quadfac - Mathbox for Luke Murphy

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Theorem quadfac 42972

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 11564	. . 3 ⊢ (𝜑 → (𝑋 − 𝑀) ∈ ℂ)
4		quadfac.n	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℂ)
5	1, 4	subcld 11564	. . 3 ⊢ (𝜑 → (𝑋 − 𝑁) ∈ ℂ)
6	3, 5	mul0ord 11857	. 2 ⊢ (𝜑 → (((𝑋 − 𝑀) · (𝑋 − 𝑁)) = 0 ↔ ((𝑋 − 𝑀) = 0 ∨ (𝑋 − 𝑁) = 0)))
7		olc 881	. . . . . 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 2963	. . . . . . . . 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 929	. . . . . . 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 870	. . . . . . . 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 11668	. . . . . . . 8 ⊢ (𝜑 → ((𝑋 − 𝑀) · (𝑋 − 𝑁)) = (((𝑋 · 𝑋) + (𝑁 · 𝑀)) − ((𝑋 · 𝑁) + (𝑋 · 𝑀))))
22	1	sqvald 14175	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑋↑2) = (𝑋 · 𝑋))
23	22	eqcomd 2769	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋 · 𝑋) = (𝑋↑2))
24	2, 4	mulcomd 11225	. . . . . . . . . . 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 7428	. . . . . . . . 9 ⊢ (𝜑 → ((𝑋 · 𝑋) + (𝑁 · 𝑀)) = ((𝑋↑2) + (𝐶 / 𝐴)))
28	4, 2	addcomd 11407	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁 + 𝑀) = (𝑀 + 𝑁))
29	28	oveq1d 7425	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑁 + 𝑀) · 𝑋) = ((𝑀 + 𝑁) · 𝑋))
30	4, 1	mulcomd 11225	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑁 · 𝑋) = (𝑋 · 𝑁))
31	2, 1	mulcomd 11225	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑀 · 𝑋) = (𝑋 · 𝑀))
32	30, 31	oveq12d 7428	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑁 · 𝑋) + (𝑀 · 𝑋)) = ((𝑋 · 𝑁) + (𝑋 · 𝑀)))
33	4, 1, 2, 32	joinlmuladdmuld 11231	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑁 + 𝑀) · 𝑋) = ((𝑋 · 𝑁) + (𝑋 · 𝑀)))
34		quadfac.mpn	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑀 + 𝑁) = -(𝐵 / 𝐴))
35	34	oveq1d 7425	. . . . . . . . . 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 7428	. . . . . . . 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 11986	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 / 𝐴) ∈ ℂ)
43	42, 1	mulneg1d 11662	. . . . . . . 8 ⊢ (𝜑 → (-(𝐵 / 𝐴) · 𝑋) = -((𝐵 / 𝐴) · 𝑋))
44	43	oveq2d 7426	. . . . . . 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 14176	. . . . . . . . 9 ⊢ (𝜑 → (𝑋↑2) ∈ ℂ)
47		quadfac.c	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ ℂ)
48	47, 41, 8	divcld 11986	. . . . . . . . 9 ⊢ (𝜑 → (𝐶 / 𝐴) ∈ ℂ)
49	46, 48	addcld 11223	. . . . . . . 8 ⊢ (𝜑 → ((𝑋↑2) + (𝐶 / 𝐴)) ∈ ℂ)
50	40, 41, 8	divcld 11986	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 / 𝐴) ∈ ℂ)
51	50, 1	mulcld 11224	. . . . . . . 8 ⊢ (𝜑 → ((𝐵 / 𝐴) · 𝑋) ∈ ℂ)
52	49, 51	subnegd 11571	. . . . . . 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 14176	. . . . . . . 8 ⊢ (𝜑 → (𝑋↑2) ∈ ℂ)
56	47, 41, 8	divcld 11986	. . . . . . . 8 ⊢ (𝜑 → (𝐶 / 𝐴) ∈ ℂ)
57	55, 56	addcld 11223	. . . . . . 7 ⊢ (𝜑 → ((𝑋↑2) + (𝐶 / 𝐴)) ∈ ℂ)
58	40, 41, 8	divcld 11986	. . . . . . . 8 ⊢ (𝜑 → (𝐵 / 𝐴) ∈ ℂ)
59	58, 1	mulcld 11224	. . . . . . 7 ⊢ (𝜑 → ((𝐵 / 𝐴) · 𝑋) ∈ ℂ)
60	57, 59	addcld 11223	. . . . . 6 ⊢ (𝜑 → (((𝑋↑2) + (𝐶 / 𝐴)) + ((𝐵 / 𝐴) · 𝑋)) ∈ ℂ)
61	41, 60	mul0ord 11857	. . . . 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 14176	. . . . . . 7 ⊢ (𝜑 → (𝑋↑2) ∈ ℂ)
64	47, 41, 8	divcld 11986	. . . . . . 7 ⊢ (𝜑 → (𝐶 / 𝐴) ∈ ℂ)
65	63, 64	addcld 11223	. . . . . 6 ⊢ (𝜑 → ((𝑋↑2) + (𝐶 / 𝐴)) ∈ ℂ)
66	40, 41, 8	divcld 11986	. . . . . . 7 ⊢ (𝜑 → (𝐵 / 𝐴) ∈ ℂ)
67	66, 1	mulcld 11224	. . . . . 6 ⊢ (𝜑 → ((𝐵 / 𝐴) · 𝑋) ∈ ℂ)
68	41, 65, 67	adddid 11228	. . . . 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 14176	. . . . . . . 8 ⊢ (𝜑 → (𝑋↑2) ∈ ℂ)
71	47, 41, 8	divcld 11986	. . . . . . . 8 ⊢ (𝜑 → (𝐶 / 𝐴) ∈ ℂ)
72	41, 70, 71	adddid 11228	. . . . . . 7 ⊢ (𝜑 → (𝐴 · ((𝑋↑2) + (𝐶 / 𝐴))) = ((𝐴 · (𝑋↑2)) + (𝐴 · (𝐶 / 𝐴))))
73	47, 41, 8	divcan2d 11988	. . . . . . . 8 ⊢ (𝜑 → (𝐴 · (𝐶 / 𝐴)) = 𝐶)
74	73	oveq2d 7426	. . . . . . 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 11986	. . . . . . . 8 ⊢ (𝜑 → (𝐵 / 𝐴) ∈ ℂ)
77	41, 76, 1	mulassd 11227	. . . . . . 7 ⊢ (𝜑 → ((𝐴 · (𝐵 / 𝐴)) · 𝑋) = (𝐴 · ((𝐵 / 𝐴) · 𝑋)))
78	40, 41, 8	divcan2d 11988	. . . . . . . 8 ⊢ (𝜑 → (𝐴 · (𝐵 / 𝐴)) = 𝐵)
79	78	oveq1d 7425	. . . . . . 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 7428	. . . . 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 14176	. . . . . 6 ⊢ (𝜑 → (𝑋↑2) ∈ ℂ)
85	41, 84	mulcld 11224	. . . . 5 ⊢ (𝜑 → (𝐴 · (𝑋↑2)) ∈ ℂ)
86	40, 1	mulcld 11224	. . . . 5 ⊢ (𝜑 → (𝐵 · 𝑋) ∈ ℂ)
87	85, 47, 86	addassd 11226	. . . 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 11224	. . . . . 6 ⊢ (𝜑 → (𝐵 · 𝑋) ∈ ℂ)
90	47, 89	addcomd 11407	. . . . 5 ⊢ (𝜑 → (𝐶 + (𝐵 · 𝑋)) = ((𝐵 · 𝑋) + 𝐶))
91	90	oveq2d 7426	. . . 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 11574	. . 3 ⊢ (𝜑 → ((𝑋 − 𝑀) = 0 ↔ 𝑋 = 𝑀))
95	1, 4	subeq0ad 11574	. . 3 ⊢ (𝜑 → ((𝑋 − 𝑁) = 0 ↔ 𝑋 = 𝑁))
96	94, 95	orbi12d 931	. 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

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∨ wo 860 = wceq 1570 ⊥wfal 1582 ∈ wcel 2143 ≠ wne 2958 (class class class)co 7410 ℂcc 11093 0cc0 11095 + caddc 11098 · cmul 11100 − cmin 11436 -cneg 11437 / cdiv 11866 2c2 12290 ↑cexp 14093

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11151 ax-resscn 11152 ax-1cn 11153 ax-icn 11154 ax-addcl 11155 ax-addrcl 11156 ax-mulcl 11157 ax-mulrcl 11158 ax-mulcom 11159 ax-addass 11160 ax-mulass 11161 ax-distr 11162 ax-i2m1 11163 ax-1ne0 11164 ax-1rid 11165 ax-rnegex 11166 ax-rrecex 11167 ax-cnre 11168 ax-pre-lttri 11169 ax-pre-lttrn 11170 ax-pre-ltadd 11171 ax-pre-mulgt0 11172

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4873 df-iun 4958 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7859 df-2nd 7983 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-er 8690 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11240 df-mnf 11241 df-xr 11242 df-ltxr 11243 df-le 11244 df-sub 11438 df-neg 11439 df-div 11867 df-nn 12229 df-2 12298 df-n0 12500 df-z 12587 df-uz 12858 df-seq 14034 df-exp 14094

This theorem is referenced by: 25or6to4 42973

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