apdifflemf - Mathbox for Jim Kingdon

	Mathbox for Jim Kingdon		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > Mathboxes > apdifflemf		GIF version

Theorem apdifflemf 15781

Description: Lemma for apdiff 15783. Being apart from the point halfway between 𝑄 and 𝑅 suffices for 𝐴 to be a different distance from 𝑄 and from 𝑅. (Contributed by Jim Kingdon, 18-May-2024.)

**Hypotheses**
Ref	Expression
apdifflemf.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
apdifflemf.q	⊢ (𝜑 → 𝑄 ∈ ℚ)
apdifflemf.r	⊢ (𝜑 → 𝑅 ∈ ℚ)
apdifflemf.qr	⊢ (𝜑 → 𝑄 < 𝑅)
apdifflemf.ap	⊢ (𝜑 → ((𝑄 + 𝑅) / 2) # 𝐴)

**Assertion**
Ref	Expression
apdifflemf	⊢ (𝜑 → (abs‘(𝐴 − 𝑄)) # (abs‘(𝐴 − 𝑅)))

**Proof of Theorem apdifflemf**
Step	Hyp	Ref	Expression
1		apdifflemf.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ)
2	1	recnd 8074	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℂ)
3		apdifflemf.r	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ ℚ)
4		qcn 9727	. . . . . . 7 ⊢ (𝑅 ∈ ℚ → 𝑅 ∈ ℂ)
5	3, 4	syl 14	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ ℂ)
6	2, 5	subcld 8356	. . . . 5 ⊢ (𝜑 → (𝐴 − 𝑅) ∈ ℂ)
7	6	adantr 276	. . . 4 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (𝐴 − 𝑅) ∈ ℂ)
8	7	abscld 11365	. . 3 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (abs‘(𝐴 − 𝑅)) ∈ ℝ)
9		apdifflemf.q	. . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ ℚ)
10		qcn 9727	. . . . . . 7 ⊢ (𝑄 ∈ ℚ → 𝑄 ∈ ℂ)
11	9, 10	syl 14	. . . . . 6 ⊢ (𝜑 → 𝑄 ∈ ℂ)
12	2, 11	subcld 8356	. . . . 5 ⊢ (𝜑 → (𝐴 − 𝑄) ∈ ℂ)
13	12	abscld 11365	. . . 4 ⊢ (𝜑 → (abs‘(𝐴 − 𝑄)) ∈ ℝ)
14	13	adantr 276	. . 3 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (abs‘(𝐴 − 𝑄)) ∈ ℝ)
15		qre 9718	. . . . . . . . . 10 ⊢ (𝑄 ∈ ℚ → 𝑄 ∈ ℝ)
16	9, 15	syl 14	. . . . . . . . 9 ⊢ (𝜑 → 𝑄 ∈ ℝ)
17	16	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → 𝑄 ∈ ℝ)
18	1	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → 𝐴 ∈ ℝ)
19		qaddcl 9728	. . . . . . . . . . . . . 14 ⊢ ((𝑄 ∈ ℚ ∧ 𝑅 ∈ ℚ) → (𝑄 + 𝑅) ∈ ℚ)
20	9, 3, 19	syl2anc 411	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑄 + 𝑅) ∈ ℚ)
21		qre 9718	. . . . . . . . . . . . 13 ⊢ ((𝑄 + 𝑅) ∈ ℚ → (𝑄 + 𝑅) ∈ ℝ)
22	20, 21	syl 14	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑄 + 𝑅) ∈ ℝ)
23	22	rehalfcld 9257	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑄 + 𝑅) / 2) ∈ ℝ)
24	23	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → ((𝑄 + 𝑅) / 2) ∈ ℝ)
25		apdifflemf.qr	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑄 < 𝑅)
26		qre 9718	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ ℚ → 𝑅 ∈ ℝ)
27	3, 26	syl 14	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 ∈ ℝ)
28		avglt1 9249	. . . . . . . . . . . . 13 ⊢ ((𝑄 ∈ ℝ ∧ 𝑅 ∈ ℝ) → (𝑄 < 𝑅 ↔ 𝑄 < ((𝑄 + 𝑅) / 2)))
29	16, 27, 28	syl2anc 411	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑄 < 𝑅 ↔ 𝑄 < ((𝑄 + 𝑅) / 2)))
30	25, 29	mpbid 147	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑄 < ((𝑄 + 𝑅) / 2))
31	30	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → 𝑄 < ((𝑄 + 𝑅) / 2))
32		simpr 110	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → ((𝑄 + 𝑅) / 2) < 𝐴)
33	17, 24, 18, 31, 32	lttrd 8171	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → 𝑄 < 𝐴)
34	17, 18, 33	ltled 8164	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → 𝑄 ≤ 𝐴)
35	17, 18, 34	abssubge0d 11360	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (abs‘(𝐴 − 𝑄)) = (𝐴 − 𝑄))
36	35	oveq2d 5941	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (𝑅 − (abs‘(𝐴 − 𝑄))) = (𝑅 − (𝐴 − 𝑄)))
37	5	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → 𝑅 ∈ ℂ)
38	2	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → 𝐴 ∈ ℂ)
39	11	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → 𝑄 ∈ ℂ)
40	37, 38, 39	subsub3d 8386	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (𝑅 − (𝐴 − 𝑄)) = ((𝑅 + 𝑄) − 𝐴))
41	37, 39	addcomd 8196	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (𝑅 + 𝑄) = (𝑄 + 𝑅))
42	41	oveq1d 5940	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → ((𝑅 + 𝑄) − 𝐴) = ((𝑄 + 𝑅) − 𝐴))
43	36, 40, 42	3eqtrd 2233	. . . . 5 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (𝑅 − (abs‘(𝐴 − 𝑄))) = ((𝑄 + 𝑅) − 𝐴))
44	22	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (𝑄 + 𝑅) ∈ ℝ)
45		2rp 9752	. . . . . . . . . 10 ⊢ 2 ∈ ℝ⁺
46	45	a1i 9	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → 2 ∈ ℝ⁺)
47	44, 18, 46	ltdivmuld 9842	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (((𝑄 + 𝑅) / 2) < 𝐴 ↔ (𝑄 + 𝑅) < (2 · 𝐴)))
48	32, 47	mpbid 147	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (𝑄 + 𝑅) < (2 · 𝐴))
49	38	2timesd 9253	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (2 · 𝐴) = (𝐴 + 𝐴))
50	48, 49	breqtrd 4060	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (𝑄 + 𝑅) < (𝐴 + 𝐴))
51	44, 18, 18	ltsubaddd 8587	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (((𝑄 + 𝑅) − 𝐴) < 𝐴 ↔ (𝑄 + 𝑅) < (𝐴 + 𝐴)))
52	50, 51	mpbird 167	. . . . 5 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → ((𝑄 + 𝑅) − 𝐴) < 𝐴)
53	43, 52	eqbrtrd 4056	. . . 4 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (𝑅 − (abs‘(𝐴 − 𝑄))) < 𝐴)
54	25	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → 𝑄 < 𝑅)
55	27	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → 𝑅 ∈ ℝ)
56		difrp 9786	. . . . . . . 8 ⊢ ((𝑄 ∈ ℝ ∧ 𝑅 ∈ ℝ) → (𝑄 < 𝑅 ↔ (𝑅 − 𝑄) ∈ ℝ⁺))
57	17, 55, 56	syl2anc 411	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (𝑄 < 𝑅 ↔ (𝑅 − 𝑄) ∈ ℝ⁺))
58	54, 57	mpbid 147	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (𝑅 − 𝑄) ∈ ℝ⁺)
59	18, 58	ltaddrpd 9824	. . . . 5 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → 𝐴 < (𝐴 + (𝑅 − 𝑄)))
60	35	oveq2d 5941	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (𝑅 + (abs‘(𝐴 − 𝑄))) = (𝑅 + (𝐴 − 𝑄)))
61	37, 38, 39	addsub12d 8379	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (𝑅 + (𝐴 − 𝑄)) = (𝐴 + (𝑅 − 𝑄)))
62	60, 61	eqtrd 2229	. . . . 5 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (𝑅 + (abs‘(𝐴 − 𝑄))) = (𝐴 + (𝑅 − 𝑄)))
63	59, 62	breqtrrd 4062	. . . 4 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → 𝐴 < (𝑅 + (abs‘(𝐴 − 𝑄))))
64	18, 55, 14	absdifltd 11362	. . . 4 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → ((abs‘(𝐴 − 𝑅)) < (abs‘(𝐴 − 𝑄)) ↔ ((𝑅 − (abs‘(𝐴 − 𝑄))) < 𝐴 ∧ 𝐴 < (𝑅 + (abs‘(𝐴 − 𝑄))))))
65	53, 63, 64	mpbir2and 946	. . 3 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (abs‘(𝐴 − 𝑅)) < (abs‘(𝐴 − 𝑄)))
66	8, 14, 65	gtapd 8683	. 2 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (abs‘(𝐴 − 𝑄)) # (abs‘(𝐴 − 𝑅)))
67	13	adantr 276	. . 3 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → (abs‘(𝐴 − 𝑄)) ∈ ℝ)
68	6	adantr 276	. . . 4 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → (𝐴 − 𝑅) ∈ ℂ)
69	68	abscld 11365	. . 3 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → (abs‘(𝐴 − 𝑅)) ∈ ℝ)
70	11, 5, 2	subsubd 8384	. . . . . . 7 ⊢ (𝜑 → (𝑄 − (𝑅 − 𝐴)) = ((𝑄 − 𝑅) + 𝐴))
71	16, 27	sublt0d 8616	. . . . . . . . 9 ⊢ (𝜑 → ((𝑄 − 𝑅) < 0 ↔ 𝑄 < 𝑅))
72	25, 71	mpbird 167	. . . . . . . 8 ⊢ (𝜑 → (𝑄 − 𝑅) < 0)
73	16, 27	resubcld 8426	. . . . . . . . 9 ⊢ (𝜑 → (𝑄 − 𝑅) ∈ ℝ)
74		ltaddnegr 8471	. . . . . . . . 9 ⊢ (((𝑄 − 𝑅) ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((𝑄 − 𝑅) < 0 ↔ ((𝑄 − 𝑅) + 𝐴) < 𝐴))
75	73, 1, 74	syl2anc 411	. . . . . . . 8 ⊢ (𝜑 → ((𝑄 − 𝑅) < 0 ↔ ((𝑄 − 𝑅) + 𝐴) < 𝐴))
76	72, 75	mpbid 147	. . . . . . 7 ⊢ (𝜑 → ((𝑄 − 𝑅) + 𝐴) < 𝐴)
77	70, 76	eqbrtrd 4056	. . . . . 6 ⊢ (𝜑 → (𝑄 − (𝑅 − 𝐴)) < 𝐴)
78	77	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → (𝑄 − (𝑅 − 𝐴)) < 𝐴)
79	1	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → 𝐴 ∈ ℝ)
80	22	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → (𝑄 + 𝑅) ∈ ℝ)
81		simpr 110	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → 𝐴 < ((𝑄 + 𝑅) / 2))
82	79, 79, 80, 81, 81	lt2halvesd 9258	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → (𝐴 + 𝐴) < (𝑄 + 𝑅))
83	79, 79, 80	ltaddsub2d 8592	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → ((𝐴 + 𝐴) < (𝑄 + 𝑅) ↔ 𝐴 < ((𝑄 + 𝑅) − 𝐴)))
84	82, 83	mpbid 147	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → 𝐴 < ((𝑄 + 𝑅) − 𝐴))
85	11	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → 𝑄 ∈ ℂ)
86	5	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → 𝑅 ∈ ℂ)
87	2	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → 𝐴 ∈ ℂ)
88	85, 86, 87	addsubassd 8376	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → ((𝑄 + 𝑅) − 𝐴) = (𝑄 + (𝑅 − 𝐴)))
89	84, 88	breqtrd 4060	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → 𝐴 < (𝑄 + (𝑅 − 𝐴)))
90	16	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → 𝑄 ∈ ℝ)
91	27	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → 𝑅 ∈ ℝ)
92	91, 79	resubcld 8426	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → (𝑅 − 𝐴) ∈ ℝ)
93	79, 90, 92	absdifltd 11362	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → ((abs‘(𝐴 − 𝑄)) < (𝑅 − 𝐴) ↔ ((𝑄 − (𝑅 − 𝐴)) < 𝐴 ∧ 𝐴 < (𝑄 + (𝑅 − 𝐴)))))
94	78, 89, 93	mpbir2and 946	. . . 4 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → (abs‘(𝐴 − 𝑄)) < (𝑅 − 𝐴))
95	23	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → ((𝑄 + 𝑅) / 2) ∈ ℝ)
96		avglt2 9250	. . . . . . . . . 10 ⊢ ((𝑄 ∈ ℝ ∧ 𝑅 ∈ ℝ) → (𝑄 < 𝑅 ↔ ((𝑄 + 𝑅) / 2) < 𝑅))
97	16, 27, 96	syl2anc 411	. . . . . . . . 9 ⊢ (𝜑 → (𝑄 < 𝑅 ↔ ((𝑄 + 𝑅) / 2) < 𝑅))
98	25, 97	mpbid 147	. . . . . . . 8 ⊢ (𝜑 → ((𝑄 + 𝑅) / 2) < 𝑅)
99	98	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → ((𝑄 + 𝑅) / 2) < 𝑅)
100	79, 95, 91, 81, 99	lttrd 8171	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → 𝐴 < 𝑅)
101	79, 91, 100	ltled 8164	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → 𝐴 ≤ 𝑅)
102	79, 91, 101	abssuble0d 11361	. . . 4 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → (abs‘(𝐴 − 𝑅)) = (𝑅 − 𝐴))
103	94, 102	breqtrrd 4062	. . 3 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → (abs‘(𝐴 − 𝑄)) < (abs‘(𝐴 − 𝑅)))
104	67, 69, 103	ltapd 8684	. 2 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → (abs‘(𝐴 − 𝑄)) # (abs‘(𝐴 − 𝑅)))
105		apdifflemf.ap	. . 3 ⊢ (𝜑 → ((𝑄 + 𝑅) / 2) # 𝐴)
106		reaplt 8634	. . . 4 ⊢ ((((𝑄 + 𝑅) / 2) ∈ ℝ ∧ 𝐴 ∈ ℝ) → (((𝑄 + 𝑅) / 2) # 𝐴 ↔ (((𝑄 + 𝑅) / 2) < 𝐴 ∨ 𝐴 < ((𝑄 + 𝑅) / 2))))
107	23, 1, 106	syl2anc 411	. . 3 ⊢ (𝜑 → (((𝑄 + 𝑅) / 2) # 𝐴 ↔ (((𝑄 + 𝑅) / 2) < 𝐴 ∨ 𝐴 < ((𝑄 + 𝑅) / 2))))
108	105, 107	mpbid 147	. 2 ⊢ (𝜑 → (((𝑄 + 𝑅) / 2) < 𝐴 ∨ 𝐴 < ((𝑄 + 𝑅) / 2)))
109	66, 104, 108	mpjaodan 799	1 ⊢ (𝜑 → (abs‘(𝐴 − 𝑄)) # (abs‘(𝐴 − 𝑅)))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 709 ∈ wcel 2167 class class class wbr 4034 ‘cfv 5259 (class class class)co 5925 ℂcc 7896 ℝcr 7897 0cc0 7898 + caddc 7901 · cmul 7903 < clt 8080 − cmin 8216 # cap 8627 / cdiv 8718 2c2 9060 ℚcq 9712 ℝ⁺crp 9747 abscabs 11181

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-coll 4149 ax-sep 4152 ax-nul 4160 ax-pow 4208 ax-pr 4243 ax-un 4469 ax-setind 4574 ax-iinf 4625 ax-cnex 7989 ax-resscn 7990 ax-1cn 7991 ax-1re 7992 ax-icn 7993 ax-addcl 7994 ax-addrcl 7995 ax-mulcl 7996 ax-mulrcl 7997 ax-addcom 7998 ax-mulcom 7999 ax-addass 8000 ax-mulass 8001 ax-distr 8002 ax-i2m1 8003 ax-0lt1 8004 ax-1rid 8005 ax-0id 8006 ax-rnegex 8007 ax-precex 8008 ax-cnre 8009 ax-pre-ltirr 8010 ax-pre-ltwlin 8011 ax-pre-lttrn 8012 ax-pre-apti 8013 ax-pre-ltadd 8014 ax-pre-mulgt0 8015 ax-pre-mulext 8016 ax-arch 8017 ax-caucvg 8018

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rmo 2483 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3452 df-if 3563 df-pw 3608 df-sn 3629 df-pr 3630 df-op 3632 df-uni 3841 df-int 3876 df-iun 3919 df-br 4035 df-opab 4096 df-mpt 4097 df-tr 4133 df-id 4329 df-po 4332 df-iso 4333 df-iord 4402 df-on 4404 df-ilim 4405 df-suc 4407 df-iom 4628 df-xp 4670 df-rel 4671 df-cnv 4672 df-co 4673 df-dm 4674 df-rn 4675 df-res 4676 df-ima 4677 df-iota 5220 df-fun 5261 df-fn 5262 df-f 5263 df-f1 5264 df-fo 5265 df-f1o 5266 df-fv 5267 df-riota 5880 df-ov 5928 df-oprab 5929 df-mpo 5930 df-1st 6207 df-2nd 6208 df-recs 6372 df-frec 6458 df-pnf 8082 df-mnf 8083 df-xr 8084 df-ltxr 8085 df-le 8086 df-sub 8218 df-neg 8219 df-reap 8621 df-ap 8628 df-div 8719 df-inn 9010 df-2 9068 df-3 9069 df-4 9070 df-n0 9269 df-z 9346 df-uz 9621 df-q 9713 df-rp 9748 df-seqfrec 10559 df-exp 10650 df-cj 11026 df-re 11027 df-im 11028 df-rsqrt 11182 df-abs 11183

This theorem is referenced by: apdiff 15783

W3C validator