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 13925

Description: Lemma for apdiff 13927. 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 7927	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℂ)
3		apdifflemf.r	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ ℚ)
4		qcn 9572	. . . . . . 7 ⊢ (𝑅 ∈ ℚ → 𝑅 ∈ ℂ)
5	3, 4	syl 14	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ ℂ)
6	2, 5	subcld 8209	. . . . 5 ⊢ (𝜑 → (𝐴 − 𝑅) ∈ ℂ)
7	6	adantr 274	. . . 4 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (𝐴 − 𝑅) ∈ ℂ)
8	7	abscld 11123	. . 3 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (abs‘(𝐴 − 𝑅)) ∈ ℝ)
9		apdifflemf.q	. . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ ℚ)
10		qcn 9572	. . . . . . 7 ⊢ (𝑄 ∈ ℚ → 𝑄 ∈ ℂ)
11	9, 10	syl 14	. . . . . 6 ⊢ (𝜑 → 𝑄 ∈ ℂ)
12	2, 11	subcld 8209	. . . . 5 ⊢ (𝜑 → (𝐴 − 𝑄) ∈ ℂ)
13	12	abscld 11123	. . . 4 ⊢ (𝜑 → (abs‘(𝐴 − 𝑄)) ∈ ℝ)
14	13	adantr 274	. . 3 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (abs‘(𝐴 − 𝑄)) ∈ ℝ)
15		qre 9563	. . . . . . . . . 10 ⊢ (𝑄 ∈ ℚ → 𝑄 ∈ ℝ)
16	9, 15	syl 14	. . . . . . . . 9 ⊢ (𝜑 → 𝑄 ∈ ℝ)
17	16	adantr 274	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → 𝑄 ∈ ℝ)
18	1	adantr 274	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → 𝐴 ∈ ℝ)
19		qaddcl 9573	. . . . . . . . . . . . . 14 ⊢ ((𝑄 ∈ ℚ ∧ 𝑅 ∈ ℚ) → (𝑄 + 𝑅) ∈ ℚ)
20	9, 3, 19	syl2anc 409	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑄 + 𝑅) ∈ ℚ)
21		qre 9563	. . . . . . . . . . . . 13 ⊢ ((𝑄 + 𝑅) ∈ ℚ → (𝑄 + 𝑅) ∈ ℝ)
22	20, 21	syl 14	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑄 + 𝑅) ∈ ℝ)
23	22	rehalfcld 9103	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑄 + 𝑅) / 2) ∈ ℝ)
24	23	adantr 274	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → ((𝑄 + 𝑅) / 2) ∈ ℝ)
25		apdifflemf.qr	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑄 < 𝑅)
26		qre 9563	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ ℚ → 𝑅 ∈ ℝ)
27	3, 26	syl 14	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 ∈ ℝ)
28		avglt1 9095	. . . . . . . . . . . . 13 ⊢ ((𝑄 ∈ ℝ ∧ 𝑅 ∈ ℝ) → (𝑄 < 𝑅 ↔ 𝑄 < ((𝑄 + 𝑅) / 2)))
29	16, 27, 28	syl2anc 409	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑄 < 𝑅 ↔ 𝑄 < ((𝑄 + 𝑅) / 2)))
30	25, 29	mpbid 146	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑄 < ((𝑄 + 𝑅) / 2))
31	30	adantr 274	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → 𝑄 < ((𝑄 + 𝑅) / 2))
32		simpr 109	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → ((𝑄 + 𝑅) / 2) < 𝐴)
33	17, 24, 18, 31, 32	lttrd 8024	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → 𝑄 < 𝐴)
34	17, 18, 33	ltled 8017	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → 𝑄 ≤ 𝐴)
35	17, 18, 34	abssubge0d 11118	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (abs‘(𝐴 − 𝑄)) = (𝐴 − 𝑄))
36	35	oveq2d 5858	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (𝑅 − (abs‘(𝐴 − 𝑄))) = (𝑅 − (𝐴 − 𝑄)))
37	5	adantr 274	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → 𝑅 ∈ ℂ)
38	2	adantr 274	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → 𝐴 ∈ ℂ)
39	11	adantr 274	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → 𝑄 ∈ ℂ)
40	37, 38, 39	subsub3d 8239	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (𝑅 − (𝐴 − 𝑄)) = ((𝑅 + 𝑄) − 𝐴))
41	37, 39	addcomd 8049	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (𝑅 + 𝑄) = (𝑄 + 𝑅))
42	41	oveq1d 5857	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → ((𝑅 + 𝑄) − 𝐴) = ((𝑄 + 𝑅) − 𝐴))
43	36, 40, 42	3eqtrd 2202	. . . . 5 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (𝑅 − (abs‘(𝐴 − 𝑄))) = ((𝑄 + 𝑅) − 𝐴))
44	22	adantr 274	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (𝑄 + 𝑅) ∈ ℝ)
45		2rp 9594	. . . . . . . . . 10 ⊢ 2 ∈ ℝ⁺
46	45	a1i 9	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → 2 ∈ ℝ⁺)
47	44, 18, 46	ltdivmuld 9684	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (((𝑄 + 𝑅) / 2) < 𝐴 ↔ (𝑄 + 𝑅) < (2 · 𝐴)))
48	32, 47	mpbid 146	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (𝑄 + 𝑅) < (2 · 𝐴))
49	38	2timesd 9099	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (2 · 𝐴) = (𝐴 + 𝐴))
50	48, 49	breqtrd 4008	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (𝑄 + 𝑅) < (𝐴 + 𝐴))
51	44, 18, 18	ltsubaddd 8439	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (((𝑄 + 𝑅) − 𝐴) < 𝐴 ↔ (𝑄 + 𝑅) < (𝐴 + 𝐴)))
52	50, 51	mpbird 166	. . . . 5 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → ((𝑄 + 𝑅) − 𝐴) < 𝐴)
53	43, 52	eqbrtrd 4004	. . . 4 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (𝑅 − (abs‘(𝐴 − 𝑄))) < 𝐴)
54	25	adantr 274	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → 𝑄 < 𝑅)
55	27	adantr 274	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → 𝑅 ∈ ℝ)
56		difrp 9628	. . . . . . . 8 ⊢ ((𝑄 ∈ ℝ ∧ 𝑅 ∈ ℝ) → (𝑄 < 𝑅 ↔ (𝑅 − 𝑄) ∈ ℝ⁺))
57	17, 55, 56	syl2anc 409	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (𝑄 < 𝑅 ↔ (𝑅 − 𝑄) ∈ ℝ⁺))
58	54, 57	mpbid 146	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (𝑅 − 𝑄) ∈ ℝ⁺)
59	18, 58	ltaddrpd 9666	. . . . 5 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → 𝐴 < (𝐴 + (𝑅 − 𝑄)))
60	35	oveq2d 5858	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (𝑅 + (abs‘(𝐴 − 𝑄))) = (𝑅 + (𝐴 − 𝑄)))
61	37, 38, 39	addsub12d 8232	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (𝑅 + (𝐴 − 𝑄)) = (𝐴 + (𝑅 − 𝑄)))
62	60, 61	eqtrd 2198	. . . . 5 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (𝑅 + (abs‘(𝐴 − 𝑄))) = (𝐴 + (𝑅 − 𝑄)))
63	59, 62	breqtrrd 4010	. . . 4 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → 𝐴 < (𝑅 + (abs‘(𝐴 − 𝑄))))
64	18, 55, 14	absdifltd 11120	. . . 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 934	. . 3 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (abs‘(𝐴 − 𝑅)) < (abs‘(𝐴 − 𝑄)))
66	8, 14, 65	gtapd 8535	. 2 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (abs‘(𝐴 − 𝑄)) # (abs‘(𝐴 − 𝑅)))
67	13	adantr 274	. . 3 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → (abs‘(𝐴 − 𝑄)) ∈ ℝ)
68	6	adantr 274	. . . 4 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → (𝐴 − 𝑅) ∈ ℂ)
69	68	abscld 11123	. . 3 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → (abs‘(𝐴 − 𝑅)) ∈ ℝ)
70	11, 5, 2	subsubd 8237	. . . . . . 7 ⊢ (𝜑 → (𝑄 − (𝑅 − 𝐴)) = ((𝑄 − 𝑅) + 𝐴))
71	16, 27	sublt0d 8468	. . . . . . . . 9 ⊢ (𝜑 → ((𝑄 − 𝑅) < 0 ↔ 𝑄 < 𝑅))
72	25, 71	mpbird 166	. . . . . . . 8 ⊢ (𝜑 → (𝑄 − 𝑅) < 0)
73	16, 27	resubcld 8279	. . . . . . . . 9 ⊢ (𝜑 → (𝑄 − 𝑅) ∈ ℝ)
74		ltaddnegr 8323	. . . . . . . . 9 ⊢ (((𝑄 − 𝑅) ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((𝑄 − 𝑅) < 0 ↔ ((𝑄 − 𝑅) + 𝐴) < 𝐴))
75	73, 1, 74	syl2anc 409	. . . . . . . 8 ⊢ (𝜑 → ((𝑄 − 𝑅) < 0 ↔ ((𝑄 − 𝑅) + 𝐴) < 𝐴))
76	72, 75	mpbid 146	. . . . . . 7 ⊢ (𝜑 → ((𝑄 − 𝑅) + 𝐴) < 𝐴)
77	70, 76	eqbrtrd 4004	. . . . . 6 ⊢ (𝜑 → (𝑄 − (𝑅 − 𝐴)) < 𝐴)
78	77	adantr 274	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → (𝑄 − (𝑅 − 𝐴)) < 𝐴)
79	1	adantr 274	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → 𝐴 ∈ ℝ)
80	22	adantr 274	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → (𝑄 + 𝑅) ∈ ℝ)
81		simpr 109	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → 𝐴 < ((𝑄 + 𝑅) / 2))
82	79, 79, 80, 81, 81	lt2halvesd 9104	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → (𝐴 + 𝐴) < (𝑄 + 𝑅))
83	79, 79, 80	ltaddsub2d 8444	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → ((𝐴 + 𝐴) < (𝑄 + 𝑅) ↔ 𝐴 < ((𝑄 + 𝑅) − 𝐴)))
84	82, 83	mpbid 146	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → 𝐴 < ((𝑄 + 𝑅) − 𝐴))
85	11	adantr 274	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → 𝑄 ∈ ℂ)
86	5	adantr 274	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → 𝑅 ∈ ℂ)
87	2	adantr 274	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → 𝐴 ∈ ℂ)
88	85, 86, 87	addsubassd 8229	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → ((𝑄 + 𝑅) − 𝐴) = (𝑄 + (𝑅 − 𝐴)))
89	84, 88	breqtrd 4008	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → 𝐴 < (𝑄 + (𝑅 − 𝐴)))
90	16	adantr 274	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → 𝑄 ∈ ℝ)
91	27	adantr 274	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → 𝑅 ∈ ℝ)
92	91, 79	resubcld 8279	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → (𝑅 − 𝐴) ∈ ℝ)
93	79, 90, 92	absdifltd 11120	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → ((abs‘(𝐴 − 𝑄)) < (𝑅 − 𝐴) ↔ ((𝑄 − (𝑅 − 𝐴)) < 𝐴 ∧ 𝐴 < (𝑄 + (𝑅 − 𝐴)))))
94	78, 89, 93	mpbir2and 934	. . . 4 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → (abs‘(𝐴 − 𝑄)) < (𝑅 − 𝐴))
95	23	adantr 274	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → ((𝑄 + 𝑅) / 2) ∈ ℝ)
96		avglt2 9096	. . . . . . . . . 10 ⊢ ((𝑄 ∈ ℝ ∧ 𝑅 ∈ ℝ) → (𝑄 < 𝑅 ↔ ((𝑄 + 𝑅) / 2) < 𝑅))
97	16, 27, 96	syl2anc 409	. . . . . . . . 9 ⊢ (𝜑 → (𝑄 < 𝑅 ↔ ((𝑄 + 𝑅) / 2) < 𝑅))
98	25, 97	mpbid 146	. . . . . . . 8 ⊢ (𝜑 → ((𝑄 + 𝑅) / 2) < 𝑅)
99	98	adantr 274	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → ((𝑄 + 𝑅) / 2) < 𝑅)
100	79, 95, 91, 81, 99	lttrd 8024	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → 𝐴 < 𝑅)
101	79, 91, 100	ltled 8017	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → 𝐴 ≤ 𝑅)
102	79, 91, 101	abssuble0d 11119	. . . 4 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → (abs‘(𝐴 − 𝑅)) = (𝑅 − 𝐴))
103	94, 102	breqtrrd 4010	. . 3 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → (abs‘(𝐴 − 𝑄)) < (abs‘(𝐴 − 𝑅)))
104	67, 69, 103	ltapd 8536	. 2 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → (abs‘(𝐴 − 𝑄)) # (abs‘(𝐴 − 𝑅)))
105		apdifflemf.ap	. . 3 ⊢ (𝜑 → ((𝑄 + 𝑅) / 2) # 𝐴)
106		reaplt 8486	. . . 4 ⊢ ((((𝑄 + 𝑅) / 2) ∈ ℝ ∧ 𝐴 ∈ ℝ) → (((𝑄 + 𝑅) / 2) # 𝐴 ↔ (((𝑄 + 𝑅) / 2) < 𝐴 ∨ 𝐴 < ((𝑄 + 𝑅) / 2))))
107	23, 1, 106	syl2anc 409	. . 3 ⊢ (𝜑 → (((𝑄 + 𝑅) / 2) # 𝐴 ↔ (((𝑄 + 𝑅) / 2) < 𝐴 ∨ 𝐴 < ((𝑄 + 𝑅) / 2))))
108	105, 107	mpbid 146	. 2 ⊢ (𝜑 → (((𝑄 + 𝑅) / 2) < 𝐴 ∨ 𝐴 < ((𝑄 + 𝑅) / 2)))
109	66, 104, 108	mpjaodan 788	1 ⊢ (𝜑 → (abs‘(𝐴 − 𝑄)) # (abs‘(𝐴 − 𝑅)))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∨ wo 698 ∈ wcel 2136 class class class wbr 3982 ‘cfv 5188 (class class class)co 5842 ℂcc 7751 ℝcr 7752 0cc0 7753 + caddc 7756 · cmul 7758 < clt 7933 − cmin 8069 # cap 8479 / cdiv 8568 2c2 8908 ℚcq 9557 ℝ⁺crp 9589 abscabs 10939

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-coll 4097 ax-sep 4100 ax-nul 4108 ax-pow 4153 ax-pr 4187 ax-un 4411 ax-setind 4514 ax-iinf 4565 ax-cnex 7844 ax-resscn 7845 ax-1cn 7846 ax-1re 7847 ax-icn 7848 ax-addcl 7849 ax-addrcl 7850 ax-mulcl 7851 ax-mulrcl 7852 ax-addcom 7853 ax-mulcom 7854 ax-addass 7855 ax-mulass 7856 ax-distr 7857 ax-i2m1 7858 ax-0lt1 7859 ax-1rid 7860 ax-0id 7861 ax-rnegex 7862 ax-precex 7863 ax-cnre 7864 ax-pre-ltirr 7865 ax-pre-ltwlin 7866 ax-pre-lttrn 7867 ax-pre-apti 7868 ax-pre-ltadd 7869 ax-pre-mulgt0 7870 ax-pre-mulext 7871 ax-arch 7872 ax-caucvg 7873

This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 969 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-nel 2432 df-ral 2449 df-rex 2450 df-reu 2451 df-rmo 2452 df-rab 2453 df-v 2728 df-sbc 2952 df-csb 3046 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-nul 3410 df-if 3521 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-int 3825 df-iun 3868 df-br 3983 df-opab 4044 df-mpt 4045 df-tr 4081 df-id 4271 df-po 4274 df-iso 4275 df-iord 4344 df-on 4346 df-ilim 4347 df-suc 4349 df-iom 4568 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-rn 4615 df-res 4616 df-ima 4617 df-iota 5153 df-fun 5190 df-fn 5191 df-f 5192 df-f1 5193 df-fo 5194 df-f1o 5195 df-fv 5196 df-riota 5798 df-ov 5845 df-oprab 5846 df-mpo 5847 df-1st 6108 df-2nd 6109 df-recs 6273 df-frec 6359 df-pnf 7935 df-mnf 7936 df-xr 7937 df-ltxr 7938 df-le 7939 df-sub 8071 df-neg 8072 df-reap 8473 df-ap 8480 df-div 8569 df-inn 8858 df-2 8916 df-3 8917 df-4 8918 df-n0 9115 df-z 9192 df-uz 9467 df-q 9558 df-rp 9590 df-seqfrec 10381 df-exp 10455 df-cj 10784 df-re 10785 df-im 10786 df-rsqrt 10940 df-abs 10941

This theorem is referenced by: apdiff 13927

W3C validator