apdifflemf - Mathbox for Jim Kingdon

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Theorem apdifflemf 16650

Description: Lemma for apdiff 16652. 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 8207	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℂ)
3		apdifflemf.r	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ ℚ)
4		qcn 9867	. . . . . . 7 ⊢ (𝑅 ∈ ℚ → 𝑅 ∈ ℂ)
5	3, 4	syl 14	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ ℂ)
6	2, 5	subcld 8489	. . . . 5 ⊢ (𝜑 → (𝐴 − 𝑅) ∈ ℂ)
7	6	adantr 276	. . . 4 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (𝐴 − 𝑅) ∈ ℂ)
8	7	abscld 11741	. . 3 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (abs‘(𝐴 − 𝑅)) ∈ ℝ)
9		apdifflemf.q	. . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ ℚ)
10		qcn 9867	. . . . . . 7 ⊢ (𝑄 ∈ ℚ → 𝑄 ∈ ℂ)
11	9, 10	syl 14	. . . . . 6 ⊢ (𝜑 → 𝑄 ∈ ℂ)
12	2, 11	subcld 8489	. . . . 5 ⊢ (𝜑 → (𝐴 − 𝑄) ∈ ℂ)
13	12	abscld 11741	. . . 4 ⊢ (𝜑 → (abs‘(𝐴 − 𝑄)) ∈ ℝ)
14	13	adantr 276	. . 3 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (abs‘(𝐴 − 𝑄)) ∈ ℝ)
15		qre 9858	. . . . . . . . . 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 9868	. . . . . . . . . . . . . 14 ⊢ ((𝑄 ∈ ℚ ∧ 𝑅 ∈ ℚ) → (𝑄 + 𝑅) ∈ ℚ)
20	9, 3, 19	syl2anc 411	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑄 + 𝑅) ∈ ℚ)
21		qre 9858	. . . . . . . . . . . . 13 ⊢ ((𝑄 + 𝑅) ∈ ℚ → (𝑄 + 𝑅) ∈ ℝ)
22	20, 21	syl 14	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑄 + 𝑅) ∈ ℝ)
23	22	rehalfcld 9390	. . . . . . . . . . 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 9858	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ ℚ → 𝑅 ∈ ℝ)
27	3, 26	syl 14	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 ∈ ℝ)
28		avglt1 9382	. . . . . . . . . . . . 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 8304	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → 𝑄 < 𝐴)
34	17, 18, 33	ltled 8297	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → 𝑄 ≤ 𝐴)
35	17, 18, 34	abssubge0d 11736	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (abs‘(𝐴 − 𝑄)) = (𝐴 − 𝑄))
36	35	oveq2d 6033	. . . . . 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 8519	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (𝑅 − (𝐴 − 𝑄)) = ((𝑅 + 𝑄) − 𝐴))
41	37, 39	addcomd 8329	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (𝑅 + 𝑄) = (𝑄 + 𝑅))
42	41	oveq1d 6032	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → ((𝑅 + 𝑄) − 𝐴) = ((𝑄 + 𝑅) − 𝐴))
43	36, 40, 42	3eqtrd 2268	. . . . 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 9892	. . . . . . . . . 10 ⊢ 2 ∈ ℝ⁺
46	45	a1i 9	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → 2 ∈ ℝ⁺)
47	44, 18, 46	ltdivmuld 9982	. . . . . . . 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 9386	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (2 · 𝐴) = (𝐴 + 𝐴))
50	48, 49	breqtrd 4114	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (𝑄 + 𝑅) < (𝐴 + 𝐴))
51	44, 18, 18	ltsubaddd 8720	. . . . . 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 4110	. . . 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 9926	. . . . . . . 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 9964	. . . . 5 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → 𝐴 < (𝐴 + (𝑅 − 𝑄)))
60	35	oveq2d 6033	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (𝑅 + (abs‘(𝐴 − 𝑄))) = (𝑅 + (𝐴 − 𝑄)))
61	37, 38, 39	addsub12d 8512	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (𝑅 + (𝐴 − 𝑄)) = (𝐴 + (𝑅 − 𝑄)))
62	60, 61	eqtrd 2264	. . . . 5 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (𝑅 + (abs‘(𝐴 − 𝑄))) = (𝐴 + (𝑅 − 𝑄)))
63	59, 62	breqtrrd 4116	. . . 4 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → 𝐴 < (𝑅 + (abs‘(𝐴 − 𝑄))))
64	18, 55, 14	absdifltd 11738	. . . 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 952	. . 3 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (abs‘(𝐴 − 𝑅)) < (abs‘(𝐴 − 𝑄)))
66	8, 14, 65	gtapd 8816	. 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 11741	. . 3 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → (abs‘(𝐴 − 𝑅)) ∈ ℝ)
70	11, 5, 2	subsubd 8517	. . . . . . 7 ⊢ (𝜑 → (𝑄 − (𝑅 − 𝐴)) = ((𝑄 − 𝑅) + 𝐴))
71	16, 27	sublt0d 8749	. . . . . . . . 9 ⊢ (𝜑 → ((𝑄 − 𝑅) < 0 ↔ 𝑄 < 𝑅))
72	25, 71	mpbird 167	. . . . . . . 8 ⊢ (𝜑 → (𝑄 − 𝑅) < 0)
73	16, 27	resubcld 8559	. . . . . . . . 9 ⊢ (𝜑 → (𝑄 − 𝑅) ∈ ℝ)
74		ltaddnegr 8604	. . . . . . . . 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 4110	. . . . . 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 9391	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → (𝐴 + 𝐴) < (𝑄 + 𝑅))
83	79, 79, 80	ltaddsub2d 8725	. . . . . . 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 8509	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → ((𝑄 + 𝑅) − 𝐴) = (𝑄 + (𝑅 − 𝐴)))
89	84, 88	breqtrd 4114	. . . . 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 8559	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → (𝑅 − 𝐴) ∈ ℝ)
93	79, 90, 92	absdifltd 11738	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → ((abs‘(𝐴 − 𝑄)) < (𝑅 − 𝐴) ↔ ((𝑄 − (𝑅 − 𝐴)) < 𝐴 ∧ 𝐴 < (𝑄 + (𝑅 − 𝐴)))))
94	78, 89, 93	mpbir2and 952	. . . 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 9383	. . . . . . . . . 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 8304	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → 𝐴 < 𝑅)
101	79, 91, 100	ltled 8297	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → 𝐴 ≤ 𝑅)
102	79, 91, 101	abssuble0d 11737	. . . 4 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → (abs‘(𝐴 − 𝑅)) = (𝑅 − 𝐴))
103	94, 102	breqtrrd 4116	. . 3 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → (abs‘(𝐴 − 𝑄)) < (abs‘(𝐴 − 𝑅)))
104	67, 69, 103	ltapd 8817	. 2 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → (abs‘(𝐴 − 𝑄)) # (abs‘(𝐴 − 𝑅)))
105		apdifflemf.ap	. . 3 ⊢ (𝜑 → ((𝑄 + 𝑅) / 2) # 𝐴)
106		reaplt 8767	. . . 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 805	1 ⊢ (𝜑 → (abs‘(𝐴 − 𝑄)) # (abs‘(𝐴 − 𝑅)))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 715 ∈ wcel 2202 class class class wbr 4088 ‘cfv 5326 (class class class)co 6017 ℂcc 8029 ℝcr 8030 0cc0 8031 + caddc 8034 · cmul 8036 < clt 8213 − cmin 8349 # cap 8760 / cdiv 8851 2c2 9193 ℚcq 9852 ℝ⁺crp 9887 abscabs 11557

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 619 ax-in2 620 ax-io 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4204 ax-sep 4207 ax-nul 4215 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-setind 4635 ax-iinf 4686 ax-cnex 8122 ax-resscn 8123 ax-1cn 8124 ax-1re 8125 ax-icn 8126 ax-addcl 8127 ax-addrcl 8128 ax-mulcl 8129 ax-mulrcl 8130 ax-addcom 8131 ax-mulcom 8132 ax-addass 8133 ax-mulass 8134 ax-distr 8135 ax-i2m1 8136 ax-0lt1 8137 ax-1rid 8138 ax-0id 8139 ax-rnegex 8140 ax-precex 8141 ax-cnre 8142 ax-pre-ltirr 8143 ax-pre-ltwlin 8144 ax-pre-lttrn 8145 ax-pre-apti 8146 ax-pre-ltadd 8147 ax-pre-mulgt0 8148 ax-pre-mulext 8149 ax-arch 8150 ax-caucvg 8151

This theorem depends on definitions: df-bi 117 df-dc 842 df-3or 1005 df-3an 1006 df-tru 1400 df-fal 1403 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-nel 2498 df-ral 2515 df-rex 2516 df-reu 2517 df-rmo 2518 df-rab 2519 df-v 2804 df-sbc 3032 df-csb 3128 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-nul 3495 df-if 3606 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-int 3929 df-iun 3972 df-br 4089 df-opab 4151 df-mpt 4152 df-tr 4188 df-id 4390 df-po 4393 df-iso 4394 df-iord 4463 df-on 4465 df-ilim 4466 df-suc 4468 df-iom 4689 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-rn 4736 df-res 4737 df-ima 4738 df-iota 5286 df-fun 5328 df-fn 5329 df-f 5330 df-f1 5331 df-fo 5332 df-f1o 5333 df-fv 5334 df-riota 5970 df-ov 6020 df-oprab 6021 df-mpo 6022 df-1st 6302 df-2nd 6303 df-recs 6470 df-frec 6556 df-pnf 8215 df-mnf 8216 df-xr 8217 df-ltxr 8218 df-le 8219 df-sub 8351 df-neg 8352 df-reap 8754 df-ap 8761 df-div 8852 df-inn 9143 df-2 9201 df-3 9202 df-4 9203 df-n0 9402 df-z 9479 df-uz 9755 df-q 9853 df-rp 9888 df-seqfrec 10709 df-exp 10800 df-cj 11402 df-re 11403 df-im 11404 df-rsqrt 11558 df-abs 11559

This theorem is referenced by: apdiff 16652

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