = [⟨A, B⟩] ~R )
45	44	eqeq1d 2045	. . . . . . 7 ⊢ ((x = [⟨A, B⟩] ~R ∧ y = [⟨𝐶, 𝐷⟩] ~R ∧ z = [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R ) → (x = [⟨w, v⟩] ~R ↔ [⟨A, B⟩] ~R = [⟨w, v⟩] ~R ))
46		simp2 904	. . . . . . . 8 ⊢ ((x = [⟨A, B⟩] ~R ∧ y = [⟨𝐶, 𝐷⟩] ~R ∧ z = [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R ) → y = [⟨𝐶, 𝐷⟩] ~R )
47	46	eqeq1d 2045	. . . . . . 7 ⊢ ((x = [⟨A, B⟩] ~R ∧ y = [⟨𝐶, 𝐷⟩] ~R ∧ z = [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R ) → (y = [⟨u, 𝑡⟩] ~R ↔ [⟨𝐶, 𝐷⟩] ~R = [⟨u, 𝑡⟩] ~R ))
48	45, 47	anbi12d 442	. . . . . 6 ⊢ ((x = [⟨A, B⟩] ~R ∧ y = [⟨𝐶, 𝐷⟩] ~R ∧ z = [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R ) → ((x = [⟨w, v⟩] ~R ∧ y = [⟨u, 𝑡⟩] ~R ) ↔ ([⟨A, B⟩] ~R = [⟨w, v⟩] ~R ∧ [⟨𝐶, 𝐷⟩] ~R = [⟨u, 𝑡⟩] ~R )))
49		simp3 905	. . . . . . 7 ⊢ ((x = [⟨A, B⟩] ~R ∧ y = [⟨𝐶, 𝐷⟩] ~R ∧ z = [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R ) → z = [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R )
50	49	eqeq1d 2045	. . . . . 6 ⊢ ((x = [⟨A, B⟩] ~R ∧ y = [⟨𝐶, 𝐷⟩] ~R ∧ z = [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R ) → (z = [⟨(w +P u), (v +P 𝑡)⟩] ~R ↔ [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R = [⟨(w +P u), (v +P 𝑡)⟩] ~R ))
51	48, 50	anbi12d 442	. . . . 5 ⊢ ((x = [⟨A, B⟩] ~R ∧ y = [⟨𝐶, 𝐷⟩] ~R ∧ z = [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R ) → (((x = [⟨w, v⟩] ~R ∧ y = [⟨u, 𝑡⟩] ~R ) ∧ z = [⟨(w +P u), (v +P 𝑡)⟩] ~R ) ↔ (([⟨A, B⟩] ~R = [⟨w, v⟩] ~R ∧ [⟨𝐶, 𝐷⟩] ~R = [⟨u, 𝑡⟩] ~R ) ∧ [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R = [⟨(w +P u), (v +P 𝑡)⟩] ~R )))
52	51	4exbidv 1747	. . . 4 ⊢ ((x = [⟨A, B⟩] ~R ∧ y = [⟨𝐶, 𝐷⟩] ~R ∧ z = [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R ) → (∃w∃v∃u∃𝑡((x = [⟨w, v⟩] ~R ∧ y = [⟨u, 𝑡⟩] ~R ) ∧ z = [⟨(w +P u), (v +P 𝑡)⟩] ~R ) ↔ ∃w∃v∃u∃𝑡(([⟨A, B⟩] ~R = [⟨w, v⟩] ~R ∧ [⟨𝐶, 𝐷⟩] ~R = [⟨u, 𝑡⟩] ~R ) ∧ [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R = [⟨(w +P u), (v +P 𝑡)⟩] ~R )))
53		addsrmo 6671	. . . 4 ⊢ ((x ∈ ((P × P) / ~R ) ∧ y ∈ ((P × P) / ~R )) → ∃*z∃w∃v∃u∃𝑡((x = [⟨w, v⟩] ~R ∧ y = [⟨u, 𝑡⟩] ~R ) ∧ z = [⟨(w +P u), (v +P 𝑡)⟩] ~R ))
54		df-plr 6656	. . . . 5 ⊢ +R = {⟨⟨x, y⟩, z⟩ ∣ ((x ∈ R ∧ y ∈ R) ∧ ∃w∃v∃u∃𝑡((x = [⟨w, v⟩] ~R ∧ y = [⟨u, 𝑡⟩] ~R ) ∧ z = [⟨(w +P u), (v +P 𝑡)⟩] ~R ))}
55		df-nr 6655	. . . . . . . . 9 ⊢ R = ((P × P) / ~R )
56	55	eleq2i 2101	. . . . . . . 8 ⊢ (x ∈ R ↔ x ∈ ((P × P) / ~R ))
57	55	eleq2i 2101	. . . . . . . 8 ⊢ (y ∈ R ↔ y ∈ ((P × P) / ~R ))
58	56, 57	anbi12i 433	. . . . . . 7 ⊢ ((x ∈ R ∧ y ∈ R) ↔ (x ∈ ((P × P) / ~R ) ∧ y ∈ ((P × P) / ~R )))
59	58	anbi1i 431	. . . . . 6 ⊢ (((x ∈ R ∧ y ∈ R) ∧ ∃w∃v∃u∃𝑡((x = [⟨w, v⟩] ~R ∧ y = [⟨u, 𝑡⟩] ~R ) ∧ z = [⟨(w +P u), (v +P 𝑡)⟩] ~R )) ↔ ((x ∈ ((P × P) / ~R ) ∧ y ∈ ((P × P) / ~R )) ∧ ∃w∃v∃u∃𝑡((x = [⟨w, v⟩] ~R ∧ y = [⟨u, 𝑡⟩] ~R ) ∧ z = [⟨(w +P u), (v +P 𝑡)⟩] ~R )))
60	59	oprabbii 5502	. . . . 5 ⊢ {⟨⟨x, y⟩, z⟩ ∣ ((x ∈ R ∧ y ∈ R) ∧ ∃w∃v∃u∃𝑡((x = [⟨w, v⟩] ~R ∧ y = [⟨u, 𝑡⟩] ~R ) ∧ z = [⟨(w +P u), (v +P 𝑡)⟩] ~R ))} = {⟨⟨x, y⟩, z⟩ ∣ ((x ∈ ((P × P) / ~R ) ∧ y ∈ ((P × P) / ~R )) ∧ ∃w∃v∃u∃𝑡((x = [⟨w, v⟩] ~R ∧ y = [⟨u, 𝑡⟩] ~R ) ∧ z = [⟨(w +P u), (v +P 𝑡)⟩] ~R ))}
61	54, 60	eqtri 2057	. . . 4 ⊢ +R = {⟨⟨x, y⟩, z⟩ ∣ ((x ∈ ((P × P) / ~R ) ∧ y ∈ ((P × P) / ~R )) ∧ ∃w∃v∃u∃𝑡((x = [⟨w, v⟩] ~R ∧ y = [⟨u, 𝑡⟩] ~R ) ∧ z = [⟨(w +P u), (v +P 𝑡)⟩] ~R ))}
62	52, 53, 61	ovig 5564	. . 3 ⊢ (([⟨A, B⟩] ~R ∈ ((P × P) / ~R ) ∧ [⟨𝐶, 𝐷⟩] ~R ∈ ((P × P) / ~R ) ∧ [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R ∈ V) → (∃w∃v∃u∃𝑡(([⟨A, B⟩] ~R = [⟨w, v⟩] ~R ∧ [⟨𝐶, 𝐷⟩] ~R = [⟨u, 𝑡⟩] ~R ) ∧ [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R = [⟨(w +P u), (v +P 𝑡)⟩] ~R ) → ([⟨A, B⟩] ~R +R [⟨𝐶, 𝐷⟩] ~R ) = [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R ))
63	43, 62	mp3an3 1220	. 2 ⊢ (([⟨A, B⟩] ~R ∈ ((P × P) / ~R ) ∧ [⟨𝐶, 𝐷⟩] ~R ∈ ((P × P) / ~R )) → (∃w∃v∃u∃𝑡(([⟨A, B⟩] ~R = [⟨w, v⟩] ~R ∧ [⟨𝐶, 𝐷⟩] ~R = [⟨u, 𝑡⟩] ~R ) ∧ [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R = [⟨(w +P u), (v +P 𝑡)⟩] ~R ) → ([⟨A, B⟩] ~R +R [⟨𝐶, 𝐷⟩] ~R ) = [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R ))
64	8, 41, 63	sylc 56	1 ⊢ (((A ∈ P ∧ B ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) → ([⟨A, B⟩] ~R +R [⟨𝐶, 𝐷⟩] ~R ) = [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R )

Syntax hints:   → wi 4   ∧ wa 97   ∧ w3a 884   = wceq 1242  ∃wex 1378   ∈ wcel 1390  Vcvv 2551  ⟨cop 3370   × cxp 4286  (class class class)co 5455  {coprab 5456  [cec 6040   / cqs 6041  Pcnp 6275   +_P cpp 6277   ~_R cer 6280  Rcnr 6281   +_R cplr 6285

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254

This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-eprel 4017  df-id 4021  df-po 4024  df-iso 4025  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-1o 5940  df-2o 5941  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-pli 6289  df-mi 6290  df-lti 6291  df-plpq 6328  df-mpq 6329  df-enq 6331  df-nqqs 6332  df-plqqs 6333  df-mqqs 6334  df-1nqqs 6335  df-rq 6336  df-ltnqqs 6337  df-enq0 6407  df-nq0 6408  df-0nq0 6409  df-plq0 6410  df-mq0 6411  df-inp 6449  df-iplp 6451  df-enr 6654  df-nr 6655  df-plr 6656

This theorem is referenced by:  addclsr  6681  addcomsrg  6683  addasssrg  6684  distrsrg  6687  m1p1sr  6688  0idsr  6695  ltasrg  6698  pitonnlem2  6743