mirbtwnhl - Metamath Proof Explorer

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Theorem mirbtwnhl 28837

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 488	. . . 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 28765	. . . . 5 ⊢ (𝜑 → ¬ 𝐴(𝐾‘𝐴)𝑋)
9	8	adantr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑍 = 𝐴) → ¬ 𝐴(𝐾‘𝐴)𝑋)
10	1, 9	eqnbrtrd 5115	. . 3 ⊢ ((𝜑 ∧ 𝑍 = 𝐴) → ¬ 𝑍(𝐾‘𝐴)𝑋)
11	1	fveq2d 6866	. . . . 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 28825	. . . . . 6 ⊢ (𝜑 → (𝑀‘𝐴) = 𝐴)
17	16	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑍 = 𝐴) → (𝑀‘𝐴) = 𝐴)
18	11, 17	eqtrd 2796	. . . 4 ⊢ ((𝜑 ∧ 𝑍 = 𝐴) → (𝑀‘𝑍) = 𝐴)
19		mirhl.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
20	2, 3, 4, 5, 19, 5, 7	hleqnid 28765	. . . . 5 ⊢ (𝜑 → ¬ 𝐴(𝐾‘𝐴)𝑌)
21	20	adantr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑍 = 𝐴) → ¬ 𝐴(𝐾‘𝐴)𝑌)
22	18, 21	eqnbrtrd 5115	. . 3 ⊢ ((𝜑 ∧ 𝑍 = 𝐴) → ¬ (𝑀‘𝑍)(𝐾‘𝐴)𝑌)
23	10, 22	2falsed 378	. 2 ⊢ ((𝜑 ∧ 𝑍 = 𝐴) → (𝑍(𝐾‘𝐴)𝑋 ↔ (𝑀‘𝑍)(𝐾‘𝐴)𝑌))
24		simplr 778	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ 𝑍(𝐾‘𝐴)𝑋) → 𝑍 ≠ 𝐴)
25	24	neneqd 2961	. . . . . 6 ⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ 𝑍(𝐾‘𝐴)𝑋) → ¬ 𝑍 = 𝐴)
26	7	ad3antrrr 740	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ 𝑍(𝐾‘𝐴)𝑋) ∧ (𝑀‘𝑍) = 𝐴) → 𝐺 ∈ TarskiG)
27	5	ad3antrrr 740	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ 𝑍(𝐾‘𝐴)𝑋) ∧ (𝑀‘𝑍) = 𝐴) → 𝐴 ∈ 𝑃)
28		mirhl.z	. . . . . . . 8 ⊢ (𝜑 → 𝑍 ∈ 𝑃)
29	28	ad3antrrr 740	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ 𝑍(𝐾‘𝐴)𝑋) ∧ (𝑀‘𝑍) = 𝐴) → 𝑍 ∈ 𝑃)
30		simpr 488	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ 𝑍(𝐾‘𝐴)𝑋) ∧ (𝑀‘𝑍) = 𝐴) → (𝑀‘𝑍) = 𝐴)
31	16	ad3antrrr 740	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ 𝑍(𝐾‘𝐴)𝑋) ∧ (𝑀‘𝑍) = 𝐴) → (𝑀‘𝐴) = 𝐴)
32	30, 31	eqtr4d 2799	. . . . . . 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 28822	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ 𝑍(𝐾‘𝐴)𝑋) ∧ (𝑀‘𝑍) = 𝐴) → 𝑍 = 𝐴)
34	25, 33	mtand 825	. . . . 5 ⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ 𝑍(𝐾‘𝐴)𝑋) → ¬ (𝑀‘𝑍) = 𝐴)
35	34	neqned 2963	. . . 4 ⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ 𝑍(𝐾‘𝐴)𝑋) → (𝑀‘𝑍) ≠ 𝐴)
36		mirbtwnhl.2	. . . . 5 ⊢ (𝜑 → 𝑌 ≠ 𝐴)
37	36	ad2antrr 736	. . . 4 ⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ 𝑍(𝐾‘𝐴)𝑋) → 𝑌 ≠ 𝐴)
38	7	ad2antrr 736	. . . . 5 ⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ 𝑍(𝐾‘𝐴)𝑋) → 𝐺 ∈ TarskiG)
39	6	ad2antrr 736	. . . . 5 ⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ 𝑍(𝐾‘𝐴)𝑋) → 𝑋 ∈ 𝑃)
40	5	ad2antrr 736	. . . . 5 ⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ 𝑍(𝐾‘𝐴)𝑋) → 𝐴 ∈ 𝑃)
41	2, 12, 3, 13, 14, 7, 5, 15, 28	mircl 28818	. . . . . 6 ⊢ (𝜑 → (𝑀‘𝑍) ∈ 𝑃)
42	41	ad2antrr 736	. . . . 5 ⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ 𝑍(𝐾‘𝐴)𝑋) → (𝑀‘𝑍) ∈ 𝑃)
43	19	ad2antrr 736	. . . . 5 ⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ 𝑍(𝐾‘𝐴)𝑋) → 𝑌 ∈ 𝑃)
44		mirbtwnhl.1	. . . . . 6 ⊢ (𝜑 → 𝑋 ≠ 𝐴)
45	44	ad2antrr 736	. . . . 5 ⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ 𝑍(𝐾‘𝐴)𝑋) → 𝑋 ≠ 𝐴)
46	28	ad2antrr 736	. . . . . 6 ⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ 𝑍(𝐾‘𝐴)𝑋) → 𝑍 ∈ 𝑃)
47	2, 3, 4, 28, 6, 5, 7	ishlg 28759	. . . . . . . . . 10 ⊢ (𝜑 → (𝑍(𝐾‘𝐴)𝑋 ↔ (𝑍 ≠ 𝐴 ∧ 𝑋 ≠ 𝐴 ∧ (𝑍 ∈ (𝐴𝐼𝑋) ∨ 𝑋 ∈ (𝐴𝐼𝑍)))))
48	47	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑍 ≠ 𝐴) → (𝑍(𝐾‘𝐴)𝑋 ↔ (𝑍 ≠ 𝐴 ∧ 𝑋 ≠ 𝐴 ∧ (𝑍 ∈ (𝐴𝐼𝑋) ∨ 𝑋 ∈ (𝐴𝐼𝑍)))))
49	48	biimpa 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ 𝑍(𝐾‘𝐴)𝑋) → (𝑍 ≠ 𝐴 ∧ 𝑋 ≠ 𝐴 ∧ (𝑍 ∈ (𝐴𝐼𝑋) ∨ 𝑋 ∈ (𝐴𝐼𝑍))))
50	49	simp3d 1156	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ 𝑍(𝐾‘𝐴)𝑋) → (𝑍 ∈ (𝐴𝐼𝑋) ∨ 𝑋 ∈ (𝐴𝐼𝑍)))
51	50	orcomd 882	. . . . . 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 28835	. . . . 5 ⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ 𝑍(𝐾‘𝐴)𝑋) → 𝐴 ∈ (𝑋𝐼(𝑀‘𝑍)))
53		mirbtwnhl.3	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (𝑋𝐼𝑌))
54	53	ad2antrr 736	. . . . 5 ⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ 𝑍(𝐾‘𝐴)𝑋) → 𝐴 ∈ (𝑋𝐼𝑌))
55	2, 3, 38, 39, 40, 42, 43, 45, 52, 54	tgbtwnconn2 28733	. . . 4 ⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ 𝑍(𝐾‘𝐴)𝑋) → ((𝑀‘𝑍) ∈ (𝐴𝐼𝑌) ∨ 𝑌 ∈ (𝐴𝐼(𝑀‘𝑍))))
56	2, 3, 4, 41, 19, 5, 7	ishlg 28759	. . . . . 6 ⊢ (𝜑 → ((𝑀‘𝑍)(𝐾‘𝐴)𝑌 ↔ ((𝑀‘𝑍) ≠ 𝐴 ∧ 𝑌 ≠ 𝐴 ∧ ((𝑀‘𝑍) ∈ (𝐴𝐼𝑌) ∨ 𝑌 ∈ (𝐴𝐼(𝑀‘𝑍))))))
57	56	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑍 ≠ 𝐴) → ((𝑀‘𝑍)(𝐾‘𝐴)𝑌 ↔ ((𝑀‘𝑍) ≠ 𝐴 ∧ 𝑌 ≠ 𝐴 ∧ ((𝑀‘𝑍) ∈ (𝐴𝐼𝑌) ∨ 𝑌 ∈ (𝐴𝐼(𝑀‘𝑍))))))
58	57	adantr 484	. . . 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 1355	. . 3 ⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ 𝑍(𝐾‘𝐴)𝑋) → (𝑀‘𝑍)(𝐾‘𝐴)𝑌)
60		simplr 778	. . . 4 ⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ (𝑀‘𝑍)(𝐾‘𝐴)𝑌) → 𝑍 ≠ 𝐴)
61	44	ad2antrr 736	. . . 4 ⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ (𝑀‘𝑍)(𝐾‘𝐴)𝑌) → 𝑋 ≠ 𝐴)
62	7	ad2antrr 736	. . . . 5 ⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ (𝑀‘𝑍)(𝐾‘𝐴)𝑌) → 𝐺 ∈ TarskiG)
63	19	ad2antrr 736	. . . . 5 ⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ (𝑀‘𝑍)(𝐾‘𝐴)𝑌) → 𝑌 ∈ 𝑃)
64	5	ad2antrr 736	. . . . 5 ⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ (𝑀‘𝑍)(𝐾‘𝐴)𝑌) → 𝐴 ∈ 𝑃)
65	28	ad2antrr 736	. . . . 5 ⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ (𝑀‘𝑍)(𝐾‘𝐴)𝑌) → 𝑍 ∈ 𝑃)
66	6	ad2antrr 736	. . . . 5 ⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ (𝑀‘𝑍)(𝐾‘𝐴)𝑌) → 𝑋 ∈ 𝑃)
67	36	ad2antrr 736	. . . . 5 ⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ (𝑀‘𝑍)(𝐾‘𝐴)𝑌) → 𝑌 ≠ 𝐴)
68	16	ad2antrr 736	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ (𝑀‘𝑍)(𝐾‘𝐴)𝑌) → (𝑀‘𝐴) = 𝐴)
69	41	ad2antrr 736	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ (𝑀‘𝑍)(𝐾‘𝐴)𝑌) → (𝑀‘𝑍) ∈ 𝑃)
70	2, 12, 3, 13, 14, 62, 64, 15, 63	mircl 28818	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ (𝑀‘𝑍)(𝐾‘𝐴)𝑌) → (𝑀‘𝑌) ∈ 𝑃)
71	57	biimpa 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ (𝑀‘𝑍)(𝐾‘𝐴)𝑌) → ((𝑀‘𝑍) ≠ 𝐴 ∧ 𝑌 ≠ 𝐴 ∧ ((𝑀‘𝑍) ∈ (𝐴𝐼𝑌) ∨ 𝑌 ∈ (𝐴𝐼(𝑀‘𝑍)))))
72	71	simp3d 1156	. . . . . . . . 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 28835	. . . . . . . 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 28645	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ (𝑀‘𝑍)(𝐾‘𝐴)𝑌) → 𝐴 ∈ ((𝑀‘𝑌)𝐼(𝑀‘𝑍)))
75	68, 74	eqeltrd 2861	. . . . . 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 28829	. . . . . 6 ⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ (𝑀‘𝑍)(𝐾‘𝐴)𝑌) → (𝐴 ∈ (𝑌𝐼𝑍) ↔ (𝑀‘𝐴) ∈ ((𝑀‘𝑌)𝐼(𝑀‘𝑍))))
77	75, 76	mpbird 259	. . . . 5 ⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ (𝑀‘𝑍)(𝐾‘𝐴)𝑌) → 𝐴 ∈ (𝑌𝐼𝑍))
78	2, 12, 3, 7, 6, 5, 19, 53	tgbtwncom 28645	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (𝑌𝐼𝑋))
79	78	ad2antrr 736	. . . . 5 ⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ (𝑀‘𝑍)(𝐾‘𝐴)𝑌) → 𝐴 ∈ (𝑌𝐼𝑋))
80	2, 3, 62, 63, 64, 65, 66, 67, 77, 79	tgbtwnconn2 28733	. . . 4 ⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ (𝑀‘𝑍)(𝐾‘𝐴)𝑌) → (𝑍 ∈ (𝐴𝐼𝑋) ∨ 𝑋 ∈ (𝐴𝐼𝑍)))
81	48	adantr 484	. . . 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 1355	. . 3 ⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ (𝑀‘𝑍)(𝐾‘𝐴)𝑌) → 𝑍(𝐾‘𝐴)𝑋)
83	59, 82	impbida 810	. 2 ⊢ ((𝜑 ∧ 𝑍 ≠ 𝐴) → (𝑍(𝐾‘𝐴)𝑋 ↔ (𝑀‘𝑍)(𝐾‘𝐴)𝑌))
84	23, 83	pm2.61dane 3043	1 ⊢ (𝜑 → (𝑍(𝐾‘𝐴)𝑋 ↔ (𝑀‘𝑍)(𝐾‘𝐴)𝑌))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 399 ∨ wo 858 ∧ w3a 1097 = wceq 1559 ∈ wcel 2141 ≠ wne 2956 class class class wbr 5097 ‘cfv 6516 (class class class)co 7391 Basecbs 17236 distcds 17286 TarskiGcstrkg 28584 Itvcitv 28590 LineGclng 28591 hlGchlg 28757 pInvGcmir 28809

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5224 ax-sep 5243 ax-nul 5253 ax-pow 5319 ax-pr 5387 ax-un 7713 ax-cnex 11123 ax-resscn 11124 ax-1cn 11125 ax-icn 11126 ax-addcl 11127 ax-addrcl 11128 ax-mulcl 11129 ax-mulrcl 11130 ax-mulcom 11131 ax-addass 11132 ax-mulass 11133 ax-distr 11134 ax-i2m1 11135 ax-1ne0 11136 ax-1rid 11137 ax-rnegex 11138 ax-rrecex 11139 ax-cnre 11140 ax-pre-lttri 11141 ax-pre-lttrn 11142 ax-pre-ltadd 11143 ax-pre-mulgt0 11144

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4863 df-int 4903 df-iun 4948 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5538 df-eprel 5543 df-po 5551 df-so 5552 df-fr 5596 df-we 5598 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-res 5655 df-ima 5656 df-pred 6283 df-ord 6344 df-on 6345 df-lim 6346 df-suc 6347 df-iota 6472 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-riota 7348 df-ov 7394 df-oprab 7395 df-mpo 7396 df-om 7842 df-1st 7965 df-2nd 7966 df-frecs 8256 df-wrecs 8287 df-recs 8336 df-rdg 8375 df-1o 8431 df-oadd 8435 df-er 8672 df-pm 8805 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-dju 9853 df-card 9891 df-pnf 11212 df-mnf 11213 df-xr 11214 df-ltxr 11215 df-le 11216 df-sub 11410 df-neg 11411 df-nn 12205 df-2 12274 df-3 12275 df-n0 12476 df-xnn0 12549 df-z 12563 df-uz 12834 df-fz 13507 df-fzo 13654 df-hash 14338 df-word 14521 df-concat 14578 df-s1 14604 df-s2 14855 df-s3 14856 df-trkgc 28605 df-trkgb 28606 df-trkgcb 28607 df-trkg 28610 df-cgrg 28668 df-hlg 28758 df-mir 28810

This theorem is referenced by: opphllem6 28909

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