mirbtwnhl - Metamath Proof Explorer

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Theorem mirbtwnhl 28195

Description: If the center of the point inversion 𝐴 is between two points 𝑋 and 𝑌, then the half lines are mirrored. (Contributed by Thierry Arnoux, 3-Mar-2020.)

**Hypotheses**
Ref	Expression
mirval.p	⊢ 𝑃 = (Base‘𝐺)
mirval.d	⊢ − = (dist‘𝐺)
mirval.i	⊢ 𝐼 = (Itv‘𝐺)
mirval.l	⊢ 𝐿 = (LineG‘𝐺)
mirval.s	⊢ 𝑆 = (pInvG‘𝐺)
mirval.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
mirhl.m	⊢ 𝑀 = (𝑆‘𝐴)
mirhl.k	⊢ 𝐾 = (hlG‘𝐺)
mirhl.a	⊢ (𝜑 → 𝐴 ∈ 𝑃)
mirhl.x	⊢ (𝜑 → 𝑋 ∈ 𝑃)
mirhl.y	⊢ (𝜑 → 𝑌 ∈ 𝑃)
mirhl.z	⊢ (𝜑 → 𝑍 ∈ 𝑃)
mirbtwnhl.1	⊢ (𝜑 → 𝑋 ≠ 𝐴)
mirbtwnhl.2	⊢ (𝜑 → 𝑌 ≠ 𝐴)
mirbtwnhl.3	⊢ (𝜑 → 𝐴 ∈ (𝑋𝐼𝑌))

**Assertion**
Ref	Expression
mirbtwnhl	⊢ (𝜑 → (𝑍(𝐾‘𝐴)𝑋 ↔ (𝑀‘𝑍)(𝐾‘𝐴)𝑌))

**Proof of Theorem mirbtwnhl**
Step	Hyp	Ref	Expression
1		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑍 = 𝐴) → 𝑍 = 𝐴)
2		mirval.p	. . . . . 6 ⊢ 𝑃 = (Base‘𝐺)
3		mirval.i	. . . . . 6 ⊢ 𝐼 = (Itv‘𝐺)
4		mirhl.k	. . . . . 6 ⊢ 𝐾 = (hlG‘𝐺)
5		mirhl.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
6		mirhl.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
7		mirval.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
8	2, 3, 4, 5, 6, 5, 7	hleqnid 28123	. . . . 5 ⊢ (𝜑 → ¬ 𝐴(𝐾‘𝐴)𝑋)
9	8	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑍 = 𝐴) → ¬ 𝐴(𝐾‘𝐴)𝑋)
10	1, 9	eqnbrtrd 5167	. . 3 ⊢ ((𝜑 ∧ 𝑍 = 𝐴) → ¬ 𝑍(𝐾‘𝐴)𝑋)
11	1	fveq2d 6896	. . . . 5 ⊢ ((𝜑 ∧ 𝑍 = 𝐴) → (𝑀‘𝑍) = (𝑀‘𝐴))
12		mirval.d	. . . . . . 7 ⊢ − = (dist‘𝐺)
13		mirval.l	. . . . . . 7 ⊢ 𝐿 = (LineG‘𝐺)
14		mirval.s	. . . . . . 7 ⊢ 𝑆 = (pInvG‘𝐺)
15		mirhl.m	. . . . . . 7 ⊢ 𝑀 = (𝑆‘𝐴)
16	2, 12, 3, 13, 14, 7, 5, 15	mircinv 28183	. . . . . 6 ⊢ (𝜑 → (𝑀‘𝐴) = 𝐴)
17	16	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑍 = 𝐴) → (𝑀‘𝐴) = 𝐴)
18	11, 17	eqtrd 2771	. . . 4 ⊢ ((𝜑 ∧ 𝑍 = 𝐴) → (𝑀‘𝑍) = 𝐴)
19		mirhl.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
20	2, 3, 4, 5, 19, 5, 7	hleqnid 28123	. . . . 5 ⊢ (𝜑 → ¬ 𝐴(𝐾‘𝐴)𝑌)
21	20	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑍 = 𝐴) → ¬ 𝐴(𝐾‘𝐴)𝑌)
22	18, 21	eqnbrtrd 5167	. . 3 ⊢ ((𝜑 ∧ 𝑍 = 𝐴) → ¬ (𝑀‘𝑍)(𝐾‘𝐴)𝑌)
23	10, 22	2falsed 375	. 2 ⊢ ((𝜑 ∧ 𝑍 = 𝐴) → (𝑍(𝐾‘𝐴)𝑋 ↔ (𝑀‘𝑍)(𝐾‘𝐴)𝑌))
24		simplr 766	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ 𝑍(𝐾‘𝐴)𝑋) → 𝑍 ≠ 𝐴)
25	24	neneqd 2944	. . . . . 6 ⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ 𝑍(𝐾‘𝐴)𝑋) → ¬ 𝑍 = 𝐴)
26	7	ad3antrrr 727	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ 𝑍(𝐾‘𝐴)𝑋) ∧ (𝑀‘𝑍) = 𝐴) → 𝐺 ∈ TarskiG)
27	5	ad3antrrr 727	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ 𝑍(𝐾‘𝐴)𝑋) ∧ (𝑀‘𝑍) = 𝐴) → 𝐴 ∈ 𝑃)
28		mirhl.z	. . . . . . . 8 ⊢ (𝜑 → 𝑍 ∈ 𝑃)
29	28	ad3antrrr 727	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ 𝑍(𝐾‘𝐴)𝑋) ∧ (𝑀‘𝑍) = 𝐴) → 𝑍 ∈ 𝑃)
30		simpr 484	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ 𝑍(𝐾‘𝐴)𝑋) ∧ (𝑀‘𝑍) = 𝐴) → (𝑀‘𝑍) = 𝐴)
31	16	ad3antrrr 727	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ 𝑍(𝐾‘𝐴)𝑋) ∧ (𝑀‘𝑍) = 𝐴) → (𝑀‘𝐴) = 𝐴)
32	30, 31	eqtr4d 2774	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ 𝑍(𝐾‘𝐴)𝑋) ∧ (𝑀‘𝑍) = 𝐴) → (𝑀‘𝑍) = (𝑀‘𝐴))
33	2, 12, 3, 13, 14, 26, 27, 15, 29, 27, 32	mireq 28180	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ 𝑍(𝐾‘𝐴)𝑋) ∧ (𝑀‘𝑍) = 𝐴) → 𝑍 = 𝐴)
34	25, 33	mtand 813	. . . . 5 ⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ 𝑍(𝐾‘𝐴)𝑋) → ¬ (𝑀‘𝑍) = 𝐴)
35	34	neqned 2946	. . . 4 ⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ 𝑍(𝐾‘𝐴)𝑋) → (𝑀‘𝑍) ≠ 𝐴)
36		mirbtwnhl.2	. . . . 5 ⊢ (𝜑 → 𝑌 ≠ 𝐴)
37	36	ad2antrr 723	. . . 4 ⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ 𝑍(𝐾‘𝐴)𝑋) → 𝑌 ≠ 𝐴)
38	7	ad2antrr 723	. . . . 5 ⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ 𝑍(𝐾‘𝐴)𝑋) → 𝐺 ∈ TarskiG)
39	6	ad2antrr 723	. . . . 5 ⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ 𝑍(𝐾‘𝐴)𝑋) → 𝑋 ∈ 𝑃)
40	5	ad2antrr 723	. . . . 5 ⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ 𝑍(𝐾‘𝐴)𝑋) → 𝐴 ∈ 𝑃)
41	2, 12, 3, 13, 14, 7, 5, 15, 28	mircl 28176	. . . . . 6 ⊢ (𝜑 → (𝑀‘𝑍) ∈ 𝑃)
42	41	ad2antrr 723	. . . . 5 ⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ 𝑍(𝐾‘𝐴)𝑋) → (𝑀‘𝑍) ∈ 𝑃)
43	19	ad2antrr 723	. . . . 5 ⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ 𝑍(𝐾‘𝐴)𝑋) → 𝑌 ∈ 𝑃)
44		mirbtwnhl.1	. . . . . 6 ⊢ (𝜑 → 𝑋 ≠ 𝐴)
45	44	ad2antrr 723	. . . . 5 ⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ 𝑍(𝐾‘𝐴)𝑋) → 𝑋 ≠ 𝐴)
46	28	ad2antrr 723	. . . . . 6 ⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ 𝑍(𝐾‘𝐴)𝑋) → 𝑍 ∈ 𝑃)
47	2, 3, 4, 28, 6, 5, 7	ishlg 28117	. . . . . . . . . 10 ⊢ (𝜑 → (𝑍(𝐾‘𝐴)𝑋 ↔ (𝑍 ≠ 𝐴 ∧ 𝑋 ≠ 𝐴 ∧ (𝑍 ∈ (𝐴𝐼𝑋) ∨ 𝑋 ∈ (𝐴𝐼𝑍)))))
48	47	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑍 ≠ 𝐴) → (𝑍(𝐾‘𝐴)𝑋 ↔ (𝑍 ≠ 𝐴 ∧ 𝑋 ≠ 𝐴 ∧ (𝑍 ∈ (𝐴𝐼𝑋) ∨ 𝑋 ∈ (𝐴𝐼𝑍)))))
49	48	biimpa 476	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ 𝑍(𝐾‘𝐴)𝑋) → (𝑍 ≠ 𝐴 ∧ 𝑋 ≠ 𝐴 ∧ (𝑍 ∈ (𝐴𝐼𝑋) ∨ 𝑋 ∈ (𝐴𝐼𝑍))))
50	49	simp3d 1143	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ 𝑍(𝐾‘𝐴)𝑋) → (𝑍 ∈ (𝐴𝐼𝑋) ∨ 𝑋 ∈ (𝐴𝐼𝑍)))
51	50	orcomd 868	. . . . . 6 ⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ 𝑍(𝐾‘𝐴)𝑋) → (𝑋 ∈ (𝐴𝐼𝑍) ∨ 𝑍 ∈ (𝐴𝐼𝑋)))
52	2, 12, 3, 13, 14, 38, 15, 40, 39, 46, 51	mirconn 28193	. . . . 5 ⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ 𝑍(𝐾‘𝐴)𝑋) → 𝐴 ∈ (𝑋𝐼(𝑀‘𝑍)))
53		mirbtwnhl.3	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (𝑋𝐼𝑌))
54	53	ad2antrr 723	. . . . 5 ⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ 𝑍(𝐾‘𝐴)𝑋) → 𝐴 ∈ (𝑋𝐼𝑌))
55	2, 3, 38, 39, 40, 42, 43, 45, 52, 54	tgbtwnconn2 28091	. . . 4 ⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ 𝑍(𝐾‘𝐴)𝑋) → ((𝑀‘𝑍) ∈ (𝐴𝐼𝑌) ∨ 𝑌 ∈ (𝐴𝐼(𝑀‘𝑍))))
56	2, 3, 4, 41, 19, 5, 7	ishlg 28117	. . . . . 6 ⊢ (𝜑 → ((𝑀‘𝑍)(𝐾‘𝐴)𝑌 ↔ ((𝑀‘𝑍) ≠ 𝐴 ∧ 𝑌 ≠ 𝐴 ∧ ((𝑀‘𝑍) ∈ (𝐴𝐼𝑌) ∨ 𝑌 ∈ (𝐴𝐼(𝑀‘𝑍))))))
57	56	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑍 ≠ 𝐴) → ((𝑀‘𝑍)(𝐾‘𝐴)𝑌 ↔ ((𝑀‘𝑍) ≠ 𝐴 ∧ 𝑌 ≠ 𝐴 ∧ ((𝑀‘𝑍) ∈ (𝐴𝐼𝑌) ∨ 𝑌 ∈ (𝐴𝐼(𝑀‘𝑍))))))
58	57	adantr 480	. . . 4 ⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ 𝑍(𝐾‘𝐴)𝑋) → ((𝑀‘𝑍)(𝐾‘𝐴)𝑌 ↔ ((𝑀‘𝑍) ≠ 𝐴 ∧ 𝑌 ≠ 𝐴 ∧ ((𝑀‘𝑍) ∈ (𝐴𝐼𝑌) ∨ 𝑌 ∈ (𝐴𝐼(𝑀‘𝑍))))))
59	35, 37, 55, 58	mpbir3and 1341	. . 3 ⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ 𝑍(𝐾‘𝐴)𝑋) → (𝑀‘𝑍)(𝐾‘𝐴)𝑌)
60		simplr 766	. . . 4 ⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ (𝑀‘𝑍)(𝐾‘𝐴)𝑌) → 𝑍 ≠ 𝐴)
61	44	ad2antrr 723	. . . 4 ⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ (𝑀‘𝑍)(𝐾‘𝐴)𝑌) → 𝑋 ≠ 𝐴)
62	7	ad2antrr 723	. . . . 5 ⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ (𝑀‘𝑍)(𝐾‘𝐴)𝑌) → 𝐺 ∈ TarskiG)
63	19	ad2antrr 723	. . . . 5 ⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ (𝑀‘𝑍)(𝐾‘𝐴)𝑌) → 𝑌 ∈ 𝑃)
64	5	ad2antrr 723	. . . . 5 ⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ (𝑀‘𝑍)(𝐾‘𝐴)𝑌) → 𝐴 ∈ 𝑃)
65	28	ad2antrr 723	. . . . 5 ⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ (𝑀‘𝑍)(𝐾‘𝐴)𝑌) → 𝑍 ∈ 𝑃)
66	6	ad2antrr 723	. . . . 5 ⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ (𝑀‘𝑍)(𝐾‘𝐴)𝑌) → 𝑋 ∈ 𝑃)
67	36	ad2antrr 723	. . . . 5 ⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ (𝑀‘𝑍)(𝐾‘𝐴)𝑌) → 𝑌 ≠ 𝐴)
68	16	ad2antrr 723	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ (𝑀‘𝑍)(𝐾‘𝐴)𝑌) → (𝑀‘𝐴) = 𝐴)
69	41	ad2antrr 723	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ (𝑀‘𝑍)(𝐾‘𝐴)𝑌) → (𝑀‘𝑍) ∈ 𝑃)
70	2, 12, 3, 13, 14, 62, 64, 15, 63	mircl 28176	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ (𝑀‘𝑍)(𝐾‘𝐴)𝑌) → (𝑀‘𝑌) ∈ 𝑃)
71	57	biimpa 476	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ (𝑀‘𝑍)(𝐾‘𝐴)𝑌) → ((𝑀‘𝑍) ≠ 𝐴 ∧ 𝑌 ≠ 𝐴 ∧ ((𝑀‘𝑍) ∈ (𝐴𝐼𝑌) ∨ 𝑌 ∈ (𝐴𝐼(𝑀‘𝑍)))))
72	71	simp3d 1143	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ (𝑀‘𝑍)(𝐾‘𝐴)𝑌) → ((𝑀‘𝑍) ∈ (𝐴𝐼𝑌) ∨ 𝑌 ∈ (𝐴𝐼(𝑀‘𝑍))))
73	2, 12, 3, 13, 14, 62, 15, 64, 69, 63, 72	mirconn 28193	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ (𝑀‘𝑍)(𝐾‘𝐴)𝑌) → 𝐴 ∈ ((𝑀‘𝑍)𝐼(𝑀‘𝑌)))
74	2, 12, 3, 62, 69, 64, 70, 73	tgbtwncom 28003	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ (𝑀‘𝑍)(𝐾‘𝐴)𝑌) → 𝐴 ∈ ((𝑀‘𝑌)𝐼(𝑀‘𝑍)))
75	68, 74	eqeltrd 2832	. . . . . 6 ⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ (𝑀‘𝑍)(𝐾‘𝐴)𝑌) → (𝑀‘𝐴) ∈ ((𝑀‘𝑌)𝐼(𝑀‘𝑍)))
76	2, 12, 3, 13, 14, 62, 64, 15, 63, 64, 65	mirbtwnb 28187	. . . . . 6 ⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ (𝑀‘𝑍)(𝐾‘𝐴)𝑌) → (𝐴 ∈ (𝑌𝐼𝑍) ↔ (𝑀‘𝐴) ∈ ((𝑀‘𝑌)𝐼(𝑀‘𝑍))))
77	75, 76	mpbird 256	. . . . 5 ⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ (𝑀‘𝑍)(𝐾‘𝐴)𝑌) → 𝐴 ∈ (𝑌𝐼𝑍))
78	2, 12, 3, 7, 6, 5, 19, 53	tgbtwncom 28003	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (𝑌𝐼𝑋))
79	78	ad2antrr 723	. . . . 5 ⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ (𝑀‘𝑍)(𝐾‘𝐴)𝑌) → 𝐴 ∈ (𝑌𝐼𝑋))
80	2, 3, 62, 63, 64, 65, 66, 67, 77, 79	tgbtwnconn2 28091	. . . 4 ⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ (𝑀‘𝑍)(𝐾‘𝐴)𝑌) → (𝑍 ∈ (𝐴𝐼𝑋) ∨ 𝑋 ∈ (𝐴𝐼𝑍)))
81	48	adantr 480	. . . 4 ⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ (𝑀‘𝑍)(𝐾‘𝐴)𝑌) → (𝑍(𝐾‘𝐴)𝑋 ↔ (𝑍 ≠ 𝐴 ∧ 𝑋 ≠ 𝐴 ∧ (𝑍 ∈ (𝐴𝐼𝑋) ∨ 𝑋 ∈ (𝐴𝐼𝑍)))))
82	60, 61, 80, 81	mpbir3and 1341	. . 3 ⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ (𝑀‘𝑍)(𝐾‘𝐴)𝑌) → 𝑍(𝐾‘𝐴)𝑋)
83	59, 82	impbida 798	. 2 ⊢ ((𝜑 ∧ 𝑍 ≠ 𝐴) → (𝑍(𝐾‘𝐴)𝑋 ↔ (𝑀‘𝑍)(𝐾‘𝐴)𝑌))
84	23, 83	pm2.61dane 3028	1 ⊢ (𝜑 → (𝑍(𝐾‘𝐴)𝑋 ↔ (𝑀‘𝑍)(𝐾‘𝐴)𝑌))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∨ wo 844 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 ≠ wne 2939 class class class wbr 5149 ‘cfv 6544 (class class class)co 7412 Basecbs 17149 distcds 17211 TarskiGcstrkg 27942 Itvcitv 27948 LineGclng 27949 hlGchlg 28115 pInvGcmir 28167

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7728 ax-cnex 11169 ax-resscn 11170 ax-1cn 11171 ax-icn 11172 ax-addcl 11173 ax-addrcl 11174 ax-mulcl 11175 ax-mulrcl 11176 ax-mulcom 11177 ax-addass 11178 ax-mulass 11179 ax-distr 11180 ax-i2m1 11181 ax-1ne0 11182 ax-1rid 11183 ax-rnegex 11184 ax-rrecex 11185 ax-cnre 11186 ax-pre-lttri 11187 ax-pre-lttrn 11188 ax-pre-ltadd 11189 ax-pre-mulgt0 11190

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7859 df-1st 7978 df-2nd 7979 df-frecs 8269 df-wrecs 8300 df-recs 8374 df-rdg 8413 df-1o 8469 df-oadd 8473 df-er 8706 df-pm 8826 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-dju 9899 df-card 9937 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-nn 12218 df-2 12280 df-3 12281 df-n0 12478 df-xnn0 12550 df-z 12564 df-uz 12828 df-fz 13490 df-fzo 13633 df-hash 14296 df-word 14470 df-concat 14526 df-s1 14551 df-s2 14804 df-s3 14805 df-trkgc 27963 df-trkgb 27964 df-trkgcb 27965 df-trkg 27968 df-cgrg 28026 df-hlg 28116 df-mir 28168

This theorem is referenced by: opphllem6 28267

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