quadfac - Mathbox for Luke Murphy

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Theorem quadfac 43005

Description: The solution of a quadratic equation via factoring. (Contributed by Luke Murphy, 10-Jul-2026.)

**Hypotheses**
Ref	Expression
quadfac.a	⊢ (𝜑 → 𝐴 ∈ ℂ)
quadfac.z	⊢ (𝜑 → 𝐴 ≠ 0)
quadfac.b	⊢ (𝜑 → 𝐵 ∈ ℂ)
quadfac.c	⊢ (𝜑 → 𝐶 ∈ ℂ)
quadfac.x	⊢ (𝜑 → 𝑋 ∈ ℂ)
quadfac.m	⊢ (𝜑 → 𝑀 ∈ ℂ)
quadfac.n	⊢ (𝜑 → 𝑁 ∈ ℂ)
quadfac.mpn	⊢ (𝜑 → (𝑀 + 𝑁) = -(𝐵 / 𝐴))
quadfac.mtn	⊢ (𝜑 → (𝑀 · 𝑁) = (𝐶 / 𝐴))

**Assertion**
Ref	Expression
quadfac	⊢ (𝜑 → (((𝐴 · (𝑋↑2)) + ((𝐵 · 𝑋) + 𝐶)) = 0 ↔ (𝑋 = 𝑀 ∨ 𝑋 = 𝑁)))

**Proof of Theorem quadfac**
Step	Hyp	Ref	Expression
1		quadfac.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ ℂ)
2		quadfac.m	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℂ)
3	1, 2	subcld 11580	. . 3 ⊢ (𝜑 → (𝑋 − 𝑀) ∈ ℂ)
4		quadfac.n	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℂ)
5	1, 4	subcld 11580	. . 3 ⊢ (𝜑 → (𝑋 − 𝑁) ∈ ℂ)
6	3, 5	mul0ord 11873	. 2 ⊢ (𝜑 → (((𝑋 − 𝑀) · (𝑋 − 𝑁)) = 0 ↔ ((𝑋 − 𝑀) = 0 ∨ (𝑋 − 𝑁) = 0)))
7		olc 882	. . . . . 6 ⊢ ((((𝑋↑2) + (𝐶 / 𝐴)) + ((𝐵 / 𝐴) · 𝑋)) = 0 → (𝐴 = 0 ∨ (((𝑋↑2) + (𝐶 / 𝐴)) + ((𝐵 / 𝐴) · 𝑋)) = 0))
8		quadfac.z	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ≠ 0)
9	8	neneqd 2965	. . . . . . . . 9 ⊢ (𝜑 → ¬ 𝐴 = 0)
10		id 23	. . . . . . . . . 10 ⊢ (𝐴 = 0 → 𝐴 = 0)
11		falim 1587	. . . . . . . . . 10 ⊢ (⊥ → 𝐴 = 0)
12	10, 11	pm5.21ni 380	. . . . . . . . 9 ⊢ (¬ 𝐴 = 0 → (𝐴 = 0 ↔ ⊥))
13	9, 12	syl 18	. . . . . . . 8 ⊢ (𝜑 → (𝐴 = 0 ↔ ⊥))
14	13	orbi1d 930	. . . . . . 7 ⊢ (𝜑 → ((𝐴 = 0 ∨ (((𝑋↑2) + (𝐶 / 𝐴)) + ((𝐵 / 𝐴) · 𝑋)) = 0) ↔ (⊥ ∨ (((𝑋↑2) + (𝐶 / 𝐴)) + ((𝐵 / 𝐴) · 𝑋)) = 0)))
15		falim 1587	. . . . . . . . 9 ⊢ (⊥ → (((𝑋↑2) + (𝐶 / 𝐴)) + ((𝐵 / 𝐴) · 𝑋)) = 0)
16		id 23	. . . . . . . . 9 ⊢ ((((𝑋↑2) + (𝐶 / 𝐴)) + ((𝐵 / 𝐴) · 𝑋)) = 0 → (((𝑋↑2) + (𝐶 / 𝐴)) + ((𝐵 / 𝐴) · 𝑋)) = 0)
17	15, 16	jaoi 871	. . . . . . . 8 ⊢ ((⊥ ∨ (((𝑋↑2) + (𝐶 / 𝐴)) + ((𝐵 / 𝐴) · 𝑋)) = 0) → (((𝑋↑2) + (𝐶 / 𝐴)) + ((𝐵 / 𝐴) · 𝑋)) = 0)
18	17	a1i 11	. . . . . . 7 ⊢ (𝜑 → ((⊥ ∨ (((𝑋↑2) + (𝐶 / 𝐴)) + ((𝐵 / 𝐴) · 𝑋)) = 0) → (((𝑋↑2) + (𝐶 / 𝐴)) + ((𝐵 / 𝐴) · 𝑋)) = 0))
19	14, 18	sylbid 243	. . . . . 6 ⊢ (𝜑 → ((𝐴 = 0 ∨ (((𝑋↑2) + (𝐶 / 𝐴)) + ((𝐵 / 𝐴) · 𝑋)) = 0) → (((𝑋↑2) + (𝐶 / 𝐴)) + ((𝐵 / 𝐴) · 𝑋)) = 0))
20	7, 19	impbid2 229	. . . . 5 ⊢ (𝜑 → ((((𝑋↑2) + (𝐶 / 𝐴)) + ((𝐵 / 𝐴) · 𝑋)) = 0 ↔ (𝐴 = 0 ∨ (((𝑋↑2) + (𝐶 / 𝐴)) + ((𝐵 / 𝐴) · 𝑋)) = 0)))
21	1, 2, 1, 4	mulsubd 11684	. . . . . . . 8 ⊢ (𝜑 → ((𝑋 − 𝑀) · (𝑋 − 𝑁)) = (((𝑋 · 𝑋) + (𝑁 · 𝑀)) − ((𝑋 · 𝑁) + (𝑋 · 𝑀))))
22	1	sqvald 14193	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑋↑2) = (𝑋 · 𝑋))
23	22	eqcomd 2771	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋 · 𝑋) = (𝑋↑2))
24	2, 4	mulcomd 11241	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑀 · 𝑁) = (𝑁 · 𝑀))
25		quadfac.mtn	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑀 · 𝑁) = (𝐶 / 𝐴))
26	24, 25	eqtr3d 2802	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁 · 𝑀) = (𝐶 / 𝐴))
27	23, 26	oveq12d 7434	. . . . . . . . 9 ⊢ (𝜑 → ((𝑋 · 𝑋) + (𝑁 · 𝑀)) = ((𝑋↑2) + (𝐶 / 𝐴)))
28	4, 2	addcomd 11423	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁 + 𝑀) = (𝑀 + 𝑁))
29	28	oveq1d 7431	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑁 + 𝑀) · 𝑋) = ((𝑀 + 𝑁) · 𝑋))
30	4, 1	mulcomd 11241	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑁 · 𝑋) = (𝑋 · 𝑁))
31	2, 1	mulcomd 11241	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑀 · 𝑋) = (𝑋 · 𝑀))
32	30, 31	oveq12d 7434	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑁 · 𝑋) + (𝑀 · 𝑋)) = ((𝑋 · 𝑁) + (𝑋 · 𝑀)))
33	4, 1, 2, 32	joinlmuladdmuld 11247	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑁 + 𝑀) · 𝑋) = ((𝑋 · 𝑁) + (𝑋 · 𝑀)))
34		quadfac.mpn	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑀 + 𝑁) = -(𝐵 / 𝐴))
35	34	oveq1d 7431	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑀 + 𝑁) · 𝑋) = (-(𝐵 / 𝐴) · 𝑋))
36	29, 33, 35	3eqtr3d 2808	. . . . . . . . 9 ⊢ (𝜑 → ((𝑋 · 𝑁) + (𝑋 · 𝑀)) = (-(𝐵 / 𝐴) · 𝑋))
37	27, 36	oveq12d 7434	. . . . . . . 8 ⊢ (𝜑 → (((𝑋 · 𝑋) + (𝑁 · 𝑀)) − ((𝑋 · 𝑁) + (𝑋 · 𝑀))) = (((𝑋↑2) + (𝐶 / 𝐴)) − (-(𝐵 / 𝐴) · 𝑋)))
38	21, 37	eqtrd 2800	. . . . . . 7 ⊢ (𝜑 → ((𝑋 − 𝑀) · (𝑋 − 𝑁)) = (((𝑋↑2) + (𝐶 / 𝐴)) − (-(𝐵 / 𝐴) · 𝑋)))
39	38	eqeq1d 2767	. . . . . 6 ⊢ (𝜑 → (((𝑋 − 𝑀) · (𝑋 − 𝑁)) = 0 ↔ (((𝑋↑2) + (𝐶 / 𝐴)) − (-(𝐵 / 𝐴) · 𝑋)) = 0))
40		quadfac.b	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ ℂ)
41		quadfac.a	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ℂ)
42	40, 41, 8	divcld 12002	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 / 𝐴) ∈ ℂ)
43	42, 1	mulneg1d 11678	. . . . . . . 8 ⊢ (𝜑 → (-(𝐵 / 𝐴) · 𝑋) = -((𝐵 / 𝐴) · 𝑋))
44	43	oveq2d 7432	. . . . . . 7 ⊢ (𝜑 → (((𝑋↑2) + (𝐶 / 𝐴)) − (-(𝐵 / 𝐴) · 𝑋)) = (((𝑋↑2) + (𝐶 / 𝐴)) − -((𝐵 / 𝐴) · 𝑋)))
45	44	eqeq1d 2767	. . . . . 6 ⊢ (𝜑 → ((((𝑋↑2) + (𝐶 / 𝐴)) − (-(𝐵 / 𝐴) · 𝑋)) = 0 ↔ (((𝑋↑2) + (𝐶 / 𝐴)) − -((𝐵 / 𝐴) · 𝑋)) = 0))
46	1	sqcld 14194	. . . . . . . . 9 ⊢ (𝜑 → (𝑋↑2) ∈ ℂ)
47		quadfac.c	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ ℂ)
48	47, 41, 8	divcld 12002	. . . . . . . . 9 ⊢ (𝜑 → (𝐶 / 𝐴) ∈ ℂ)
49	46, 48	addcld 11239	. . . . . . . 8 ⊢ (𝜑 → ((𝑋↑2) + (𝐶 / 𝐴)) ∈ ℂ)
50	40, 41, 8	divcld 12002	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 / 𝐴) ∈ ℂ)
51	50, 1	mulcld 11240	. . . . . . . 8 ⊢ (𝜑 → ((𝐵 / 𝐴) · 𝑋) ∈ ℂ)
52	49, 51	subnegd 11587	. . . . . . 7 ⊢ (𝜑 → (((𝑋↑2) + (𝐶 / 𝐴)) − -((𝐵 / 𝐴) · 𝑋)) = (((𝑋↑2) + (𝐶 / 𝐴)) + ((𝐵 / 𝐴) · 𝑋)))
53	52	eqeq1d 2767	. . . . . 6 ⊢ (𝜑 → ((((𝑋↑2) + (𝐶 / 𝐴)) − -((𝐵 / 𝐴) · 𝑋)) = 0 ↔ (((𝑋↑2) + (𝐶 / 𝐴)) + ((𝐵 / 𝐴) · 𝑋)) = 0))
54	39, 45, 53	3bitrd 308	. . . . 5 ⊢ (𝜑 → (((𝑋 − 𝑀) · (𝑋 − 𝑁)) = 0 ↔ (((𝑋↑2) + (𝐶 / 𝐴)) + ((𝐵 / 𝐴) · 𝑋)) = 0))
55	1	sqcld 14194	. . . . . . . 8 ⊢ (𝜑 → (𝑋↑2) ∈ ℂ)
56	47, 41, 8	divcld 12002	. . . . . . . 8 ⊢ (𝜑 → (𝐶 / 𝐴) ∈ ℂ)
57	55, 56	addcld 11239	. . . . . . 7 ⊢ (𝜑 → ((𝑋↑2) + (𝐶 / 𝐴)) ∈ ℂ)
58	40, 41, 8	divcld 12002	. . . . . . . 8 ⊢ (𝜑 → (𝐵 / 𝐴) ∈ ℂ)
59	58, 1	mulcld 11240	. . . . . . 7 ⊢ (𝜑 → ((𝐵 / 𝐴) · 𝑋) ∈ ℂ)
60	57, 59	addcld 11239	. . . . . 6 ⊢ (𝜑 → (((𝑋↑2) + (𝐶 / 𝐴)) + ((𝐵 / 𝐴) · 𝑋)) ∈ ℂ)
61	41, 60	mul0ord 11873	. . . . 5 ⊢ (𝜑 → ((𝐴 · (((𝑋↑2) + (𝐶 / 𝐴)) + ((𝐵 / 𝐴) · 𝑋))) = 0 ↔ (𝐴 = 0 ∨ (((𝑋↑2) + (𝐶 / 𝐴)) + ((𝐵 / 𝐴) · 𝑋)) = 0)))
62	20, 54, 61	3bitr4d 314	. . . 4 ⊢ (𝜑 → (((𝑋 − 𝑀) · (𝑋 − 𝑁)) = 0 ↔ (𝐴 · (((𝑋↑2) + (𝐶 / 𝐴)) + ((𝐵 / 𝐴) · 𝑋))) = 0))
63	1	sqcld 14194	. . . . . . 7 ⊢ (𝜑 → (𝑋↑2) ∈ ℂ)
64	47, 41, 8	divcld 12002	. . . . . . 7 ⊢ (𝜑 → (𝐶 / 𝐴) ∈ ℂ)
65	63, 64	addcld 11239	. . . . . 6 ⊢ (𝜑 → ((𝑋↑2) + (𝐶 / 𝐴)) ∈ ℂ)
66	40, 41, 8	divcld 12002	. . . . . . 7 ⊢ (𝜑 → (𝐵 / 𝐴) ∈ ℂ)
67	66, 1	mulcld 11240	. . . . . 6 ⊢ (𝜑 → ((𝐵 / 𝐴) · 𝑋) ∈ ℂ)
68	41, 65, 67	adddid 11244	. . . . 5 ⊢ (𝜑 → (𝐴 · (((𝑋↑2) + (𝐶 / 𝐴)) + ((𝐵 / 𝐴) · 𝑋))) = ((𝐴 · ((𝑋↑2) + (𝐶 / 𝐴))) + (𝐴 · ((𝐵 / 𝐴) · 𝑋))))
69	68	eqeq1d 2767	. . . 4 ⊢ (𝜑 → ((𝐴 · (((𝑋↑2) + (𝐶 / 𝐴)) + ((𝐵 / 𝐴) · 𝑋))) = 0 ↔ ((𝐴 · ((𝑋↑2) + (𝐶 / 𝐴))) + (𝐴 · ((𝐵 / 𝐴) · 𝑋))) = 0))
70	1	sqcld 14194	. . . . . . . 8 ⊢ (𝜑 → (𝑋↑2) ∈ ℂ)
71	47, 41, 8	divcld 12002	. . . . . . . 8 ⊢ (𝜑 → (𝐶 / 𝐴) ∈ ℂ)
72	41, 70, 71	adddid 11244	. . . . . . 7 ⊢ (𝜑 → (𝐴 · ((𝑋↑2) + (𝐶 / 𝐴))) = ((𝐴 · (𝑋↑2)) + (𝐴 · (𝐶 / 𝐴))))
73	47, 41, 8	divcan2d 12004	. . . . . . . 8 ⊢ (𝜑 → (𝐴 · (𝐶 / 𝐴)) = 𝐶)
74	73	oveq2d 7432	. . . . . . 7 ⊢ (𝜑 → ((𝐴 · (𝑋↑2)) + (𝐴 · (𝐶 / 𝐴))) = ((𝐴 · (𝑋↑2)) + 𝐶))
75	72, 74	eqtrd 2800	. . . . . 6 ⊢ (𝜑 → (𝐴 · ((𝑋↑2) + (𝐶 / 𝐴))) = ((𝐴 · (𝑋↑2)) + 𝐶))
76	40, 41, 8	divcld 12002	. . . . . . . 8 ⊢ (𝜑 → (𝐵 / 𝐴) ∈ ℂ)
77	41, 76, 1	mulassd 11243	. . . . . . 7 ⊢ (𝜑 → ((𝐴 · (𝐵 / 𝐴)) · 𝑋) = (𝐴 · ((𝐵 / 𝐴) · 𝑋)))
78	40, 41, 8	divcan2d 12004	. . . . . . . 8 ⊢ (𝜑 → (𝐴 · (𝐵 / 𝐴)) = 𝐵)
79	78	oveq1d 7431	. . . . . . 7 ⊢ (𝜑 → ((𝐴 · (𝐵 / 𝐴)) · 𝑋) = (𝐵 · 𝑋))
80	77, 79	eqtr3d 2802	. . . . . 6 ⊢ (𝜑 → (𝐴 · ((𝐵 / 𝐴) · 𝑋)) = (𝐵 · 𝑋))
81	75, 80	oveq12d 7434	. . . . 5 ⊢ (𝜑 → ((𝐴 · ((𝑋↑2) + (𝐶 / 𝐴))) + (𝐴 · ((𝐵 / 𝐴) · 𝑋))) = (((𝐴 · (𝑋↑2)) + 𝐶) + (𝐵 · 𝑋)))
82	81	eqeq1d 2767	. . . 4 ⊢ (𝜑 → (((𝐴 · ((𝑋↑2) + (𝐶 / 𝐴))) + (𝐴 · ((𝐵 / 𝐴) · 𝑋))) = 0 ↔ (((𝐴 · (𝑋↑2)) + 𝐶) + (𝐵 · 𝑋)) = 0))
83	62, 69, 82	3bitrd 308	. . 3 ⊢ (𝜑 → (((𝑋 − 𝑀) · (𝑋 − 𝑁)) = 0 ↔ (((𝐴 · (𝑋↑2)) + 𝐶) + (𝐵 · 𝑋)) = 0))
84	1	sqcld 14194	. . . . . 6 ⊢ (𝜑 → (𝑋↑2) ∈ ℂ)
85	41, 84	mulcld 11240	. . . . 5 ⊢ (𝜑 → (𝐴 · (𝑋↑2)) ∈ ℂ)
86	40, 1	mulcld 11240	. . . . 5 ⊢ (𝜑 → (𝐵 · 𝑋) ∈ ℂ)
87	85, 47, 86	addassd 11242	. . . 4 ⊢ (𝜑 → (((𝐴 · (𝑋↑2)) + 𝐶) + (𝐵 · 𝑋)) = ((𝐴 · (𝑋↑2)) + (𝐶 + (𝐵 · 𝑋))))
88	87	eqeq1d 2767	. . 3 ⊢ (𝜑 → ((((𝐴 · (𝑋↑2)) + 𝐶) + (𝐵 · 𝑋)) = 0 ↔ ((𝐴 · (𝑋↑2)) + (𝐶 + (𝐵 · 𝑋))) = 0))
89	40, 1	mulcld 11240	. . . . . 6 ⊢ (𝜑 → (𝐵 · 𝑋) ∈ ℂ)
90	47, 89	addcomd 11423	. . . . 5 ⊢ (𝜑 → (𝐶 + (𝐵 · 𝑋)) = ((𝐵 · 𝑋) + 𝐶))
91	90	oveq2d 7432	. . . 4 ⊢ (𝜑 → ((𝐴 · (𝑋↑2)) + (𝐶 + (𝐵 · 𝑋))) = ((𝐴 · (𝑋↑2)) + ((𝐵 · 𝑋) + 𝐶)))
92	91	eqeq1d 2767	. . 3 ⊢ (𝜑 → (((𝐴 · (𝑋↑2)) + (𝐶 + (𝐵 · 𝑋))) = 0 ↔ ((𝐴 · (𝑋↑2)) + ((𝐵 · 𝑋) + 𝐶)) = 0))
93	83, 88, 92	3bitrd 308	. 2 ⊢ (𝜑 → (((𝑋 − 𝑀) · (𝑋 − 𝑁)) = 0 ↔ ((𝐴 · (𝑋↑2)) + ((𝐵 · 𝑋) + 𝐶)) = 0))
94	1, 2	subeq0ad 11590	. . 3 ⊢ (𝜑 → ((𝑋 − 𝑀) = 0 ↔ 𝑋 = 𝑀))
95	1, 4	subeq0ad 11590	. . 3 ⊢ (𝜑 → ((𝑋 − 𝑁) = 0 ↔ 𝑋 = 𝑁))
96	94, 95	orbi12d 932	. 2 ⊢ (𝜑 → (((𝑋 − 𝑀) = 0 ∨ (𝑋 − 𝑁) = 0) ↔ (𝑋 = 𝑀 ∨ 𝑋 = 𝑁)))
97	6, 93, 96	3bitr3d 312	1 ⊢ (𝜑 → (((𝐴 · (𝑋↑2)) + ((𝐵 · 𝑋) + 𝐶)) = 0 ↔ (𝑋 = 𝑀 ∨ 𝑋 = 𝑁)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∨ wo 861 = wceq 1570 ⊥wfal 1582 ∈ wcel 2146 ≠ wne 2960 (class class class)co 7416 ℂcc 11109 0cc0 11111 + caddc 11114 · cmul 11116 − cmin 11452 -cneg 11453 / cdiv 11882 2c2 12306 ↑cexp 14111

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 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7738 ax-cnex 11167 ax-resscn 11168 ax-1cn 11169 ax-icn 11170 ax-addcl 11171 ax-addrcl 11172 ax-mulcl 11173 ax-mulrcl 11174 ax-mulcom 11175 ax-addass 11176 ax-mulass 11177 ax-distr 11178 ax-i2m1 11179 ax-1ne0 11180 ax-1rid 11181 ax-rnegex 11182 ax-rrecex 11183 ax-cnre 11184 ax-pre-lttri 11185 ax-pre-lttrn 11186 ax-pre-ltadd 11187 ax-pre-mulgt0 11188

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 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7865 df-2nd 7989 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-er 8696 df-en 8946 df-dom 8947 df-sdom 8948 df-pnf 11256 df-mnf 11257 df-xr 11258 df-ltxr 11259 df-le 11260 df-sub 11454 df-neg 11455 df-div 11883 df-nn 12245 df-2 12314 df-n0 12516 df-z 12603 df-uz 12875 df-seq 14052 df-exp 14112

This theorem is used by: 25or6to4 43006

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