Theorem tgextancgra 33104

Description: The exterior angle is NOT congruent to the remote interior angle -- the strictness tgextale 33101 cannot give, and what tgaas 33116's contradiction branch needs. The exterior angle is ⟨“𝐴𝐶𝐷”⟩, at the extension of 𝐵𝐶 beyond 𝐶. The tail of GeoCoq's l11_41_aux , run through acopyeu 29372 rather than conga_cop_or_out_ts : a congruent exterior angle would, by angle-construction uniqueness, put the midpoint-reflected point of tgmircgra 33096 on the ray 𝐶𝐷-- the two rays make congruent angles with the shared base ray 𝐶𝐴 and, by tgextahp 33103 and hpgid 29274, lie on the same open half-plane of line 𝐴𝐶-- which tgextamirnhl 33102 forbids. Both of acopyeu 29372's non-collinearity hypotheses are discharged, not assumed: source side by rotating the triangle's own ncol , target side by tgextancd 33098. (Contributed by JAG, 12-Aug-2026.)

Hypotheses

Ref	Expression
tgextalem1inag.p	⊢ 𝑃 = (Base‘𝐺)
tgextalem1inag.m	⊢ − = (dist‘𝐺)
tgextalem1inag.i	⊢ 𝐼 = (Itv‘𝐺)
tgextalem1inag.l	⊢ 𝐿 = (LineG‘𝐺)
tgextalem1inag.k	⊢ 𝐾 = (hlG‘𝐺)
tgextalem1inag.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
tgextalem1inag.a	⊢ (𝜑 → 𝐴 ∈ 𝑃)
tgextalem1inag.b	⊢ (𝜑 → 𝐵 ∈ 𝑃)
tgextalem1inag.c	⊢ (𝜑 → 𝐶 ∈ 𝑃)
tgextalem1inag.d	⊢ (𝜑 → 𝐷 ∈ 𝑃)
tgextalem1inag.ext	⊢ (𝜑 → 𝐶 ∈ (𝐵𝐼𝐷))
tgextalem1inag.ncol	⊢ (𝜑 → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))
tgextalem1inag.ne	⊢ (𝜑 → 𝐶 ≠ 𝐷)

Assertion

Ref	Expression
tgextancgra	⊢ (𝜑 → ¬ ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝐴𝐶𝐷”⟩)

Proof of Theorem tgextancgra

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

Step	Hyp	Ref	Expression
1		tgextalem1inag.p	. . 3 ⊢ 𝑃 = (Base‘𝐺)
2		tgextalem1inag.m	. . 3 ⊢ − = (dist‘𝐺)
3		tgextalem1inag.i	. . 3 ⊢ 𝐼 = (Itv‘𝐺)
4		tgextalem1inag.l	. . 3 ⊢ 𝐿 = (LineG‘𝐺)
5		tgextalem1inag.k	. . 3 ⊢ 𝐾 = (hlG‘𝐺)
6		tgextalem1inag.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
7		tgextalem1inag.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
8		tgextalem1inag.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
9		tgextalem1inag.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
10		tgextalem1inag.d	. . 3 ⊢ (𝜑 → 𝐷 ∈ 𝑃)
11		tgextalem1inag.ext	. . 3 ⊢ (𝜑 → 𝐶 ∈ (𝐵𝐼𝐷))
12		tgextalem1inag.ncol	. . 3 ⊢ (𝜑 → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))
13		tgextalem1inag.ne	. . 3 ⊢ (𝜑 → 𝐶 ≠ 𝐷)
14	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13	tgextamirnhl 33102	. 2 ⊢ (𝜑 → ¬ 𝐷(𝐾‘𝐶)(((pInvG‘𝐺)‘(𝐴(midG‘𝐺)𝐶))‘𝐵))
15	6	adantr 486	. . 3 ⊢ ((𝜑 ∧ ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝐴𝐶𝐷”⟩) → 𝐺 ∈ TarskiG)
16	9	adantr 486	. . 3 ⊢ ((𝜑 ∧ ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝐴𝐶𝐷”⟩) → 𝐶 ∈ 𝑃)
17	7	adantr 486	. . 3 ⊢ ((𝜑 ∧ ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝐴𝐶𝐷”⟩) → 𝐴 ∈ 𝑃)
18	8	adantr 486	. . 3 ⊢ ((𝜑 ∧ ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝐴𝐶𝐷”⟩) → 𝐵 ∈ 𝑃)
19	7	adantr 486	. . 3 ⊢ ((𝜑 ∧ ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝐴𝐶𝐷”⟩) → 𝐴 ∈ 𝑃)
20	9	adantr 486	. . 3 ⊢ ((𝜑 ∧ ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝐴𝐶𝐷”⟩) → 𝐶 ∈ 𝑃)
21	10	adantr 486	. . 3 ⊢ ((𝜑 ∧ ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝐴𝐶𝐷”⟩) → 𝐷 ∈ 𝑃)
22	1, 4, 3, 6, 8, 9, 7, 12	ncolrot1 29055	. . . . 5 ⊢ (𝜑 → ¬ (𝐵 ∈ (𝐶𝐿𝐴) ∨ 𝐶 = 𝐴))
23	1, 4, 3, 6, 9, 7, 8, 22	ncolrot1 29055	. . . 4 ⊢ (𝜑 → ¬ (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
24	23	adantr 486	. . 3 ⊢ ((𝜑 ∧ ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝐴𝐶𝐷”⟩) → ¬ (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
25	1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 12, 13	tgextancd 33098	. . . 4 ⊢ (𝜑 → ¬ (𝐴 ∈ (𝐶𝐿𝐷) ∨ 𝐶 = 𝐷))
26	25	adantr 486	. . 3 ⊢ ((𝜑 ∧ ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝐴𝐶𝐷”⟩) → ¬ (𝐴 ∈ (𝐶𝐿𝐷) ∨ 𝐶 = 𝐷))
27	10	adantr 486	. . 3 ⊢ ((𝜑 ∧ ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝐴𝐶𝐷”⟩) → 𝐷 ∈ 𝑃)
28		eqid 2763	. . . . 5 ⊢ (pInvG‘𝐺) = (pInvG‘𝐺)
29	1, 4, 3, 6, 8, 9, 7, 12	ncoltgdim2 29058	. . . . . 6 ⊢ (𝜑 → 𝐺DimTarskiG≥2)
30	1, 2, 3, 6, 29, 7, 9	midcl 29312	. . . . 5 ⊢ (𝜑 → (𝐴(midG‘𝐺)𝐶) ∈ 𝑃)
31		eqid 2763	. . . . 5 ⊢ ((pInvG‘𝐺)‘(𝐴(midG‘𝐺)𝐶)) = ((pInvG‘𝐺)‘(𝐴(midG‘𝐺)𝐶))
32	1, 2, 3, 4, 28, 6, 30, 31, 8	mircl 29163	. . . 4 ⊢ (𝜑 → (((pInvG‘𝐺)‘(𝐴(midG‘𝐺)𝐶))‘𝐵) ∈ 𝑃)
33	32	adantr 486	. . 3 ⊢ ((𝜑 ∧ ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝐴𝐶𝐷”⟩) → (((pInvG‘𝐺)‘(𝐴(midG‘𝐺)𝐶))‘𝐵) ∈ 𝑃)
34		simpr 490	. . 3 ⊢ ((𝜑 ∧ ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝐴𝐶𝐷”⟩) → ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝐴𝐶𝐷”⟩)
35	1, 2, 3, 4, 6, 7, 8, 9, 12	tgmircgra 33096	. . . 4 ⊢ (𝜑 → ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝐴𝐶(((pInvG‘𝐺)‘(𝐴(midG‘𝐺)𝐶))‘𝐵)”⟩)
36	35	adantr 486	. . 3 ⊢ ((𝜑 ∧ ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝐴𝐶𝐷”⟩) → ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝐴𝐶(((pInvG‘𝐺)‘(𝐴(midG‘𝐺)𝐶))‘𝐵)”⟩)
37	1, 3, 4, 6, 7, 8, 9, 12	ncolne2 29124	. . . . . 6 ⊢ (𝜑 → 𝐴 ≠ 𝐶)
38	1, 3, 4, 6, 7, 9, 37	tgelrnln 29128	. . . . 5 ⊢ (𝜑 → (𝐴𝐿𝐶) ∈ ran 𝐿)
39		simpl 488	. . . . . . . . 9 ⊢ ((𝑢 = 𝑎 ∧ 𝑣 = 𝑏) → 𝑢 = 𝑎)
40	39	eleq1d 2849	. . . . . . . 8 ⊢ ((𝑢 = 𝑎 ∧ 𝑣 = 𝑏) → (𝑢 ∈ (𝑃 ∖ (𝐴𝐿𝐶)) ↔ 𝑎 ∈ (𝑃 ∖ (𝐴𝐿𝐶))))
41		simpr 490	. . . . . . . . 9 ⊢ ((𝑢 = 𝑎 ∧ 𝑣 = 𝑏) → 𝑣 = 𝑏)
42	41	eleq1d 2849	. . . . . . . 8 ⊢ ((𝑢 = 𝑎 ∧ 𝑣 = 𝑏) → (𝑣 ∈ (𝑃 ∖ (𝐴𝐿𝐶)) ↔ 𝑏 ∈ (𝑃 ∖ (𝐴𝐿𝐶))))
43	40, 42	anbi12d 644	. . . . . . 7 ⊢ ((𝑢 = 𝑎 ∧ 𝑣 = 𝑏) → ((𝑢 ∈ (𝑃 ∖ (𝐴𝐿𝐶)) ∧ 𝑣 ∈ (𝑃 ∖ (𝐴𝐿𝐶))) ↔ (𝑎 ∈ (𝑃 ∖ (𝐴𝐿𝐶)) ∧ 𝑏 ∈ (𝑃 ∖ (𝐴𝐿𝐶)))))
44		simpr 490	. . . . . . . . 9 ⊢ (((𝑢 = 𝑎 ∧ 𝑣 = 𝑏) ∧ 𝑤 = 𝑡) → 𝑤 = 𝑡)
45		simpll 779	. . . . . . . . . 10 ⊢ (((𝑢 = 𝑎 ∧ 𝑣 = 𝑏) ∧ 𝑤 = 𝑡) → 𝑢 = 𝑎)
46		simplr 781	. . . . . . . . . 10 ⊢ (((𝑢 = 𝑎 ∧ 𝑣 = 𝑏) ∧ 𝑤 = 𝑡) → 𝑣 = 𝑏)
47	45, 46	oveq12d 7461	. . . . . . . . 9 ⊢ (((𝑢 = 𝑎 ∧ 𝑣 = 𝑏) ∧ 𝑤 = 𝑡) → (𝑢𝐼𝑣) = (𝑎𝐼𝑏))
48	44, 47	eleq12d 2858	. . . . . . . 8 ⊢ (((𝑢 = 𝑎 ∧ 𝑣 = 𝑏) ∧ 𝑤 = 𝑡) → (𝑤 ∈ (𝑢𝐼𝑣) ↔ 𝑡 ∈ (𝑎𝐼𝑏)))
49	48	cbvrexdva 3247	. . . . . . 7 ⊢ ((𝑢 = 𝑎 ∧ 𝑣 = 𝑏) → (∃𝑤 ∈ (𝐴𝐿𝐶)𝑤 ∈ (𝑢𝐼𝑣) ↔ ∃𝑡 ∈ (𝐴𝐿𝐶)𝑡 ∈ (𝑎𝐼𝑏)))
50	43, 49	anbi12d 644	. . . . . 6 ⊢ ((𝑢 = 𝑎 ∧ 𝑣 = 𝑏) → (((𝑢 ∈ (𝑃 ∖ (𝐴𝐿𝐶)) ∧ 𝑣 ∈ (𝑃 ∖ (𝐴𝐿𝐶))) ∧ ∃𝑤 ∈ (𝐴𝐿𝐶)𝑤 ∈ (𝑢𝐼𝑣)) ↔ ((𝑎 ∈ (𝑃 ∖ (𝐴𝐿𝐶)) ∧ 𝑏 ∈ (𝑃 ∖ (𝐴𝐿𝐶))) ∧ ∃𝑡 ∈ (𝐴𝐿𝐶)𝑡 ∈ (𝑎𝐼𝑏))))
51	50	cbvopabv 5181	. . . . 5 ⊢ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ (𝑃 ∖ (𝐴𝐿𝐶)) ∧ 𝑣 ∈ (𝑃 ∖ (𝐴𝐿𝐶))) ∧ ∃𝑤 ∈ (𝐴𝐿𝐶)𝑤 ∈ (𝑢𝐼𝑣))} = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ (𝐴𝐿𝐶)) ∧ 𝑏 ∈ (𝑃 ∖ (𝐴𝐿𝐶))) ∧ ∃𝑡 ∈ (𝐴𝐿𝐶)𝑡 ∈ (𝑎𝐼𝑏))}
52	1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 12, 13	tgextancd 33098	. . . . . . . 8 ⊢ (𝜑 → ¬ (𝐴 ∈ (𝐶𝐿𝐷) ∨ 𝐶 = 𝐷))
53		ioran 999	. . . . . . . 8 ⊢ (¬ (𝐴 ∈ (𝐶𝐿𝐷) ∨ 𝐶 = 𝐷) ↔ (¬ 𝐴 ∈ (𝐶𝐿𝐷) ∧ ¬ 𝐶 = 𝐷))
54	52, 53	sylib 221	. . . . . . 7 ⊢ (𝜑 → (¬ 𝐴 ∈ (𝐶𝐿𝐷) ∧ ¬ 𝐶 = 𝐷))
55	54	simpld 500	. . . . . 6 ⊢ (𝜑 → ¬ 𝐴 ∈ (𝐶𝐿𝐷))
56	1, 3, 4, 6, 7, 8, 9, 12	ncolne2 29124	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ≠ 𝐶)
57	56	necomd 3014	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ≠ 𝐴)
58	1, 3, 4, 6, 9, 7, 57	tglinerflx2 29132	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ (𝐶𝐿𝐴))
59	58	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐴𝐿𝐶)) → 𝐴 ∈ (𝐶𝐿𝐴))
60	6	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐴𝐿𝐶)) → 𝐺 ∈ TarskiG)
61	9	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐴𝐿𝐶)) → 𝐶 ∈ 𝑃)
62	7	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐴𝐿𝐶)) → 𝐴 ∈ 𝑃)
63	1, 3, 4, 6, 7, 8, 9, 12	ncolne2 29124	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ≠ 𝐶)
64	63	necomd 3014	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ≠ 𝐴)
65	64	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐴𝐿𝐶)) → 𝐶 ≠ 𝐴)
66	10	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐴𝐿𝐶)) → 𝐷 ∈ 𝑃)
67	13	necomd 3014	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 ≠ 𝐶)
68	67	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐴𝐿𝐶)) → 𝐷 ≠ 𝐶)
69		simpr 490	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐴𝐿𝐶)) → 𝐷 ∈ (𝐴𝐿𝐶))
70	6	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐴𝐿𝐶)) → 𝐺 ∈ TarskiG)
71	7	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐴𝐿𝐶)) → 𝐴 ∈ 𝑃)
72	9	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐴𝐿𝐶)) → 𝐶 ∈ 𝑃)
73	1, 3, 4, 6, 7, 8, 9, 12	ncolne2 29124	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ≠ 𝐶)
74	73	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐴𝐿𝐶)) → 𝐴 ≠ 𝐶)
75	1, 3, 4, 70, 71, 72, 74	tglinecom 29133	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐴𝐿𝐶)) → (𝐴𝐿𝐶) = (𝐶𝐿𝐴))
76	69, 75	eleqtrd 2866	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐴𝐿𝐶)) → 𝐷 ∈ (𝐶𝐿𝐴))
77	1, 3, 4, 60, 61, 62, 65, 66, 68, 76	tglineelsb2 29130	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐴𝐿𝐶)) → (𝐶𝐿𝐴) = (𝐶𝐿𝐷))
78	59, 77	eleqtrd 2866	. . . . . 6 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐴𝐿𝐶)) → 𝐴 ∈ (𝐶𝐿𝐷))
79	55, 78	mtand 828	. . . . 5 ⊢ (𝜑 → ¬ 𝐷 ∈ (𝐴𝐿𝐶))
80	1, 3, 4, 6, 38, 10, 51, 79	hpgid 29274	. . . 4 ⊢ (𝜑 → 𝐷((hpG‘𝐺)‘(𝐴𝐿𝐶))𝐷)
81	80	adantr 486	. . 3 ⊢ ((𝜑 ∧ ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝐴𝐶𝐷”⟩) → 𝐷((hpG‘𝐺)‘(𝐴𝐿𝐶))𝐷)
82	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13	tgextahp 33103	. . . 4 ⊢ (𝜑 → (((pInvG‘𝐺)‘(𝐴(midG‘𝐺)𝐶))‘𝐵)((hpG‘𝐺)‘(𝐴𝐿𝐶))𝐷)
83	82	adantr 486	. . 3 ⊢ ((𝜑 ∧ ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝐴𝐶𝐷”⟩) → (((pInvG‘𝐺)‘(𝐴(midG‘𝐺)𝐶))‘𝐵)((hpG‘𝐺)‘(𝐴𝐿𝐶))𝐷)
84	1, 3, 2, 15, 16, 17, 18, 19, 20, 21, 4, 24, 26, 27, 33, 5, 34, 36, 81, 83	acopyeu 29372	. 2 ⊢ ((𝜑 ∧ ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝐴𝐶𝐷”⟩) → 𝐷(𝐾‘𝐶)(((pInvG‘𝐺)‘(𝐴(midG‘𝐺)𝐶))‘𝐵))
85	14, 84	mtand 828	1 ⊢ (𝜑 → ¬ ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝐴𝐶𝐷”⟩)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  ∃wrex 3090   ∖ cdif 4048   class class class wbr 5106  {copab 5170  ‘cfv 6558  (class class class)co 7443  ⟨“cs3 15021  Basecbs 17416  distcds 17466  TarskiGcstrkg 28919  Itvcitv 28925  LineGclng 28926  hlGchlg 29093  pInvGcmir 29154  hpGchpg 29265  midGcmid 29307  cgrAccgra 29344

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  df-mid 29309  df-lmi 29310  df-cgra 29345  df-inag 29386

This theorem is used by:  tgextalt1  33106  tgextalt2  33107