tgaltai - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > tgaltai		Structured version Visualization version GIF version

Theorem tgaltai 29254

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 ∈ TarskiG_E)
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 2766	. . 3 ⊢ (hlG‘𝐺) = (hlG‘𝐺)
4		tgaltai.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5	4	ad2antrr 739	. . 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 739	. . 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 739	. . 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 739	. . 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 739	. . 3 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑃) ∧ (𝑡((hlG‘𝐺)‘𝑍)𝑊 ∧ (𝑍(dist‘𝐺)𝑡) = (𝑋(dist‘𝐺)𝑌))) → 𝑊 ∈ 𝑃)
14		simplr 781	. . 3 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑃) ∧ (𝑡((hlG‘𝐺)‘𝑍)𝑊 ∧ (𝑍(dist‘𝐺)𝑡) = (𝑋(dist‘𝐺)𝑌))) → 𝑡 ∈ 𝑃)
15		eqid 2766	. . . 4 ⊢ (dist‘𝐺) = (dist‘𝐺)
16		eqid 2766	. . . 4 ⊢ (cgrG‘𝐺) = (cgrG‘𝐺)
17		simprr 785	. . . . . 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 2772	. . . . 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 28786	. . . 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 28768	. . . 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 2849	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑠 → (𝑡 ∈ (𝑎𝐼𝑏) ↔ 𝑠 ∈ (𝑎𝐼𝑏)))
25	24	cbvrexvw 3247	. . . . . . . . . 10 ⊢ (∃𝑡 ∈ (𝑋𝐿𝑍)𝑡 ∈ (𝑎𝐼𝑏) ↔ ∃𝑠 ∈ (𝑋𝐿𝑍)𝑠 ∈ (𝑎𝐼𝑏))
26	25	anbi2i 635	. . . . . . . . 9 ⊢ (((𝑎 ∈ (𝑃 ∖ (𝑋𝐿𝑍)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑋𝐿𝑍))) ∧ ∃𝑡 ∈ (𝑋𝐿𝑍)𝑡 ∈ (𝑎𝐼𝑏)) ↔ ((𝑎 ∈ (𝑃 ∖ (𝑋𝐿𝑍)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑋𝐿𝑍))) ∧ ∃𝑠 ∈ (𝑋𝐿𝑍)𝑠 ∈ (𝑎𝐼𝑏)))
27	26	opabbii 5183	. . . . . . . 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 2789	. . . . . . 7 ⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ (𝑋𝐿𝑍)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑋𝐿𝑍))) ∧ ∃𝑠 ∈ (𝑋𝐿𝑍)𝑠 ∈ (𝑎𝐼𝑏))}
29		tgaltai.1	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ TarskiG_E)
30	29	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑃) ∧ (𝑡((hlG‘𝐺)‘𝑍)𝑊 ∧ (𝑍(dist‘𝐺)𝑡) = (𝑋(dist‘𝐺)𝑌))) → 𝐺 ∈ TarskiG_E)
31		tgaltai.4	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑋 ≠ 𝑍)
32	1, 2, 21, 4, 8, 10, 31	tgelrnln 28940	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑋𝐿𝑍) ∈ ran 𝐿)
33		tgaltai.3	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑌𝑂𝑊)
34	1, 15, 2, 23, 21, 32, 4, 6, 12, 33	oppne1 29059	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ 𝑌 ∈ (𝑋𝐿𝑍))
35	31	neneqd 2966	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ 𝑋 = 𝑍)
36		ioran 999	. . . . . . . . . . 11 ⊢ (¬ (𝑌 ∈ (𝑋𝐿𝑍) ∨ 𝑋 = 𝑍) ↔ (¬ 𝑌 ∈ (𝑋𝐿𝑍) ∧ ¬ 𝑋 = 𝑍))
37	34, 35, 36	sylanbrc 595	. . . . . . . . . 10 ⊢ (𝜑 → ¬ (𝑌 ∈ (𝑋𝐿𝑍) ∨ 𝑋 = 𝑍))
38	1, 21, 2, 4, 8, 10, 6, 37	ncolcom 28867	. . . . . . . . 9 ⊢ (𝜑 → ¬ (𝑌 ∈ (𝑍𝐿𝑋) ∨ 𝑍 = 𝑋))
39	1, 21, 2, 4, 10, 8, 6, 38	ncolrot2 28869	. . . . . . . 8 ⊢ (𝜑 → ¬ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))
40	39	ad2antrr 739	. . . . . . 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 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑃) ∧ (𝑡((hlG‘𝐺)‘𝑍)𝑊 ∧ (𝑍(dist‘𝐺)𝑡) = (𝑋(dist‘𝐺)𝑌))) → (𝑋𝐿𝑌) ∥ (𝑍𝐿𝑊))
43		simprl 783	. . . . . . . . . . 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 28914	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑃) ∧ (𝑡((hlG‘𝐺)‘𝑍)𝑊 ∧ (𝑍(dist‘𝐺)𝑡) = (𝑋(dist‘𝐺)𝑌))) → 𝑡 ≠ 𝑍)
45	44	necomd 3016	. . . . . . . . 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 29230	. . . . . . . . . 10 ⊢ (𝜑 → (𝑍𝐿𝑊) ∈ ran 𝐿)
47	46	ad2antrr 739	. . . . . . . . 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 28938	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑍 ≠ 𝑊)
49	1, 2, 21, 4, 10, 12, 48	tglinerflx1 28943	. . . . . . . . . 10 ⊢ (𝜑 → 𝑍 ∈ (𝑍𝐿𝑊))
50	49	ad2antrr 739	. . . . . . . . 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 28916	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑃) ∧ (𝑡((hlG‘𝐺)‘𝑍)𝑊 ∧ (𝑍(dist‘𝐺)𝑡) = (𝑋(dist‘𝐺)𝑌))) → 𝑡 ∈ (𝑊𝐿𝑍))
52	48	necomd 3016	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑊 ≠ 𝑍)
53	1, 2, 21, 4, 12, 10, 52	tglinecom 28945	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑊𝐿𝑍) = (𝑍𝐿𝑊))
54	53	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑃) ∧ (𝑡((hlG‘𝐺)‘𝑍)𝑊 ∧ (𝑍(dist‘𝐺)𝑡) = (𝑋(dist‘𝐺)𝑌))) → (𝑊𝐿𝑍) = (𝑍𝐿𝑊))
55	51, 54	eleqtrd 2868	. . . . . . . . 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 28946	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑃) ∧ (𝑡((hlG‘𝐺)‘𝑍)𝑊 ∧ (𝑍(dist‘𝐺)𝑡) = (𝑋(dist‘𝐺)𝑌))) → (𝑍𝐿𝑊) = (𝑍𝐿𝑡))
57	42, 56	breqtrd 5142	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑃) ∧ (𝑡((hlG‘𝐺)‘𝑍)𝑊 ∧ (𝑍(dist‘𝐺)𝑡) = (𝑋(dist‘𝐺)𝑌))) → (𝑋𝐿𝑌) ∥ (𝑍𝐿𝑡))
58	32	ad2antrr 739	. . . . . . . 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 29062	. . . . . . . . . 10 ⊢ (𝜑 → 𝑊𝑂𝑌)
60	59	ad2antrr 739	. . . . . . . . 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 28944	. . . . . . . . . 10 ⊢ (𝜑 → 𝑍 ∈ (𝑋𝐿𝑍))
62	61	ad2antrr 739	. . . . . . . . 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 28913	. . . . . . . . 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 29072	. . . . . . . 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 29062	. . . . . . 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 29253	. . . . . 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 501	. . . . 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 28786	. . . 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 28822	. . 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 28918	. . . 4 ⊢ (𝜑 → 𝑋((hlG‘𝐺)‘𝑍)𝑋)
71	70	ad2antrr 739	. . 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 29159	. 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 29229	. . . 4 ⊢ (𝜑 → (𝑋𝐿𝑌) ∈ ran 𝐿)
74	1, 2, 21, 4, 8, 6, 73	tglnne 28938	. . 3 ⊢ (𝜑 → 𝑋 ≠ 𝑌)
75	1, 2, 3, 10, 8, 6, 4, 12, 15, 52, 74	hlcgrex 28925	. 2 ⊢ (𝜑 → ∃𝑡 ∈ 𝑃 (𝑡((hlG‘𝐺)‘𝑍)𝑊 ∧ (𝑍(dist‘𝐺)𝑡) = (𝑋(dist‘𝐺)𝑌)))
76	72, 75	r19.29a 3176	1 ⊢ (𝜑 → ⟨“𝑌𝑋𝑍”⟩(cgrA‘𝐺)⟨“𝑊𝑍𝑋”⟩)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 ∨ wo 861 = wceq 1570 ∈ wcel 2146 ≠ wne 2961 ∃wrex 3092 ∖ cdif 3905 class class class wbr 5114 {copab 5178 ran crn 5667 ‘cfv 6543 (class class class)co 7423 ⟨“cs3 14905 Basecbs 17294 distcds 17344 TarskiGcstrkg 28733 TarskiG_Ecstrkge 28738 Itvcitv 28739 LineGclng 28740 cgrGccgrg 28816 hlGchlg 28906 cgrAccgra 29155 parlnGcprlng 29223

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 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-rep 5243 ax-sep 5262 ax-nul 5274 ax-pow 5341 ax-pr 5409 ax-un 7745 ax-cnex 11174 ax-resscn 11175 ax-1cn 11176 ax-icn 11177 ax-addcl 11178 ax-addrcl 11179 ax-mulcl 11180 ax-mulrcl 11181 ax-mulcom 11182 ax-addass 11183 ax-mulass 11184 ax-distr 11185 ax-i2m1 11186 ax-1ne0 11187 ax-1rid 11188 ax-rnegex 11189 ax-rrecex 11190 ax-cnre 11191 ax-pre-lttri 11192 ax-pre-lttrn 11193 ax-pre-ltadd 11194 ax-pre-mulgt0 11195

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 2100 df-mo 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-nel 3068 df-ral 3083 df-rex 3093 df-rmo 3372 df-reu 3373 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3928 df-nul 4290 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-uni 4878 df-int 4918 df-iun 4963 df-br 5115 df-opab 5179 df-mpt 5198 df-tr 5224 df-id 5561 df-eprel 5566 df-po 5574 df-so 5575 df-fr 5619 df-we 5621 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6309 df-ord 6370 df-on 6371 df-lim 6372 df-suc 6373 df-iota 6499 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7380 df-ov 7426 df-oprab 7427 df-mpo 7428 df-om 7872 df-1st 7995 df-2nd 7996 df-frecs 8287 df-wrecs 8318 df-recs 8367 df-rdg 8406 df-1o 8462 df-oadd 8466 df-er 8703 df-map 8835 df-pm 8836 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-dju 9906 df-card 9944 df-pnf 11263 df-mnf 11264 df-xr 11265 df-ltxr 11266 df-le 11267 df-sub 11461 df-neg 11462 df-nn 12252 df-2 12321 df-3 12322 df-n0 12523 df-xnn0 12596 df-z 12610 df-uz 12881 df-fz 13554 df-fzo 13702 df-hash 14387 df-word 14571 df-concat 14628 df-s1 14655 df-s2 14911 df-s3 14912 df-trkgc 28754 df-trkgb 28755 df-trkgcb 28756 df-trkge 28757 df-trkgld 28758 df-trkg 28759 df-cgrg 28817 df-ismt 28839 df-leg 28889 df-hlg 28907 df-mir 28967 df-rag 29011 df-perpg 29013 df-hpg 29077 df-plng 29093 df-mid 29120 df-lmi 29121 df-cgra 29156 df-prlng 29224

This theorem is used by: (None)

W3C validator