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Theorem tgaltai 29427

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 2761	. . 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 2761	. . . 4 ⊢ (dist‘𝐺) = (dist‘𝐺)
16		eqid 2761	. . . 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 2767	. . . . 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 28924	. . . 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 28906	. . . 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 2844	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑠 → (𝑡 ∈ (𝑎𝐼𝑏) ↔ 𝑠 ∈ (𝑎𝐼𝑏)))
25	24	cbvrexvw 3242	. . . . . . . . . 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 5172	. . . . . . . 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 2784	. . . . . . 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 29080	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑋𝐿𝑍) ∈ ran 𝐿)
33		tgaltai.3	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑌𝑂𝑊)
34	1, 15, 2, 23, 21, 32, 4, 6, 12, 33	oppne1 29199	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ 𝑌 ∈ (𝑋𝐿𝑍))
35	31	neneqd 2961	. . . . . . . . . . 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 29006	. . . . . . . . 9 ⊢ (𝜑 → ¬ (𝑌 ∈ (𝑍𝐿𝑋) ∨ 𝑍 = 𝑋))
39	1, 21, 2, 4, 10, 8, 6, 38	ncolrot2 29008	. . . . . . . 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 29053	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑃) ∧ (𝑡((hlG‘𝐺)‘𝑍)𝑊 ∧ (𝑍(dist‘𝐺)𝑡) = (𝑋(dist‘𝐺)𝑌))) → 𝑡 ≠ 𝑍)
45	44	necomd 3011	. . . . . . . . 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 29403	. . . . . . . . . 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 29078	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑍 ≠ 𝑊)
49	1, 2, 21, 4, 10, 12, 48	tglinerflx1 29083	. . . . . . . . . 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 29055	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑃) ∧ (𝑡((hlG‘𝐺)‘𝑍)𝑊 ∧ (𝑍(dist‘𝐺)𝑡) = (𝑋(dist‘𝐺)𝑌))) → 𝑡 ∈ (𝑊𝐿𝑍))
52	48	necomd 3011	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑊 ≠ 𝑍)
53	1, 2, 21, 4, 12, 10, 52	tglinecom 29085	. . . . . . . . . . 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 2863	. . . . . . . . 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 29086	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑃) ∧ (𝑡((hlG‘𝐺)‘𝑍)𝑊 ∧ (𝑍(dist‘𝐺)𝑡) = (𝑋(dist‘𝐺)𝑌))) → (𝑍𝐿𝑊) = (𝑍𝐿𝑡))
57	42, 56	breqtrd 5131	. . . . . . 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 29202	. . . . . . . . . 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 29084	. . . . . . . . . 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 29052	. . . . . . . . 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 29212	. . . . . . . 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 29202	. . . . . . 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 29426	. . . . . 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 28924	. . . 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 28961	. . 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 29057	. . . 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 29300	. 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 29402	. . . 4 ⊢ (𝜑 → (𝑋𝐿𝑌) ∈ ran 𝐿)
74	1, 2, 21, 4, 8, 6, 73	tglnne 29078	. . 3 ⊢ (𝜑 → 𝑋 ≠ 𝑌)
75	1, 2, 3, 10, 8, 6, 4, 12, 15, 52, 74	hlcgrex 29064	. 2 ⊢ (𝜑 → ∃𝑡 ∈ 𝑃 (𝑡((hlG‘𝐺)‘𝑍)𝑊 ∧ (𝑍(dist‘𝐺)𝑡) = (𝑋(dist‘𝐺)𝑌)))
76	72, 75	r19.29a 3171	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 2956 ∃wrex 3087 ∖ cdif 3896 class class class wbr 5103 {copab 5167 ran crn 5652 ‘cfv 6531 (class class class)co 7412 ⟨“cs3 14973 Basecbs 17367 distcds 17417 TarskiGcstrkg 28871 TarskiG_Ecstrkge 28876 Itvcitv 28877 LineGclng 28878 cgrGccgrg 28955 hlGchlg 29045 cgrAccgra 29296 parlnGcprlng 29396

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 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7740 ax-cnex 11237 ax-resscn 11238 ax-1cn 11239 ax-icn 11240 ax-addcl 11241 ax-addrcl 11242 ax-mulcl 11243 ax-mulrcl 11244 ax-mulcom 11245 ax-addass 11246 ax-mulass 11247 ax-distr 11248 ax-i2m1 11249 ax-1ne0 11250 ax-1rid 11251 ax-rnegex 11252 ax-rrecex 11253 ax-cnre 11254 ax-pre-lttri 11255 ax-pre-lttrn 11256 ax-pre-ltadd 11257 ax-pre-mulgt0 11258

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 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6297 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6487 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7867 df-1st 7990 df-2nd 7991 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8460 df-oadd 8464 df-er 8701 df-map 8833 df-pm 8834 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-dju 9963 df-card 10001 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11524 df-neg 11525 df-nn 12317 df-2 12386 df-3 12387 df-n0 12588 df-xnn0 12661 df-z 12675 df-uz 12947 df-fz 13621 df-fzo 13769 df-hash 14455 df-word 14639 df-concat 14696 df-s1 14723 df-s2 14979 df-s3 14980 df-trkgc 28892 df-trkgb 28893 df-trkgcb 28894 df-trkge 28895 df-trkgld 28896 df-trkg 28897 df-cgrg 28956 df-ismt 28978 df-leg 29028 df-hlg 29046 df-mir 29107 df-rag 29151 df-perpg 29153 df-hpg 29218 df-plng 29234 df-mid 29261 df-lmi 29262 df-cgra 29297 df-prlng 29397

This theorem is used by: (None)

W3C validator