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Theorem tgaltai 29332

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 2762	. . 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 2762	. . . 4 ⊢ (dist‘𝐺) = (dist‘𝐺)
16		eqid 2762	. . . 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 2768	. . . . 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 28829	. . . 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 28811	. . . 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 2845	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑠 → (𝑡 ∈ (𝑎𝐼𝑏) ↔ 𝑠 ∈ (𝑎𝐼𝑏)))
25	24	cbvrexvw 3243	. . . . . . . . . 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 5176	. . . . . . . 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 2785	. . . . . . 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 28985	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑋𝐿𝑍) ∈ ran 𝐿)
33		tgaltai.3	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑌𝑂𝑊)
34	1, 15, 2, 23, 21, 32, 4, 6, 12, 33	oppne1 29104	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ 𝑌 ∈ (𝑋𝐿𝑍))
35	31	neneqd 2962	. . . . . . . . . . 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 28911	. . . . . . . . 9 ⊢ (𝜑 → ¬ (𝑌 ∈ (𝑍𝐿𝑋) ∨ 𝑍 = 𝑋))
39	1, 21, 2, 4, 10, 8, 6, 38	ncolrot2 28913	. . . . . . . 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 28958	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑃) ∧ (𝑡((hlG‘𝐺)‘𝑍)𝑊 ∧ (𝑍(dist‘𝐺)𝑡) = (𝑋(dist‘𝐺)𝑌))) → 𝑡 ≠ 𝑍)
45	44	necomd 3012	. . . . . . . . 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 29308	. . . . . . . . . 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 28983	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑍 ≠ 𝑊)
49	1, 2, 21, 4, 10, 12, 48	tglinerflx1 28988	. . . . . . . . . 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 28960	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑃) ∧ (𝑡((hlG‘𝐺)‘𝑍)𝑊 ∧ (𝑍(dist‘𝐺)𝑡) = (𝑋(dist‘𝐺)𝑌))) → 𝑡 ∈ (𝑊𝐿𝑍))
52	48	necomd 3012	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑊 ≠ 𝑍)
53	1, 2, 21, 4, 12, 10, 52	tglinecom 28990	. . . . . . . . . . 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 2864	. . . . . . . . 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 28991	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑃) ∧ (𝑡((hlG‘𝐺)‘𝑍)𝑊 ∧ (𝑍(dist‘𝐺)𝑡) = (𝑋(dist‘𝐺)𝑌))) → (𝑍𝐿𝑊) = (𝑍𝐿𝑡))
57	42, 56	breqtrd 5135	. . . . . . 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 29107	. . . . . . . . . 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 28989	. . . . . . . . . 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 28957	. . . . . . . . 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 29117	. . . . . . . 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 29107	. . . . . . 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 29331	. . . . . 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 28829	. . . 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 28866	. . 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 28962	. . . 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 29205	. 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 29307	. . . 4 ⊢ (𝜑 → (𝑋𝐿𝑌) ∈ ran 𝐿)
74	1, 2, 21, 4, 8, 6, 73	tglnne 28983	. . 3 ⊢ (𝜑 → 𝑋 ≠ 𝑌)
75	1, 2, 3, 10, 8, 6, 4, 12, 15, 52, 74	hlcgrex 28969	. 2 ⊢ (𝜑 → ∃𝑡 ∈ 𝑃 (𝑡((hlG‘𝐺)‘𝑍)𝑊 ∧ (𝑍(dist‘𝐺)𝑡) = (𝑋(dist‘𝐺)𝑌)))
76	72, 75	r19.29a 3172	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 2145 ≠ wne 2957 ∃wrex 3088 ∖ cdif 3899 class class class wbr 5107 {copab 5171 ran crn 5660 ‘cfv 6537 (class class class)co 7417 ⟨“cs3 14917 Basecbs 17307 distcds 17357 TarskiGcstrkg 28776 TarskiG_Ecstrkge 28781 Itvcitv 28782 LineGclng 28783 cgrGccgrg 28860 hlGchlg 28950 cgrAccgra 29201 parlnGcprlng 29301

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 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7740 ax-cnex 11184 ax-resscn 11185 ax-1cn 11186 ax-icn 11187 ax-addcl 11188 ax-addrcl 11189 ax-mulcl 11190 ax-mulrcl 11191 ax-mulcom 11192 ax-addass 11193 ax-mulass 11194 ax-distr 11195 ax-i2m1 11196 ax-1ne0 11197 ax-1rid 11198 ax-rnegex 11199 ax-rrecex 11200 ax-cnre 11201 ax-pre-lttri 11202 ax-pre-lttrn 11203 ax-pre-ltadd 11204 ax-pre-mulgt0 11205

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 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 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 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4871 df-int 4911 df-iun 4956 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7374 df-ov 7420 df-oprab 7421 df-mpo 7422 df-om 7867 df-1st 7990 df-2nd 7991 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-oadd 8463 df-er 8700 df-map 8832 df-pm 8833 df-en 8957 df-dom 8958 df-sdom 8959 df-fin 8960 df-dju 9910 df-card 9948 df-pnf 11273 df-mnf 11274 df-xr 11275 df-ltxr 11276 df-le 11277 df-sub 11471 df-neg 11472 df-nn 12262 df-2 12331 df-3 12332 df-n0 12533 df-xnn0 12606 df-z 12620 df-uz 12892 df-fz 13566 df-fzo 13714 df-hash 14399 df-word 14583 df-concat 14640 df-s1 14667 df-s2 14923 df-s3 14924 df-trkgc 28797 df-trkgb 28798 df-trkgcb 28799 df-trkge 28800 df-trkgld 28801 df-trkg 28802 df-cgrg 28861 df-ismt 28883 df-leg 28933 df-hlg 28951 df-mir 29012 df-rag 29056 df-perpg 29058 df-hpg 29123 df-plng 29139 df-mid 29166 df-lmi 29167 df-cgra 29202 df-prlng 29302

This theorem is used by: (None)

W3C validator