apdifflemf - Mathbox for Jim Kingdon

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Theorem apdifflemf 15460

Description: Lemma for apdiff 15462. 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 8034	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℂ)
3		apdifflemf.r	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ ℚ)
4		qcn 9685	. . . . . . 7 ⊢ (𝑅 ∈ ℚ → 𝑅 ∈ ℂ)
5	3, 4	syl 14	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ ℂ)
6	2, 5	subcld 8316	. . . . 5 ⊢ (𝜑 → (𝐴 − 𝑅) ∈ ℂ)
7	6	adantr 276	. . . 4 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (𝐴 − 𝑅) ∈ ℂ)
8	7	abscld 11299	. . 3 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (abs‘(𝐴 − 𝑅)) ∈ ℝ)
9		apdifflemf.q	. . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ ℚ)
10		qcn 9685	. . . . . . 7 ⊢ (𝑄 ∈ ℚ → 𝑄 ∈ ℂ)
11	9, 10	syl 14	. . . . . 6 ⊢ (𝜑 → 𝑄 ∈ ℂ)
12	2, 11	subcld 8316	. . . . 5 ⊢ (𝜑 → (𝐴 − 𝑄) ∈ ℂ)
13	12	abscld 11299	. . . 4 ⊢ (𝜑 → (abs‘(𝐴 − 𝑄)) ∈ ℝ)
14	13	adantr 276	. . 3 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (abs‘(𝐴 − 𝑄)) ∈ ℝ)
15		qre 9676	. . . . . . . . . 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 9686	. . . . . . . . . . . . . 14 ⊢ ((𝑄 ∈ ℚ ∧ 𝑅 ∈ ℚ) → (𝑄 + 𝑅) ∈ ℚ)
20	9, 3, 19	syl2anc 411	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑄 + 𝑅) ∈ ℚ)
21		qre 9676	. . . . . . . . . . . . 13 ⊢ ((𝑄 + 𝑅) ∈ ℚ → (𝑄 + 𝑅) ∈ ℝ)
22	20, 21	syl 14	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑄 + 𝑅) ∈ ℝ)
23	22	rehalfcld 9215	. . . . . . . . . . 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 9676	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ ℚ → 𝑅 ∈ ℝ)
27	3, 26	syl 14	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 ∈ ℝ)
28		avglt1 9207	. . . . . . . . . . . . 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 8131	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → 𝑄 < 𝐴)
34	17, 18, 33	ltled 8124	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → 𝑄 ≤ 𝐴)
35	17, 18, 34	abssubge0d 11294	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (abs‘(𝐴 − 𝑄)) = (𝐴 − 𝑄))
36	35	oveq2d 5922	. . . . . 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 8346	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (𝑅 − (𝐴 − 𝑄)) = ((𝑅 + 𝑄) − 𝐴))
41	37, 39	addcomd 8156	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (𝑅 + 𝑄) = (𝑄 + 𝑅))
42	41	oveq1d 5921	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → ((𝑅 + 𝑄) − 𝐴) = ((𝑄 + 𝑅) − 𝐴))
43	36, 40, 42	3eqtrd 2226	. . . . 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 9710	. . . . . . . . . 10 ⊢ 2 ∈ ℝ⁺
46	45	a1i 9	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → 2 ∈ ℝ⁺)
47	44, 18, 46	ltdivmuld 9800	. . . . . . . 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 9211	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (2 · 𝐴) = (𝐴 + 𝐴))
50	48, 49	breqtrd 4051	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (𝑄 + 𝑅) < (𝐴 + 𝐴))
51	44, 18, 18	ltsubaddd 8546	. . . . . 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 4047	. . . 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 9744	. . . . . . . 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 9782	. . . . 5 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → 𝐴 < (𝐴 + (𝑅 − 𝑄)))
60	35	oveq2d 5922	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (𝑅 + (abs‘(𝐴 − 𝑄))) = (𝑅 + (𝐴 − 𝑄)))
61	37, 38, 39	addsub12d 8339	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (𝑅 + (𝐴 − 𝑄)) = (𝐴 + (𝑅 − 𝑄)))
62	60, 61	eqtrd 2222	. . . . 5 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (𝑅 + (abs‘(𝐴 − 𝑄))) = (𝐴 + (𝑅 − 𝑄)))
63	59, 62	breqtrrd 4053	. . . 4 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → 𝐴 < (𝑅 + (abs‘(𝐴 − 𝑄))))
64	18, 55, 14	absdifltd 11296	. . . 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 8642	. 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 11299	. . 3 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → (abs‘(𝐴 − 𝑅)) ∈ ℝ)
70	11, 5, 2	subsubd 8344	. . . . . . 7 ⊢ (𝜑 → (𝑄 − (𝑅 − 𝐴)) = ((𝑄 − 𝑅) + 𝐴))
71	16, 27	sublt0d 8575	. . . . . . . . 9 ⊢ (𝜑 → ((𝑄 − 𝑅) < 0 ↔ 𝑄 < 𝑅))
72	25, 71	mpbird 167	. . . . . . . 8 ⊢ (𝜑 → (𝑄 − 𝑅) < 0)
73	16, 27	resubcld 8386	. . . . . . . . 9 ⊢ (𝜑 → (𝑄 − 𝑅) ∈ ℝ)
74		ltaddnegr 8430	. . . . . . . . 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 4047	. . . . . 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 9216	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → (𝐴 + 𝐴) < (𝑄 + 𝑅))
83	79, 79, 80	ltaddsub2d 8551	. . . . . . 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 8336	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → ((𝑄 + 𝑅) − 𝐴) = (𝑄 + (𝑅 − 𝐴)))
89	84, 88	breqtrd 4051	. . . . 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 8386	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → (𝑅 − 𝐴) ∈ ℝ)
93	79, 90, 92	absdifltd 11296	. . . . 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 9208	. . . . . . . . . 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 8131	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → 𝐴 < 𝑅)
101	79, 91, 100	ltled 8124	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → 𝐴 ≤ 𝑅)
102	79, 91, 101	abssuble0d 11295	. . . 4 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → (abs‘(𝐴 − 𝑅)) = (𝑅 − 𝐴))
103	94, 102	breqtrrd 4053	. . 3 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → (abs‘(𝐴 − 𝑄)) < (abs‘(𝐴 − 𝑅)))
104	67, 69, 103	ltapd 8643	. 2 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → (abs‘(𝐴 − 𝑄)) # (abs‘(𝐴 − 𝑅)))
105		apdifflemf.ap	. . 3 ⊢ (𝜑 → ((𝑄 + 𝑅) / 2) # 𝐴)
106		reaplt 8593	. . . 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 2160 class class class wbr 4025 ‘cfv 5242 (class class class)co 5906 ℂcc 7856 ℝcr 7857 0cc0 7858 + caddc 7861 · cmul 7863 < clt 8040 − cmin 8176 # cap 8586 / cdiv 8677 2c2 9019 ℚcq 9670 ℝ⁺crp 9705 abscabs 11115

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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4140 ax-sep 4143 ax-nul 4151 ax-pow 4199 ax-pr 4234 ax-un 4458 ax-setind 4561 ax-iinf 4612 ax-cnex 7949 ax-resscn 7950 ax-1cn 7951 ax-1re 7952 ax-icn 7953 ax-addcl 7954 ax-addrcl 7955 ax-mulcl 7956 ax-mulrcl 7957 ax-addcom 7958 ax-mulcom 7959 ax-addass 7960 ax-mulass 7961 ax-distr 7962 ax-i2m1 7963 ax-0lt1 7964 ax-1rid 7965 ax-0id 7966 ax-rnegex 7967 ax-precex 7968 ax-cnre 7969 ax-pre-ltirr 7970 ax-pre-ltwlin 7971 ax-pre-lttrn 7972 ax-pre-apti 7973 ax-pre-ltadd 7974 ax-pre-mulgt0 7975 ax-pre-mulext 7976 ax-arch 7977 ax-caucvg 7978

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 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2758 df-sbc 2982 df-csb 3077 df-dif 3151 df-un 3153 df-in 3155 df-ss 3162 df-nul 3443 df-if 3554 df-pw 3599 df-sn 3620 df-pr 3621 df-op 3623 df-uni 3832 df-int 3867 df-iun 3910 df-br 4026 df-opab 4087 df-mpt 4088 df-tr 4124 df-id 4318 df-po 4321 df-iso 4322 df-iord 4391 df-on 4393 df-ilim 4394 df-suc 4396 df-iom 4615 df-xp 4657 df-rel 4658 df-cnv 4659 df-co 4660 df-dm 4661 df-rn 4662 df-res 4663 df-ima 4664 df-iota 5203 df-fun 5244 df-fn 5245 df-f 5246 df-f1 5247 df-fo 5248 df-f1o 5249 df-fv 5250 df-riota 5861 df-ov 5909 df-oprab 5910 df-mpo 5911 df-1st 6180 df-2nd 6181 df-recs 6345 df-frec 6431 df-pnf 8042 df-mnf 8043 df-xr 8044 df-ltxr 8045 df-le 8046 df-sub 8178 df-neg 8179 df-reap 8580 df-ap 8587 df-div 8678 df-inn 8969 df-2 9027 df-3 9028 df-4 9029 df-n0 9227 df-z 9304 df-uz 9579 df-q 9671 df-rp 9706 df-seqfrec 10505 df-exp 10584 df-cj 10960 df-re 10961 df-im 10962 df-rsqrt 11116 df-abs 11117

This theorem is referenced by: apdiff 15462

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