Theorem tgaltai 29195

Description: Parallelism implies alternate angles congruence: if a line (𝑋𝐿𝑍) crosses two parallel lines (𝑋𝐿𝑌) and (𝑍𝐿𝑊), then the alternate angles are congruent. First direction of Theorem 12.21 of [Schwabhauser] p. 126. This is also Proposition 1.29 of Euclid's Elements . (Contributed by Thierry Arnoux, 20-Jul-2026.)

Hypotheses

Ref	Expression
tgaltai.p	⊢ 𝑃 = (Base‘𝐺)
tgaltai.i	⊢ 𝐼 = (Itv‘𝐺)
tgaltai.l	⊢ 𝐿 = (LineG‘𝐺)
tgaltai.r	⊢ ∥ = (parlnG‘𝐺)
tgaltai.o	⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ (𝑋𝐿𝑍)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑋𝐿𝑍))) ∧ ∃𝑡 ∈ (𝑋𝐿𝑍)𝑡 ∈ (𝑎𝐼𝑏))}
tgaltai.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
tgaltai.1	⊢ (𝜑 → 𝐺 ∈ TarskiGE)
tgaltai.x	⊢ (𝜑 → 𝑋 ∈ 𝑃)
tgaltai.y	⊢ (𝜑 → 𝑌 ∈ 𝑃)
tgaltai.z	⊢ (𝜑 → 𝑍 ∈ 𝑃)
tgaltai.w	⊢ (𝜑 → 𝑊 ∈ 𝑃)
tgaltai.2	⊢ (𝜑 → (𝑋𝐿𝑌) ∥ (𝑍𝐿𝑊))
tgaltai.3	⊢ (𝜑 → 𝑌𝑂𝑊)
tgaltai.4	⊢ (𝜑 → 𝑋 ≠ 𝑍)

Assertion

Ref	Expression
tgaltai	⊢ (𝜑 → ⟨“𝑌𝑋𝑍”⟩(cgrA‘𝐺)⟨“𝑊𝑍𝑋”⟩)

Distinct variable groups:   ∥ ,𝑎   𝐺,𝑎,𝑏,𝑡   𝐼,𝑎,𝑏,𝑡   𝐿,𝑎,𝑏,𝑡   𝑂,𝑎,𝑏,𝑡   𝑃,𝑎,𝑏,𝑡   𝑊,𝑎,𝑡   𝑋,𝑎,𝑏,𝑡   𝑌,𝑎,𝑏,𝑡   𝑍,𝑎,𝑏,𝑡   𝜑,𝑎,𝑡

Allowed substitution hints:   𝜑(𝑏)   ∥ (𝑡,𝑏)   𝑊(𝑏)

Proof of Theorem tgaltai

Dummy variable 𝑠 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		tgaltai.p	. . 3 ⊢ 𝑃 = (Base‘𝐺)
2		tgaltai.i	. . 3 ⊢ 𝐼 = (Itv‘𝐺)
3		eqid 2763	. . 3 ⊢ (hlG‘𝐺) = (hlG‘𝐺)
4		tgaltai.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5	4	ad2antrr 738	. . 3 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑃) ∧ (𝑡((hlG‘𝐺)‘𝑍)𝑊 ∧ (𝑍(dist‘𝐺)𝑡) = (𝑋(dist‘𝐺)𝑌))) → 𝐺 ∈ TarskiG)
6		tgaltai.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
7	6	ad2antrr 738	. . 3 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑃) ∧ (𝑡((hlG‘𝐺)‘𝑍)𝑊 ∧ (𝑍(dist‘𝐺)𝑡) = (𝑋(dist‘𝐺)𝑌))) → 𝑌 ∈ 𝑃)
8		tgaltai.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
9	8	ad2antrr 738	. . 3 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑃) ∧ (𝑡((hlG‘𝐺)‘𝑍)𝑊 ∧ (𝑍(dist‘𝐺)𝑡) = (𝑋(dist‘𝐺)𝑌))) → 𝑋 ∈ 𝑃)
10		tgaltai.z	. . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝑃)
11	10	ad2antrr 738	. . 3 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑃) ∧ (𝑡((hlG‘𝐺)‘𝑍)𝑊 ∧ (𝑍(dist‘𝐺)𝑡) = (𝑋(dist‘𝐺)𝑌))) → 𝑍 ∈ 𝑃)
12		tgaltai.w	. . . 4 ⊢ (𝜑 → 𝑊 ∈ 𝑃)
13	12	ad2antrr 738	. . 3 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑃) ∧ (𝑡((hlG‘𝐺)‘𝑍)𝑊 ∧ (𝑍(dist‘𝐺)𝑡) = (𝑋(dist‘𝐺)𝑌))) → 𝑊 ∈ 𝑃)
14		simplr 780	. . 3 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑃) ∧ (𝑡((hlG‘𝐺)‘𝑍)𝑊 ∧ (𝑍(dist‘𝐺)𝑡) = (𝑋(dist‘𝐺)𝑌))) → 𝑡 ∈ 𝑃)
15		eqid 2763	. . . 4 ⊢ (dist‘𝐺) = (dist‘𝐺)
16		eqid 2763	. . . 4 ⊢ (cgrG‘𝐺) = (cgrG‘𝐺)
17		simprr 784	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑃) ∧ (𝑡((hlG‘𝐺)‘𝑍)𝑊 ∧ (𝑍(dist‘𝐺)𝑡) = (𝑋(dist‘𝐺)𝑌))) → (𝑍(dist‘𝐺)𝑡) = (𝑋(dist‘𝐺)𝑌))
18	17	eqcomd 2769	. . . . 5 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑃) ∧ (𝑡((hlG‘𝐺)‘𝑍)𝑊 ∧ (𝑍(dist‘𝐺)𝑡) = (𝑋(dist‘𝐺)𝑌))) → (𝑋(dist‘𝐺)𝑌) = (𝑍(dist‘𝐺)𝑡))
19	1, 15, 2, 5, 9, 7, 11, 14, 18	tgcgrcomlr 28727	. . . 4 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑃) ∧ (𝑡((hlG‘𝐺)‘𝑍)𝑊 ∧ (𝑍(dist‘𝐺)𝑡) = (𝑋(dist‘𝐺)𝑌))) → (𝑌(dist‘𝐺)𝑋) = (𝑡(dist‘𝐺)𝑍))
20	1, 15, 2, 5, 9, 11	axtgcgrrflx 28709	. . . 4 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑃) ∧ (𝑡((hlG‘𝐺)‘𝑍)𝑊 ∧ (𝑍(dist‘𝐺)𝑡) = (𝑋(dist‘𝐺)𝑌))) → (𝑋(dist‘𝐺)𝑍) = (𝑍(dist‘𝐺)𝑋))
21		tgaltai.l	. . . . . . 7 ⊢ 𝐿 = (LineG‘𝐺)
22		tgaltai.r	. . . . . . 7 ⊢ ∥ = (parlnG‘𝐺)
23		tgaltai.o	. . . . . . . 8 ⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ (𝑋𝐿𝑍)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑋𝐿𝑍))) ∧ ∃𝑡 ∈ (𝑋𝐿𝑍)𝑡 ∈ (𝑎𝐼𝑏))}
24		eleq1w 2846	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑠 → (𝑡 ∈ (𝑎𝐼𝑏) ↔ 𝑠 ∈ (𝑎𝐼𝑏)))
25	24	cbvrexvw 3244	. . . . . . . . . 10 ⊢ (∃𝑡 ∈ (𝑋𝐿𝑍)𝑡 ∈ (𝑎𝐼𝑏) ↔ ∃𝑠 ∈ (𝑋𝐿𝑍)𝑠 ∈ (𝑎𝐼𝑏))
26	25	anbi2i 634	. . . . . . . . 9 ⊢ (((𝑎 ∈ (𝑃 ∖ (𝑋𝐿𝑍)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑋𝐿𝑍))) ∧ ∃𝑡 ∈ (𝑋𝐿𝑍)𝑡 ∈ (𝑎𝐼𝑏)) ↔ ((𝑎 ∈ (𝑃 ∖ (𝑋𝐿𝑍)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑋𝐿𝑍))) ∧ ∃𝑠 ∈ (𝑋𝐿𝑍)𝑠 ∈ (𝑎𝐼𝑏)))
27	26	opabbii 5179	. . . . . . . 8 ⊢ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ (𝑋𝐿𝑍)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑋𝐿𝑍))) ∧ ∃𝑡 ∈ (𝑋𝐿𝑍)𝑡 ∈ (𝑎𝐼𝑏))} = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ (𝑋𝐿𝑍)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑋𝐿𝑍))) ∧ ∃𝑠 ∈ (𝑋𝐿𝑍)𝑠 ∈ (𝑎𝐼𝑏))}
28	23, 27	eqtri 2786	. . . . . . 7 ⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ (𝑋𝐿𝑍)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑋𝐿𝑍))) ∧ ∃𝑠 ∈ (𝑋𝐿𝑍)𝑠 ∈ (𝑎𝐼𝑏))}
29		tgaltai.1	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ TarskiGE)
30	29	ad2antrr 738	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑃) ∧ (𝑡((hlG‘𝐺)‘𝑍)𝑊 ∧ (𝑍(dist‘𝐺)𝑡) = (𝑋(dist‘𝐺)𝑌))) → 𝐺 ∈ TarskiGE)
31		tgaltai.4	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑋 ≠ 𝑍)
32	1, 2, 21, 4, 8, 10, 31	tgelrnln 28881	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑋𝐿𝑍) ∈ ran 𝐿)
33		tgaltai.3	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑌𝑂𝑊)
34	1, 15, 2, 23, 21, 32, 4, 6, 12, 33	oppne1 29000	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ 𝑌 ∈ (𝑋𝐿𝑍))
35	31	neneqd 2963	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ 𝑋 = 𝑍)
36		ioran 999	. . . . . . . . . . 11 ⊢ (¬ (𝑌 ∈ (𝑋𝐿𝑍) ∨ 𝑋 = 𝑍) ↔ (¬ 𝑌 ∈ (𝑋𝐿𝑍) ∧ ¬ 𝑋 = 𝑍))
37	34, 35, 36	sylanbrc 594	. . . . . . . . . 10 ⊢ (𝜑 → ¬ (𝑌 ∈ (𝑋𝐿𝑍) ∨ 𝑋 = 𝑍))
38	1, 21, 2, 4, 8, 10, 6, 37	ncolcom 28808	. . . . . . . . 9 ⊢ (𝜑 → ¬ (𝑌 ∈ (𝑍𝐿𝑋) ∨ 𝑍 = 𝑋))
39	1, 21, 2, 4, 10, 8, 6, 38	ncolrot2 28810	. . . . . . . 8 ⊢ (𝜑 → ¬ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))
40	39	ad2antrr 738	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑃) ∧ (𝑡((hlG‘𝐺)‘𝑍)𝑊 ∧ (𝑍(dist‘𝐺)𝑡) = (𝑋(dist‘𝐺)𝑌))) → ¬ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))
41		tgaltai.2	. . . . . . . . 9 ⊢ (𝜑 → (𝑋𝐿𝑌) ∥ (𝑍𝐿𝑊))
42	41	ad2antrr 738	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑃) ∧ (𝑡((hlG‘𝐺)‘𝑍)𝑊 ∧ (𝑍(dist‘𝐺)𝑡) = (𝑋(dist‘𝐺)𝑌))) → (𝑋𝐿𝑌) ∥ (𝑍𝐿𝑊))
43		simprl 782	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑃) ∧ (𝑡((hlG‘𝐺)‘𝑍)𝑊 ∧ (𝑍(dist‘𝐺)𝑡) = (𝑋(dist‘𝐺)𝑌))) → 𝑡((hlG‘𝐺)‘𝑍)𝑊)
44	1, 2, 3, 14, 13, 11, 5, 43	hlne1 28855	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑃) ∧ (𝑡((hlG‘𝐺)‘𝑍)𝑊 ∧ (𝑍(dist‘𝐺)𝑡) = (𝑋(dist‘𝐺)𝑌))) → 𝑡 ≠ 𝑍)
45	44	necomd 3013	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑃) ∧ (𝑡((hlG‘𝐺)‘𝑍)𝑊 ∧ (𝑍(dist‘𝐺)𝑡) = (𝑋(dist‘𝐺)𝑌))) → 𝑍 ≠ 𝑡)
46	21, 22, 4, 41	prlngrcl2 29171	. . . . . . . . . 10 ⊢ (𝜑 → (𝑍𝐿𝑊) ∈ ran 𝐿)
47	46	ad2antrr 738	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑃) ∧ (𝑡((hlG‘𝐺)‘𝑍)𝑊 ∧ (𝑍(dist‘𝐺)𝑡) = (𝑋(dist‘𝐺)𝑌))) → (𝑍𝐿𝑊) ∈ ran 𝐿)
48	1, 2, 21, 4, 10, 12, 46	tglnne 28879	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑍 ≠ 𝑊)
49	1, 2, 21, 4, 10, 12, 48	tglinerflx1 28884	. . . . . . . . . 10 ⊢ (𝜑 → 𝑍 ∈ (𝑍𝐿𝑊))
50	49	ad2antrr 738	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑃) ∧ (𝑡((hlG‘𝐺)‘𝑍)𝑊 ∧ (𝑍(dist‘𝐺)𝑡) = (𝑋(dist‘𝐺)𝑌))) → 𝑍 ∈ (𝑍𝐿𝑊))
51	1, 2, 3, 14, 13, 11, 5, 21, 43	hlln 28857	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑃) ∧ (𝑡((hlG‘𝐺)‘𝑍)𝑊 ∧ (𝑍(dist‘𝐺)𝑡) = (𝑋(dist‘𝐺)𝑌))) → 𝑡 ∈ (𝑊𝐿𝑍))
52	48	necomd 3013	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑊 ≠ 𝑍)
53	1, 2, 21, 4, 12, 10, 52	tglinecom 28886	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑊𝐿𝑍) = (𝑍𝐿𝑊))
54	53	ad2antrr 738	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑃) ∧ (𝑡((hlG‘𝐺)‘𝑍)𝑊 ∧ (𝑍(dist‘𝐺)𝑡) = (𝑋(dist‘𝐺)𝑌))) → (𝑊𝐿𝑍) = (𝑍𝐿𝑊))
55	51, 54	eleqtrd 2865	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑃) ∧ (𝑡((hlG‘𝐺)‘𝑍)𝑊 ∧ (𝑍(dist‘𝐺)𝑡) = (𝑋(dist‘𝐺)𝑌))) → 𝑡 ∈ (𝑍𝐿𝑊))
56	1, 2, 21, 5, 11, 14, 45, 45, 47, 50, 55	tglinethru 28887	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑃) ∧ (𝑡((hlG‘𝐺)‘𝑍)𝑊 ∧ (𝑍(dist‘𝐺)𝑡) = (𝑋(dist‘𝐺)𝑌))) → (𝑍𝐿𝑊) = (𝑍𝐿𝑡))
57	42, 56	breqtrd 5138	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑃) ∧ (𝑡((hlG‘𝐺)‘𝑍)𝑊 ∧ (𝑍(dist‘𝐺)𝑡) = (𝑋(dist‘𝐺)𝑌))) → (𝑋𝐿𝑌) ∥ (𝑍𝐿𝑡))
58	32	ad2antrr 738	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑃) ∧ (𝑡((hlG‘𝐺)‘𝑍)𝑊 ∧ (𝑍(dist‘𝐺)𝑡) = (𝑋(dist‘𝐺)𝑌))) → (𝑋𝐿𝑍) ∈ ran 𝐿)
59	1, 15, 2, 28, 21, 32, 4, 6, 12, 33	oppcom 29003	. . . . . . . . . 10 ⊢ (𝜑 → 𝑊𝑂𝑌)
60	59	ad2antrr 738	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑃) ∧ (𝑡((hlG‘𝐺)‘𝑍)𝑊 ∧ (𝑍(dist‘𝐺)𝑡) = (𝑋(dist‘𝐺)𝑌))) → 𝑊𝑂𝑌)
61	1, 2, 21, 4, 8, 10, 31	tglinerflx2 28885	. . . . . . . . . 10 ⊢ (𝜑 → 𝑍 ∈ (𝑋𝐿𝑍))
62	61	ad2antrr 738	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑃) ∧ (𝑡((hlG‘𝐺)‘𝑍)𝑊 ∧ (𝑍(dist‘𝐺)𝑡) = (𝑋(dist‘𝐺)𝑌))) → 𝑍 ∈ (𝑋𝐿𝑍))
63	1, 2, 3, 14, 13, 11, 5, 43	hlcomd 28854	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑃) ∧ (𝑡((hlG‘𝐺)‘𝑍)𝑊 ∧ (𝑍(dist‘𝐺)𝑡) = (𝑋(dist‘𝐺)𝑌))) → 𝑊((hlG‘𝐺)‘𝑍)𝑡)
64	1, 15, 2, 28, 21, 58, 5, 3, 13, 14, 7, 60, 62, 63	opphl 29013	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑃) ∧ (𝑡((hlG‘𝐺)‘𝑍)𝑊 ∧ (𝑍(dist‘𝐺)𝑡) = (𝑋(dist‘𝐺)𝑌))) → 𝑡𝑂𝑌)
65	1, 15, 2, 28, 21, 58, 5, 14, 7, 64	oppcom 29003	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑃) ∧ (𝑡((hlG‘𝐺)‘𝑍)𝑊 ∧ (𝑍(dist‘𝐺)𝑡) = (𝑋(dist‘𝐺)𝑌))) → 𝑌𝑂𝑡)
66	1, 15, 2, 21, 22, 28, 5, 30, 9, 7, 11, 14, 40, 57, 18, 65	quadcgrprlng 29194	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑃) ∧ (𝑡((hlG‘𝐺)‘𝑍)𝑊 ∧ (𝑍(dist‘𝐺)𝑡) = (𝑋(dist‘𝐺)𝑌))) → ((𝑌𝐿𝑍) ∥ (𝑡𝐿𝑋) ∧ (𝑌(dist‘𝐺)𝑍) = (𝑡(dist‘𝐺)𝑋)))
67	66	simprd 500	. . . . 5 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑃) ∧ (𝑡((hlG‘𝐺)‘𝑍)𝑊 ∧ (𝑍(dist‘𝐺)𝑡) = (𝑋(dist‘𝐺)𝑌))) → (𝑌(dist‘𝐺)𝑍) = (𝑡(dist‘𝐺)𝑋))
68	1, 15, 2, 5, 7, 11, 14, 9, 67	tgcgrcomlr 28727	. . . 4 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑃) ∧ (𝑡((hlG‘𝐺)‘𝑍)𝑊 ∧ (𝑍(dist‘𝐺)𝑡) = (𝑋(dist‘𝐺)𝑌))) → (𝑍(dist‘𝐺)𝑌) = (𝑋(dist‘𝐺)𝑡))
69	1, 15, 16, 5, 7, 9, 11, 14, 11, 9, 19, 20, 68	trgcgr 28763	. . 3 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑃) ∧ (𝑡((hlG‘𝐺)‘𝑍)𝑊 ∧ (𝑍(dist‘𝐺)𝑡) = (𝑋(dist‘𝐺)𝑌))) → ⟨“𝑌𝑋𝑍”⟩(cgrG‘𝐺)⟨“𝑡𝑍𝑋”⟩)
70	1, 2, 3, 8, 8, 10, 4, 31	hlid 28859	. . . 4 ⊢ (𝜑 → 𝑋((hlG‘𝐺)‘𝑍)𝑋)
71	70	ad2antrr 738	. . 3 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑃) ∧ (𝑡((hlG‘𝐺)‘𝑍)𝑊 ∧ (𝑍(dist‘𝐺)𝑡) = (𝑋(dist‘𝐺)𝑌))) → 𝑋((hlG‘𝐺)‘𝑍)𝑋)
72	1, 2, 3, 5, 7, 9, 11, 13, 11, 9, 14, 9, 69, 43, 71	iscgrad 29100	. 2 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑃) ∧ (𝑡((hlG‘𝐺)‘𝑍)𝑊 ∧ (𝑍(dist‘𝐺)𝑡) = (𝑋(dist‘𝐺)𝑌))) → ⟨“𝑌𝑋𝑍”⟩(cgrA‘𝐺)⟨“𝑊𝑍𝑋”⟩)
73	21, 22, 4, 41	prlngrcl1 29170	. . . 4 ⊢ (𝜑 → (𝑋𝐿𝑌) ∈ ran 𝐿)
74	1, 2, 21, 4, 8, 6, 73	tglnne 28879	. . 3 ⊢ (𝜑 → 𝑋 ≠ 𝑌)
75	1, 2, 3, 10, 8, 6, 4, 12, 15, 52, 74	hlcgrex 28866	. 2 ⊢ (𝜑 → ∃𝑡 ∈ 𝑃 (𝑡((hlG‘𝐺)‘𝑍)𝑊 ∧ (𝑍(dist‘𝐺)𝑡) = (𝑋(dist‘𝐺)𝑌)))
76	72, 75	r19.29a 3173	1 ⊢ (𝜑 → ⟨“𝑌𝑋𝑍”⟩(cgrA‘𝐺)⟨“𝑊𝑍𝑋”⟩)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   ∨ wo 860   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  ∃wrex 3089   ∖ cdif 3903   class class class wbr 5110  {copab 5174  ran crn 5664  ‘cfv 6538  (class class class)co 7412  ⟨“cs3 14881  Basecbs 17270  distcds 17320  TarskiGcstrkg 28674  TarskiG_Ecstrkge 28679  Itvcitv 28680  LineGclng 28681  cgrGccgrg 28757  hlGchlg 28847  cgrAccgra 29096  parlnGcprlng 29164

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-cnex 11157  ax-resscn 11158  ax-1cn 11159  ax-icn 11160  ax-addcl 11161  ax-addrcl 11162  ax-mulcl 11163  ax-mulrcl 11164  ax-mulcom 11165  ax-addass 11166  ax-mulass 11167  ax-distr 11168  ax-i2m1 11169  ax-1ne0 11170  ax-1rid 11171  ax-rnegex 11172  ax-rrecex 11173  ax-cnre 11174  ax-pre-lttri 11175  ax-pre-lttrn 11176  ax-pre-ltadd 11177  ax-pre-mulgt0 11178

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-tp 4595  df-op 4597  df-uni 4874  df-int 4914  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-1o 8454  df-oadd 8458  df-er 8695  df-map 8827  df-pm 8828  df-en 8945  df-dom 8946  df-sdom 8947  df-fin 8948  df-dju 9888  df-card 9926  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11444  df-neg 11445  df-nn 12235  df-2 12304  df-3 12305  df-n0 12506  df-xnn0 12579  df-z 12593  df-uz 12864  df-fz 13537  df-fzo 13685  df-hash 14369  df-word 14553  df-concat 14610  df-s1 14636  df-s2 14887  df-s3 14888  df-trkgc 28695  df-trkgb 28696  df-trkgcb 28697  df-trkge 28698  df-trkgld 28699  df-trkg 28700  df-cgrg 28758  df-ismt 28780  df-leg 28830  df-hlg 28848  df-mir 28908  df-rag 28952  df-perpg 28954  df-hpg 29018  df-plng 29034  df-mid 29061  df-lmi 29062  df-cgra 29097  df-prlng 29165

This theorem is referenced by: (None)