Theorem tgextahpl 33099

Description: The generic open half-plane core. From a point 𝑋 between 𝐴 and 𝐷 that is on the half-line from 𝐶 through 𝑊, with 𝑊 off line 𝐴𝐶, both 𝑋 and 𝐷 and both 𝑋 and 𝑊 land on the same side of line 𝐴𝐶 (two colhp 29278 applications), and hpgcom 29275 / hpgtr 29276 compose them. 𝑋 ≠ 𝐶 is read off .xk by hlne1 29101, so no 𝑥 = 𝐶 case reaches here. The $d s are on proof dummies only -- colhp 29278's class abstraction needs its own bound variables -- so nothing propagates to callers. (Contributed by JAG, 12-Aug-2026.)

Hypotheses

Ref	Expression
tgextahpl.p	⊢ 𝑃 = (Base‘𝐺)
tgextahpl.m	⊢ − = (dist‘𝐺)
tgextahpl.i	⊢ 𝐼 = (Itv‘𝐺)
tgextahpl.l	⊢ 𝐿 = (LineG‘𝐺)
tgextahpl.k	⊢ 𝐾 = (hlG‘𝐺)
tgextahpl.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
tgextahpl.a	⊢ (𝜑 → 𝐴 ∈ 𝑃)
tgextahpl.c	⊢ (𝜑 → 𝐶 ∈ 𝑃)
tgextahpl.d	⊢ (𝜑 → 𝐷 ∈ 𝑃)
tgextahpl.w	⊢ (𝜑 → 𝑊 ∈ 𝑃)
tgextahpl.x	⊢ (𝜑 → 𝑋 ∈ 𝑃)
tgextahpl.ncd	⊢ (𝜑 → ¬ (𝐴 ∈ (𝐶𝐿𝐷) ∨ 𝐶 = 𝐷))
tgextahpl.nw	⊢ (𝜑 → ¬ 𝑊 ∈ (𝐴𝐿𝐶))
tgextahpl.xad	⊢ (𝜑 → 𝑋 ∈ (𝐴𝐼𝐷))
tgextahpl.xk	⊢ (𝜑 → 𝑋(𝐾‘𝐶)𝑊)

Assertion

Ref	Expression
tgextahpl	⊢ (𝜑 → 𝑊((hpG‘𝐺)‘(𝐴𝐿𝐶))𝐷)

Proof of Theorem tgextahpl

Dummy variables 𝑎 𝑏 𝑡 𝑣 𝑢 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		tgextahpl.p	. 2 ⊢ 𝑃 = (Base‘𝐺)
2		tgextahpl.i	. 2 ⊢ 𝐼 = (Itv‘𝐺)
3		tgextahpl.l	. 2 ⊢ 𝐿 = (LineG‘𝐺)
4		tgextahpl.g	. 2 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5		tgextahpl.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
6		tgextahpl.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
7		tgextahpl.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑃)
8		tgextahpl.ncd	. . . 4 ⊢ (𝜑 → ¬ (𝐴 ∈ (𝐶𝐿𝐷) ∨ 𝐶 = 𝐷))
9	1, 2, 3, 4, 5, 6, 7, 8	ncolne1 29123	. . 3 ⊢ (𝜑 → 𝐴 ≠ 𝐶)
10	1, 2, 3, 4, 5, 6, 9	tgelrnln 29128	. 2 ⊢ (𝜑 → (𝐴𝐿𝐶) ∈ ran 𝐿)
11		tgextahpl.w	. 2 ⊢ (𝜑 → 𝑊 ∈ 𝑃)
12		simpl 488	. . . . . 6 ⊢ ((𝑢 = 𝑎 ∧ 𝑣 = 𝑏) → 𝑢 = 𝑎)
13	12	eleq1d 2849	. . . . 5 ⊢ ((𝑢 = 𝑎 ∧ 𝑣 = 𝑏) → (𝑢 ∈ (𝑃 ∖ (𝐴𝐿𝐶)) ↔ 𝑎 ∈ (𝑃 ∖ (𝐴𝐿𝐶))))
14		simpr 490	. . . . . 6 ⊢ ((𝑢 = 𝑎 ∧ 𝑣 = 𝑏) → 𝑣 = 𝑏)
15	14	eleq1d 2849	. . . . 5 ⊢ ((𝑢 = 𝑎 ∧ 𝑣 = 𝑏) → (𝑣 ∈ (𝑃 ∖ (𝐴𝐿𝐶)) ↔ 𝑏 ∈ (𝑃 ∖ (𝐴𝐿𝐶))))
16	13, 15	anbi12d 644	. . . 4 ⊢ ((𝑢 = 𝑎 ∧ 𝑣 = 𝑏) → ((𝑢 ∈ (𝑃 ∖ (𝐴𝐿𝐶)) ∧ 𝑣 ∈ (𝑃 ∖ (𝐴𝐿𝐶))) ↔ (𝑎 ∈ (𝑃 ∖ (𝐴𝐿𝐶)) ∧ 𝑏 ∈ (𝑃 ∖ (𝐴𝐿𝐶)))))
17		simpr 490	. . . . . 6 ⊢ (((𝑢 = 𝑎 ∧ 𝑣 = 𝑏) ∧ 𝑤 = 𝑡) → 𝑤 = 𝑡)
18		simpll 779	. . . . . . 7 ⊢ (((𝑢 = 𝑎 ∧ 𝑣 = 𝑏) ∧ 𝑤 = 𝑡) → 𝑢 = 𝑎)
19		simplr 781	. . . . . . 7 ⊢ (((𝑢 = 𝑎 ∧ 𝑣 = 𝑏) ∧ 𝑤 = 𝑡) → 𝑣 = 𝑏)
20	18, 19	oveq12d 7461	. . . . . 6 ⊢ (((𝑢 = 𝑎 ∧ 𝑣 = 𝑏) ∧ 𝑤 = 𝑡) → (𝑢𝐼𝑣) = (𝑎𝐼𝑏))
21	17, 20	eleq12d 2858	. . . . 5 ⊢ (((𝑢 = 𝑎 ∧ 𝑣 = 𝑏) ∧ 𝑤 = 𝑡) → (𝑤 ∈ (𝑢𝐼𝑣) ↔ 𝑡 ∈ (𝑎𝐼𝑏)))
22	21	cbvrexdva 3247	. . . 4 ⊢ ((𝑢 = 𝑎 ∧ 𝑣 = 𝑏) → (∃𝑤 ∈ (𝐴𝐿𝐶)𝑤 ∈ (𝑢𝐼𝑣) ↔ ∃𝑡 ∈ (𝐴𝐿𝐶)𝑡 ∈ (𝑎𝐼𝑏)))
23	16, 22	anbi12d 644	. . 3 ⊢ ((𝑢 = 𝑎 ∧ 𝑣 = 𝑏) → (((𝑢 ∈ (𝑃 ∖ (𝐴𝐿𝐶)) ∧ 𝑣 ∈ (𝑃 ∖ (𝐴𝐿𝐶))) ∧ ∃𝑤 ∈ (𝐴𝐿𝐶)𝑤 ∈ (𝑢𝐼𝑣)) ↔ ((𝑎 ∈ (𝑃 ∖ (𝐴𝐿𝐶)) ∧ 𝑏 ∈ (𝑃 ∖ (𝐴𝐿𝐶))) ∧ ∃𝑡 ∈ (𝐴𝐿𝐶)𝑡 ∈ (𝑎𝐼𝑏))))
24	23	cbvopabv 5181	. 2 ⊢ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ (𝑃 ∖ (𝐴𝐿𝐶)) ∧ 𝑣 ∈ (𝑃 ∖ (𝐴𝐿𝐶))) ∧ ∃𝑤 ∈ (𝐴𝐿𝐶)𝑤 ∈ (𝑢𝐼𝑣))} = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ (𝐴𝐿𝐶)) ∧ 𝑏 ∈ (𝑃 ∖ (𝐴𝐿𝐶))) ∧ ∃𝑡 ∈ (𝐴𝐿𝐶)𝑡 ∈ (𝑎𝐼𝑏))}
25		tgextahpl.x	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
26		tgextahpl.xk	. . . . 5 ⊢ (𝜑 → 𝑋(𝐾‘𝐶)𝑊)
27		tgextahpl.nw	. . . . . 6 ⊢ (𝜑 → ¬ 𝑊 ∈ (𝐴𝐿𝐶))
28		tgextahpl.k	. . . . . . . 8 ⊢ 𝐾 = (hlG‘𝐺)
29	11	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐴𝐿𝐶)) → 𝑊 ∈ 𝑃)
30	25	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐴𝐿𝐶)) → 𝑋 ∈ 𝑃)
31	6	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐴𝐿𝐶)) → 𝐶 ∈ 𝑃)
32	4	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐴𝐿𝐶)) → 𝐺 ∈ TarskiG)
33	26	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐴𝐿𝐶)) → 𝑋(𝐾‘𝐶)𝑊)
34	1, 2, 28, 30, 29, 31, 32, 33	hlcomd 29100	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐴𝐿𝐶)) → 𝑊(𝐾‘𝐶)𝑋)
35	1, 2, 28, 29, 30, 31, 32, 3, 34	hlln 29103	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐴𝐿𝐶)) → 𝑊 ∈ (𝑋𝐿𝐶))
36	5	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐴𝐿𝐶)) → 𝐴 ∈ 𝑃)
37	9	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐴𝐿𝐶)) → 𝐴 ≠ 𝐶)
38	1, 2, 3, 32, 36, 31, 37	tglinecom 29133	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐴𝐿𝐶)) → (𝐴𝐿𝐶) = (𝐶𝐿𝐴))
39	9	necomd 3014	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐶 ≠ 𝐴)
40	39	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐴𝐿𝐶)) → 𝐶 ≠ 𝐴)
41	1, 2, 28, 25, 11, 6, 4, 26	hlne1 29101	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ≠ 𝐶)
42	41	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐴𝐿𝐶)) → 𝑋 ≠ 𝐶)
43		simpr 490	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐴𝐿𝐶)) → 𝑋 ∈ (𝐴𝐿𝐶))
44	43, 38	eleqtrd 2866	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐴𝐿𝐶)) → 𝑋 ∈ (𝐶𝐿𝐴))
45	1, 2, 3, 32, 31, 36, 40, 30, 42, 44	tglineelsb2 29130	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐴𝐿𝐶)) → (𝐶𝐿𝐴) = (𝐶𝐿𝑋))
46	41	necomd 3014	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐶 ≠ 𝑋)
47	46	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐴𝐿𝐶)) → 𝐶 ≠ 𝑋)
48	1, 2, 3, 32, 31, 30, 47	tglinecom 29133	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐴𝐿𝐶)) → (𝐶𝐿𝑋) = (𝑋𝐿𝐶))
49	45, 48	eqtrd 2798	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐴𝐿𝐶)) → (𝐶𝐿𝐴) = (𝑋𝐿𝐶))
50	38, 49	eqtrd 2798	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐴𝐿𝐶)) → (𝐴𝐿𝐶) = (𝑋𝐿𝐶))
51	35, 50	eleqtrrd 2867	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐴𝐿𝐶)) → 𝑊 ∈ (𝐴𝐿𝐶))
52	27, 51	mtand 828	. . . . 5 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝐴𝐿𝐶))
53	26, 52	jca 521	. . . 4 ⊢ (𝜑 → (𝑋(𝐾‘𝐶)𝑊 ∧ ¬ 𝑋 ∈ (𝐴𝐿𝐶)))
54	1, 2, 3, 4, 5, 6, 9	tglinerflx2 29132	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ (𝐴𝐿𝐶))
55	4	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐶𝐼𝑊)) → 𝐺 ∈ TarskiG)
56	25	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐶𝐼𝑊)) → 𝑋 ∈ 𝑃)
57	11	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐶𝐼𝑊)) → 𝑊 ∈ 𝑃)
58	6	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐶𝐼𝑊)) → 𝐶 ∈ 𝑃)
59		simpr 490	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐶𝐼𝑊)) → 𝑋 ∈ (𝐶𝐼𝑊))
60	1, 3, 2, 55, 56, 57, 58, 59	btwncolg2 29049	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐶𝐼𝑊)) → (𝐶 ∈ (𝑋𝐿𝑊) ∨ 𝑋 = 𝑊))
61	4	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑊 ∈ (𝐶𝐼𝑋)) → 𝐺 ∈ TarskiG)
62	25	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑊 ∈ (𝐶𝐼𝑋)) → 𝑋 ∈ 𝑃)
63	11	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑊 ∈ (𝐶𝐼𝑋)) → 𝑊 ∈ 𝑃)
64	6	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑊 ∈ (𝐶𝐼𝑋)) → 𝐶 ∈ 𝑃)
65		tgextahpl.m	. . . . . . . 8 ⊢ − = (dist‘𝐺)
66		simpr 490	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑊 ∈ (𝐶𝐼𝑋)) → 𝑊 ∈ (𝐶𝐼𝑋))
67	1, 65, 2, 61, 64, 63, 62, 66	tgbtwncom 28981	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑊 ∈ (𝐶𝐼𝑋)) → 𝑊 ∈ (𝑋𝐼𝐶))
68	1, 3, 2, 61, 62, 63, 64, 67	btwncolg3 29050	. . . . . 6 ⊢ ((𝜑 ∧ 𝑊 ∈ (𝐶𝐼𝑋)) → (𝐶 ∈ (𝑋𝐿𝑊) ∨ 𝑋 = 𝑊))
69	1, 2, 28, 25, 11, 6, 4	ishlg 29098	. . . . . . . 8 ⊢ (𝜑 → (𝑋(𝐾‘𝐶)𝑊 ↔ (𝑋 ≠ 𝐶 ∧ 𝑊 ≠ 𝐶 ∧ (𝑋 ∈ (𝐶𝐼𝑊) ∨ 𝑊 ∈ (𝐶𝐼𝑋)))))
70	26, 69	mpbid 235	. . . . . . 7 ⊢ (𝜑 → (𝑋 ≠ 𝐶 ∧ 𝑊 ≠ 𝐶 ∧ (𝑋 ∈ (𝐶𝐼𝑊) ∨ 𝑊 ∈ (𝐶𝐼𝑋))))
71	70	simp3d 1162	. . . . . 6 ⊢ (𝜑 → (𝑋 ∈ (𝐶𝐼𝑊) ∨ 𝑊 ∈ (𝐶𝐼𝑋)))
72	60, 68, 71	mpjaodan 973	. . . . 5 ⊢ (𝜑 → (𝐶 ∈ (𝑋𝐿𝑊) ∨ 𝑋 = 𝑊))
73	1, 2, 3, 4, 10, 25, 24, 11, 54, 72, 28	colhp 29278	. . . 4 ⊢ (𝜑 → (𝑋((hpG‘𝐺)‘(𝐴𝐿𝐶))𝑊 ↔ (𝑋(𝐾‘𝐶)𝑊 ∧ ¬ 𝑋 ∈ (𝐴𝐿𝐶))))
74	53, 73	mpbird 260	. . 3 ⊢ (𝜑 → 𝑋((hpG‘𝐺)‘(𝐴𝐿𝐶))𝑊)
75	1, 2, 3, 4, 10, 25, 24, 11, 74	hpgcom 29275	. 2 ⊢ (𝜑 → 𝑊((hpG‘𝐺)‘(𝐴𝐿𝐶))𝑋)
76		tgextahpl.xad	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝐴𝐼𝐷))
77	1, 2, 3, 4, 5, 6, 7, 8	ncolne2 29124	. . . . 5 ⊢ (𝜑 → 𝐴 ≠ 𝐷)
78	1, 2, 3, 4, 5, 6, 9	tglinerflx1 29131	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ (𝐴𝐿𝐶))
79		nelne2 3057	. . . . . . 7 ⊢ ((𝐴 ∈ (𝐴𝐿𝐶) ∧ ¬ 𝑋 ∈ (𝐴𝐿𝐶)) → 𝐴 ≠ 𝑋)
80	78, 52, 79	syl2anc 596	. . . . . 6 ⊢ (𝜑 → 𝐴 ≠ 𝑋)
81	80	necomd 3014	. . . . 5 ⊢ (𝜑 → 𝑋 ≠ 𝐴)
82	1, 2, 28, 5, 7, 25, 4, 6, 76, 77, 81	btwnhl1 29108	. . . 4 ⊢ (𝜑 → 𝑋(𝐾‘𝐴)𝐷)
83	82, 52	jca 521	. . 3 ⊢ (𝜑 → (𝑋(𝐾‘𝐴)𝐷 ∧ ¬ 𝑋 ∈ (𝐴𝐿𝐶)))
84	1, 3, 2, 4, 25, 7, 5, 76	btwncolg2 29049	. . . 4 ⊢ (𝜑 → (𝐴 ∈ (𝑋𝐿𝐷) ∨ 𝑋 = 𝐷))
85	1, 2, 3, 4, 10, 25, 24, 7, 78, 84, 28	colhp 29278	. . 3 ⊢ (𝜑 → (𝑋((hpG‘𝐺)‘(𝐴𝐿𝐶))𝐷 ↔ (𝑋(𝐾‘𝐴)𝐷 ∧ ¬ 𝑋 ∈ (𝐴𝐿𝐶))))
86	83, 85	mpbird 260	. 2 ⊢ (𝜑 → 𝑋((hpG‘𝐺)‘(𝐴𝐿𝐶))𝐷)
87	1, 2, 3, 4, 10, 11, 24, 25, 75, 7, 86	hpgtr 29276	1 ⊢ (𝜑 → 𝑊((hpG‘𝐺)‘(𝐴𝐿𝐶))𝐷)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  ∃wrex 3090   ∖ cdif 4048   class class class wbr 5106  {copab 5170  ‘cfv 6558  (class class class)co 7443  Basecbs 17416  distcds 17466  TarskiGcstrkg 28919  Itvcitv 28925  LineGclng 28926  hlGchlg 29093  hpGchpg 29265

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775  ax-cnex 11283  ax-resscn 11284  ax-1cn 11285  ax-icn 11286  ax-addcl 11287  ax-addrcl 11288  ax-mulcl 11289  ax-mulrcl 11290  ax-mulcom 11291  ax-addass 11292  ax-mulass 11293  ax-distr 11294  ax-i2m1 11295  ax-1ne0 11296  ax-1rid 11297  ax-rnegex 11298  ax-rrecex 11299  ax-cnre 11300  ax-pre-lttri 11301  ax-pre-lttrn 11302  ax-pre-ltadd 11303  ax-pre-mulgt0 11304

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-om 7902  df-1st 8025  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-1o 8504  df-oadd 8508  df-er 8745  df-map 8877  df-pm 8878  df-en 9002  df-dom 9003  df-sdom 9004  df-fin 9005  df-dju 10009  df-card 10047  df-pnf 11372  df-mnf 11373  df-xr 11374  df-ltxr 11375  df-le 11376  df-sub 11570  df-neg 11571  df-nn 12363  df-2 12432  df-3 12433  df-n0 12634  df-xnn0 12707  df-z 12721  df-uz 12993  df-fz 13667  df-fzo 13816  df-hash 14502  df-word 14686  df-concat 14743  df-s1 14770  df-s2 15027  df-s3 15028  df-trkgc 28940  df-trkgb 28941  df-trkgcb 28942  df-trkgld 28944  df-trkg 28945  df-cgrg 29004  df-leg 29076  df-hlg 29094  df-mir 29155  df-rag 29199  df-perpg 29201  df-hpg 29266

This theorem is used by:  tgextahp  33103