quadfac - Mathbox for Luke Murphy

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Theorem quadfac 43514

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 11706	. . 3 ⊢ (𝜑 → (𝑋 − 𝑀) ∈ ℂ)
4		quadfac.n	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℂ)
5	1, 4	subcld 11706	. . 3 ⊢ (𝜑 → (𝑋 − 𝑁) ∈ ℂ)
6	3, 5	mul0ord 12002	. 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 2964	. . . . . . . . 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 11812	. . . . . . . 8 ⊢ (𝜑 → ((𝑋 − 𝑀) · (𝑋 − 𝑁)) = (((𝑋 · 𝑋) + (𝑁 · 𝑀)) − ((𝑋 · 𝑁) + (𝑋 · 𝑀))))
22	1	sqvald 14324	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑋↑2) = (𝑋 · 𝑋))
23	22	eqcomd 2769	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋 · 𝑋) = (𝑋↑2))
24	2, 4	mulcomd 11367	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑀 · 𝑁) = (𝑁 · 𝑀))
25		quadfac.mtn	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑀 · 𝑁) = (𝐶 / 𝐴))
26	24, 25	eqtr3d 2800	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁 · 𝑀) = (𝐶 / 𝐴))
27	23, 26	oveq12d 7467	. . . . . . . . 9 ⊢ (𝜑 → ((𝑋 · 𝑋) + (𝑁 · 𝑀)) = ((𝑋↑2) + (𝐶 / 𝐴)))
28	4, 2	addcomd 11549	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁 + 𝑀) = (𝑀 + 𝑁))
29	28	oveq1d 7464	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑁 + 𝑀) · 𝑋) = ((𝑀 + 𝑁) · 𝑋))
30	4, 1	mulcomd 11367	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑁 · 𝑋) = (𝑋 · 𝑁))
31	2, 1	mulcomd 11367	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑀 · 𝑋) = (𝑋 · 𝑀))
32	30, 31	oveq12d 7467	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑁 · 𝑋) + (𝑀 · 𝑋)) = ((𝑋 · 𝑁) + (𝑋 · 𝑀)))
33	4, 1, 2, 32	joinlmuladdmuld 11373	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑁 + 𝑀) · 𝑋) = ((𝑋 · 𝑁) + (𝑋 · 𝑀)))
34		quadfac.mpn	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑀 + 𝑁) = -(𝐵 / 𝐴))
35	34	oveq1d 7464	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑀 + 𝑁) · 𝑋) = ( -(𝐵 / 𝐴) · 𝑋))
36	29, 33, 35	3eqtr3d 2806	. . . . . . . . 9 ⊢ (𝜑 → ((𝑋 · 𝑁) + (𝑋 · 𝑀)) = ( -(𝐵 / 𝐴) · 𝑋))
37	27, 36	oveq12d 7467	. . . . . . . 8 ⊢ (𝜑 → (((𝑋 · 𝑋) + (𝑁 · 𝑀)) − ((𝑋 · 𝑁) + (𝑋 · 𝑀))) = (((𝑋↑2) + (𝐶 / 𝐴)) − ( -(𝐵 / 𝐴) · 𝑋)))
38	21, 37	eqtrd 2798	. . . . . . 7 ⊢ (𝜑 → ((𝑋 − 𝑀) · (𝑋 − 𝑁)) = (((𝑋↑2) + (𝐶 / 𝐴)) − ( -(𝐵 / 𝐴) · 𝑋)))
39	38	eqeq1d 2765	. . . . . 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 12131	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 / 𝐴) ∈ ℂ)
43	42, 1	mulneg1d 11806	. . . . . . . 8 ⊢ (𝜑 → ( -(𝐵 / 𝐴) · 𝑋) = -((𝐵 / 𝐴) · 𝑋))
44	43	oveq2d 7465	. . . . . . 7 ⊢ (𝜑 → (((𝑋↑2) + (𝐶 / 𝐴)) − ( -(𝐵 / 𝐴) · 𝑋)) = (((𝑋↑2) + (𝐶 / 𝐴)) − -((𝐵 / 𝐴) · 𝑋)))
45	44	eqeq1d 2765	. . . . . 6 ⊢ (𝜑 → ((((𝑋↑2) + (𝐶 / 𝐴)) − ( -(𝐵 / 𝐴) · 𝑋)) = 0 ↔ (((𝑋↑2) + (𝐶 / 𝐴)) − -((𝐵 / 𝐴) · 𝑋)) = 0))
46	1	sqcld 14325	. . . . . . . . 9 ⊢ (𝜑 → (𝑋↑2) ∈ ℂ)
47		quadfac.c	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ ℂ)
48	47, 41, 8	divcld 12131	. . . . . . . . 9 ⊢ (𝜑 → (𝐶 / 𝐴) ∈ ℂ)
49	46, 48	addcld 11365	. . . . . . . 8 ⊢ (𝜑 → ((𝑋↑2) + (𝐶 / 𝐴)) ∈ ℂ)
50	40, 41, 8	divcld 12131	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 / 𝐴) ∈ ℂ)
51	50, 1	mulcld 11366	. . . . . . . 8 ⊢ (𝜑 → ((𝐵 / 𝐴) · 𝑋) ∈ ℂ)
52	49, 51	subnegd 11713	. . . . . . 7 ⊢ (𝜑 → (((𝑋↑2) + (𝐶 / 𝐴)) − -((𝐵 / 𝐴) · 𝑋)) = (((𝑋↑2) + (𝐶 / 𝐴)) + ((𝐵 / 𝐴) · 𝑋)))
53	52	eqeq1d 2765	. . . . . 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 14325	. . . . . . . 8 ⊢ (𝜑 → (𝑋↑2) ∈ ℂ)
56	47, 41, 8	divcld 12131	. . . . . . . 8 ⊢ (𝜑 → (𝐶 / 𝐴) ∈ ℂ)
57	55, 56	addcld 11365	. . . . . . 7 ⊢ (𝜑 → ((𝑋↑2) + (𝐶 / 𝐴)) ∈ ℂ)
58	40, 41, 8	divcld 12131	. . . . . . . 8 ⊢ (𝜑 → (𝐵 / 𝐴) ∈ ℂ)
59	58, 1	mulcld 11366	. . . . . . 7 ⊢ (𝜑 → ((𝐵 / 𝐴) · 𝑋) ∈ ℂ)
60	57, 59	addcld 11365	. . . . . 6 ⊢ (𝜑 → (((𝑋↑2) + (𝐶 / 𝐴)) + ((𝐵 / 𝐴) · 𝑋)) ∈ ℂ)
61	41, 60	mul0ord 12002	. . . . 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 14325	. . . . . . 7 ⊢ (𝜑 → (𝑋↑2) ∈ ℂ)
64	47, 41, 8	divcld 12131	. . . . . . 7 ⊢ (𝜑 → (𝐶 / 𝐴) ∈ ℂ)
65	63, 64	addcld 11365	. . . . . 6 ⊢ (𝜑 → ((𝑋↑2) + (𝐶 / 𝐴)) ∈ ℂ)
66	40, 41, 8	divcld 12131	. . . . . . 7 ⊢ (𝜑 → (𝐵 / 𝐴) ∈ ℂ)
67	66, 1	mulcld 11366	. . . . . 6 ⊢ (𝜑 → ((𝐵 / 𝐴) · 𝑋) ∈ ℂ)
68	41, 65, 67	adddid 11370	. . . . 5 ⊢ (𝜑 → (𝐴 · (((𝑋↑2) + (𝐶 / 𝐴)) + ((𝐵 / 𝐴) · 𝑋))) = ((𝐴 · ((𝑋↑2) + (𝐶 / 𝐴))) + (𝐴 · ((𝐵 / 𝐴) · 𝑋))))
69	68	eqeq1d 2765	. . . 4 ⊢ (𝜑 → ((𝐴 · (((𝑋↑2) + (𝐶 / 𝐴)) + ((𝐵 / 𝐴) · 𝑋))) = 0 ↔ ((𝐴 · ((𝑋↑2) + (𝐶 / 𝐴))) + (𝐴 · ((𝐵 / 𝐴) · 𝑋))) = 0))
70	1	sqcld 14325	. . . . . . . 8 ⊢ (𝜑 → (𝑋↑2) ∈ ℂ)
71	47, 41, 8	divcld 12131	. . . . . . . 8 ⊢ (𝜑 → (𝐶 / 𝐴) ∈ ℂ)
72	41, 70, 71	adddid 11370	. . . . . . 7 ⊢ (𝜑 → (𝐴 · ((𝑋↑2) + (𝐶 / 𝐴))) = ((𝐴 · (𝑋↑2)) + (𝐴 · (𝐶 / 𝐴))))
73	47, 41, 8	divcan2d 12133	. . . . . . . 8 ⊢ (𝜑 → (𝐴 · (𝐶 / 𝐴)) = 𝐶)
74	73	oveq2d 7465	. . . . . . 7 ⊢ (𝜑 → ((𝐴 · (𝑋↑2)) + (𝐴 · (𝐶 / 𝐴))) = ((𝐴 · (𝑋↑2)) + 𝐶))
75	72, 74	eqtrd 2798	. . . . . 6 ⊢ (𝜑 → (𝐴 · ((𝑋↑2) + (𝐶 / 𝐴))) = ((𝐴 · (𝑋↑2)) + 𝐶))
76	40, 41, 8	divcld 12131	. . . . . . . 8 ⊢ (𝜑 → (𝐵 / 𝐴) ∈ ℂ)
77	41, 76, 1	mulassd 11369	. . . . . . 7 ⊢ (𝜑 → ((𝐴 · (𝐵 / 𝐴)) · 𝑋) = (𝐴 · ((𝐵 / 𝐴) · 𝑋)))
78	40, 41, 8	divcan2d 12133	. . . . . . . 8 ⊢ (𝜑 → (𝐴 · (𝐵 / 𝐴)) = 𝐵)
79	78	oveq1d 7464	. . . . . . 7 ⊢ (𝜑 → ((𝐴 · (𝐵 / 𝐴)) · 𝑋) = (𝐵 · 𝑋))
80	77, 79	eqtr3d 2800	. . . . . 6 ⊢ (𝜑 → (𝐴 · ((𝐵 / 𝐴) · 𝑋)) = (𝐵 · 𝑋))
81	75, 80	oveq12d 7467	. . . . 5 ⊢ (𝜑 → ((𝐴 · ((𝑋↑2) + (𝐶 / 𝐴))) + (𝐴 · ((𝐵 / 𝐴) · 𝑋))) = (((𝐴 · (𝑋↑2)) + 𝐶) + (𝐵 · 𝑋)))
82	81	eqeq1d 2765	. . . 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 14325	. . . . . 6 ⊢ (𝜑 → (𝑋↑2) ∈ ℂ)
85	41, 84	mulcld 11366	. . . . 5 ⊢ (𝜑 → (𝐴 · (𝑋↑2)) ∈ ℂ)
86	40, 1	mulcld 11366	. . . . 5 ⊢ (𝜑 → (𝐵 · 𝑋) ∈ ℂ)
87	85, 47, 86	addassd 11368	. . . 4 ⊢ (𝜑 → (((𝐴 · (𝑋↑2)) + 𝐶) + (𝐵 · 𝑋)) = ((𝐴 · (𝑋↑2)) + (𝐶 + (𝐵 · 𝑋))))
88	87	eqeq1d 2765	. . 3 ⊢ (𝜑 → ((((𝐴 · (𝑋↑2)) + 𝐶) + (𝐵 · 𝑋)) = 0 ↔ ((𝐴 · (𝑋↑2)) + (𝐶 + (𝐵 · 𝑋))) = 0))
89	40, 1	mulcld 11366	. . . . . 6 ⊢ (𝜑 → (𝐵 · 𝑋) ∈ ℂ)
90	47, 89	addcomd 11549	. . . . 5 ⊢ (𝜑 → (𝐶 + (𝐵 · 𝑋)) = ((𝐵 · 𝑋) + 𝐶))
91	90	oveq2d 7465	. . . 4 ⊢ (𝜑 → ((𝐴 · (𝑋↑2)) + (𝐶 + (𝐵 · 𝑋))) = ((𝐴 · (𝑋↑2)) + ((𝐵 · 𝑋) + 𝐶)))
92	91	eqeq1d 2765	. . 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 11714	. . 3 ⊢ (𝜑 → ((𝑋 − 𝑀) = 0 ↔ 𝑋 = 𝑀))
95	1, 4	subeq0ad 11714	. . 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 2147 ≠ wne 2959 (class class class)co 7449 ℂcc 11235 0cc0 11237 + caddc 11240 · cmul 11242 − cmin 11578 -cneg 11579 / cdiv 12011 2c2 12435 ↑cexp 14242

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 5253 ax-nul 5264 ax-pow 5332 ax-pr 5396 ax-un 7781 ax-cnex 11293 ax-resscn 11294 ax-1cn 11295 ax-icn 11296 ax-addcl 11297 ax-addrcl 11298 ax-mulcl 11299 ax-mulrcl 11300 ax-mulcom 11301 ax-addass 11302 ax-mulass 11303 ax-distr 11304 ax-i2m1 11305 ax-1ne0 11306 ax-1rid 11307 ax-rnegex 11308 ax-rrecex 11309 ax-cnre 11310 ax-pre-lttri 11311 ax-pre-lttrn 11312 ax-pre-ltadd 11313 ax-pre-mulgt0 11314

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 4839 df-uni 4871 df-iun 4956 df-br 5107 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5551 df-eprel 5556 df-po 5564 df-so 5565 df-fr 5609 df-we 5611 df-xp 5662 df-rel 5663 df-cnv 5664 df-co 5665 df-dm 5666 df-rn 5667 df-res 5668 df-ima 5669 df-pred 6328 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6518 df-fun 6566 df-fn 6567 df-f 6568 df-f1 6569 df-fo 6570 df-f1o 6571 df-fv 6572 df-riota 7406 df-ov 7452 df-oprab 7453 df-mpo 7454 df-om 7910 df-2nd 8034 df-frecs 8335 df-wrecs 8366 df-recs 8415 df-rdg 8454 df-er 8753 df-en 9010 df-dom 9011 df-sdom 9012 df-pnf 11382 df-mnf 11383 df-xr 11384 df-ltxr 11385 df-le 11386 df-sub 11580 df-neg 11581 df-div 12012 df-nn 12374 df-2 12443 df-n0 12645 df-z 12732 df-uz 13004 df-seq 14183 df-exp 14243

This theorem is used by: 25or6to4 43515

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