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Theorem cgraswap 28652
Description: Swap rays in a congruence relation. Theorem 11.9 of [Schwabhauser] p. 96. (Contributed by Thierry Arnoux, 5-Mar-2020.)
Hypotheses
Ref Expression
cgraid.p 𝑃 = (Baseβ€˜πΊ)
cgraid.i 𝐼 = (Itvβ€˜πΊ)
cgraid.g (πœ‘ β†’ 𝐺 ∈ TarskiG)
cgraid.k 𝐾 = (hlGβ€˜πΊ)
cgraid.a (πœ‘ β†’ 𝐴 ∈ 𝑃)
cgraid.b (πœ‘ β†’ 𝐡 ∈ 𝑃)
cgraid.c (πœ‘ β†’ 𝐢 ∈ 𝑃)
cgraid.1 (πœ‘ β†’ 𝐴 β‰  𝐡)
cgraid.2 (πœ‘ β†’ 𝐡 β‰  𝐢)
Assertion
Ref Expression
cgraswap (πœ‘ β†’ βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπΆπ΅π΄β€βŸ©)

Proof of Theorem cgraswap
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cgraid.p . . . . 5 𝑃 = (Baseβ€˜πΊ)
2 eqid 2728 . . . . 5 (distβ€˜πΊ) = (distβ€˜πΊ)
3 eqid 2728 . . . . 5 (cgrGβ€˜πΊ) = (cgrGβ€˜πΊ)
4 cgraid.g . . . . . 6 (πœ‘ β†’ 𝐺 ∈ TarskiG)
54ad3antrrr 728 . . . . 5 ((((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((π‘₯(πΎβ€˜π΅)𝐢 ∧ (𝐡(distβ€˜πΊ)π‘₯) = (𝐡(distβ€˜πΊ)𝐴)) ∧ (𝑦(πΎβ€˜π΅)𝐴 ∧ (𝐡(distβ€˜πΊ)𝑦) = (𝐡(distβ€˜πΊ)𝐢)))) β†’ 𝐺 ∈ TarskiG)
6 cgraid.a . . . . . 6 (πœ‘ β†’ 𝐴 ∈ 𝑃)
76ad3antrrr 728 . . . . 5 ((((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((π‘₯(πΎβ€˜π΅)𝐢 ∧ (𝐡(distβ€˜πΊ)π‘₯) = (𝐡(distβ€˜πΊ)𝐴)) ∧ (𝑦(πΎβ€˜π΅)𝐴 ∧ (𝐡(distβ€˜πΊ)𝑦) = (𝐡(distβ€˜πΊ)𝐢)))) β†’ 𝐴 ∈ 𝑃)
8 cgraid.b . . . . . 6 (πœ‘ β†’ 𝐡 ∈ 𝑃)
98ad3antrrr 728 . . . . 5 ((((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((π‘₯(πΎβ€˜π΅)𝐢 ∧ (𝐡(distβ€˜πΊ)π‘₯) = (𝐡(distβ€˜πΊ)𝐴)) ∧ (𝑦(πΎβ€˜π΅)𝐴 ∧ (𝐡(distβ€˜πΊ)𝑦) = (𝐡(distβ€˜πΊ)𝐢)))) β†’ 𝐡 ∈ 𝑃)
10 cgraid.c . . . . . 6 (πœ‘ β†’ 𝐢 ∈ 𝑃)
1110ad3antrrr 728 . . . . 5 ((((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((π‘₯(πΎβ€˜π΅)𝐢 ∧ (𝐡(distβ€˜πΊ)π‘₯) = (𝐡(distβ€˜πΊ)𝐴)) ∧ (𝑦(πΎβ€˜π΅)𝐴 ∧ (𝐡(distβ€˜πΊ)𝑦) = (𝐡(distβ€˜πΊ)𝐢)))) β†’ 𝐢 ∈ 𝑃)
12 simpllr 774 . . . . 5 ((((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((π‘₯(πΎβ€˜π΅)𝐢 ∧ (𝐡(distβ€˜πΊ)π‘₯) = (𝐡(distβ€˜πΊ)𝐴)) ∧ (𝑦(πΎβ€˜π΅)𝐴 ∧ (𝐡(distβ€˜πΊ)𝑦) = (𝐡(distβ€˜πΊ)𝐢)))) β†’ π‘₯ ∈ 𝑃)
13 simplr 767 . . . . 5 ((((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((π‘₯(πΎβ€˜π΅)𝐢 ∧ (𝐡(distβ€˜πΊ)π‘₯) = (𝐡(distβ€˜πΊ)𝐴)) ∧ (𝑦(πΎβ€˜π΅)𝐴 ∧ (𝐡(distβ€˜πΊ)𝑦) = (𝐡(distβ€˜πΊ)𝐢)))) β†’ 𝑦 ∈ 𝑃)
14 cgraid.i . . . . . . 7 𝐼 = (Itvβ€˜πΊ)
15 simprlr 778 . . . . . . 7 ((((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((π‘₯(πΎβ€˜π΅)𝐢 ∧ (𝐡(distβ€˜πΊ)π‘₯) = (𝐡(distβ€˜πΊ)𝐴)) ∧ (𝑦(πΎβ€˜π΅)𝐴 ∧ (𝐡(distβ€˜πΊ)𝑦) = (𝐡(distβ€˜πΊ)𝐢)))) β†’ (𝐡(distβ€˜πΊ)π‘₯) = (𝐡(distβ€˜πΊ)𝐴))
161, 2, 14, 5, 9, 12, 9, 7, 15tgcgrcomlr 28312 . . . . . 6 ((((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((π‘₯(πΎβ€˜π΅)𝐢 ∧ (𝐡(distβ€˜πΊ)π‘₯) = (𝐡(distβ€˜πΊ)𝐴)) ∧ (𝑦(πΎβ€˜π΅)𝐴 ∧ (𝐡(distβ€˜πΊ)𝑦) = (𝐡(distβ€˜πΊ)𝐢)))) β†’ (π‘₯(distβ€˜πΊ)𝐡) = (𝐴(distβ€˜πΊ)𝐡))
1716eqcomd 2734 . . . . 5 ((((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((π‘₯(πΎβ€˜π΅)𝐢 ∧ (𝐡(distβ€˜πΊ)π‘₯) = (𝐡(distβ€˜πΊ)𝐴)) ∧ (𝑦(πΎβ€˜π΅)𝐴 ∧ (𝐡(distβ€˜πΊ)𝑦) = (𝐡(distβ€˜πΊ)𝐢)))) β†’ (𝐴(distβ€˜πΊ)𝐡) = (π‘₯(distβ€˜πΊ)𝐡))
18 simprrr 780 . . . . . 6 ((((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((π‘₯(πΎβ€˜π΅)𝐢 ∧ (𝐡(distβ€˜πΊ)π‘₯) = (𝐡(distβ€˜πΊ)𝐴)) ∧ (𝑦(πΎβ€˜π΅)𝐴 ∧ (𝐡(distβ€˜πΊ)𝑦) = (𝐡(distβ€˜πΊ)𝐢)))) β†’ (𝐡(distβ€˜πΊ)𝑦) = (𝐡(distβ€˜πΊ)𝐢))
1918eqcomd 2734 . . . . 5 ((((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((π‘₯(πΎβ€˜π΅)𝐢 ∧ (𝐡(distβ€˜πΊ)π‘₯) = (𝐡(distβ€˜πΊ)𝐴)) ∧ (𝑦(πΎβ€˜π΅)𝐴 ∧ (𝐡(distβ€˜πΊ)𝑦) = (𝐡(distβ€˜πΊ)𝐢)))) β†’ (𝐡(distβ€˜πΊ)𝐢) = (𝐡(distβ€˜πΊ)𝑦))
20 eqid 2728 . . . . . . . 8 (LineGβ€˜πΊ) = (LineGβ€˜πΊ)
21 cgraid.k . . . . . . . . . . 11 𝐾 = (hlGβ€˜πΊ)
22 simprll 777 . . . . . . . . . . 11 ((((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((π‘₯(πΎβ€˜π΅)𝐢 ∧ (𝐡(distβ€˜πΊ)π‘₯) = (𝐡(distβ€˜πΊ)𝐴)) ∧ (𝑦(πΎβ€˜π΅)𝐴 ∧ (𝐡(distβ€˜πΊ)𝑦) = (𝐡(distβ€˜πΊ)𝐢)))) β†’ π‘₯(πΎβ€˜π΅)𝐢)
231, 14, 21, 12, 11, 9, 5, 20, 22hlln 28439 . . . . . . . . . 10 ((((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((π‘₯(πΎβ€˜π΅)𝐢 ∧ (𝐡(distβ€˜πΊ)π‘₯) = (𝐡(distβ€˜πΊ)𝐴)) ∧ (𝑦(πΎβ€˜π΅)𝐴 ∧ (𝐡(distβ€˜πΊ)𝑦) = (𝐡(distβ€˜πΊ)𝐢)))) β†’ π‘₯ ∈ (𝐢(LineGβ€˜πΊ)𝐡))
2423orcd 871 . . . . . . . . 9 ((((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((π‘₯(πΎβ€˜π΅)𝐢 ∧ (𝐡(distβ€˜πΊ)π‘₯) = (𝐡(distβ€˜πΊ)𝐴)) ∧ (𝑦(πΎβ€˜π΅)𝐴 ∧ (𝐡(distβ€˜πΊ)𝑦) = (𝐡(distβ€˜πΊ)𝐢)))) β†’ (π‘₯ ∈ (𝐢(LineGβ€˜πΊ)𝐡) ∨ 𝐢 = 𝐡))
251, 20, 14, 5, 11, 9, 12, 24colrot1 28391 . . . . . . . 8 ((((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((π‘₯(πΎβ€˜π΅)𝐢 ∧ (𝐡(distβ€˜πΊ)π‘₯) = (𝐡(distβ€˜πΊ)𝐴)) ∧ (𝑦(πΎβ€˜π΅)𝐴 ∧ (𝐡(distβ€˜πΊ)𝑦) = (𝐡(distβ€˜πΊ)𝐢)))) β†’ (𝐢 ∈ (𝐡(LineGβ€˜πΊ)π‘₯) ∨ 𝐡 = π‘₯))
26 eqid 2728 . . . . . . . . . 10 (≀Gβ€˜πΊ) = (≀Gβ€˜πΊ)
271, 14, 21, 12, 11, 9, 5ishlg 28434 . . . . . . . . . . . . 13 ((((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((π‘₯(πΎβ€˜π΅)𝐢 ∧ (𝐡(distβ€˜πΊ)π‘₯) = (𝐡(distβ€˜πΊ)𝐴)) ∧ (𝑦(πΎβ€˜π΅)𝐴 ∧ (𝐡(distβ€˜πΊ)𝑦) = (𝐡(distβ€˜πΊ)𝐢)))) β†’ (π‘₯(πΎβ€˜π΅)𝐢 ↔ (π‘₯ β‰  𝐡 ∧ 𝐢 β‰  𝐡 ∧ (π‘₯ ∈ (𝐡𝐼𝐢) ∨ 𝐢 ∈ (𝐡𝐼π‘₯)))))
2822, 27mpbid 231 . . . . . . . . . . . 12 ((((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((π‘₯(πΎβ€˜π΅)𝐢 ∧ (𝐡(distβ€˜πΊ)π‘₯) = (𝐡(distβ€˜πΊ)𝐴)) ∧ (𝑦(πΎβ€˜π΅)𝐴 ∧ (𝐡(distβ€˜πΊ)𝑦) = (𝐡(distβ€˜πΊ)𝐢)))) β†’ (π‘₯ β‰  𝐡 ∧ 𝐢 β‰  𝐡 ∧ (π‘₯ ∈ (𝐡𝐼𝐢) ∨ 𝐢 ∈ (𝐡𝐼π‘₯))))
2928simp3d 1141 . . . . . . . . . . 11 ((((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((π‘₯(πΎβ€˜π΅)𝐢 ∧ (𝐡(distβ€˜πΊ)π‘₯) = (𝐡(distβ€˜πΊ)𝐴)) ∧ (𝑦(πΎβ€˜π΅)𝐴 ∧ (𝐡(distβ€˜πΊ)𝑦) = (𝐡(distβ€˜πΊ)𝐢)))) β†’ (π‘₯ ∈ (𝐡𝐼𝐢) ∨ 𝐢 ∈ (𝐡𝐼π‘₯)))
3029orcomd 869 . . . . . . . . . 10 ((((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((π‘₯(πΎβ€˜π΅)𝐢 ∧ (𝐡(distβ€˜πΊ)π‘₯) = (𝐡(distβ€˜πΊ)𝐴)) ∧ (𝑦(πΎβ€˜π΅)𝐴 ∧ (𝐡(distβ€˜πΊ)𝑦) = (𝐡(distβ€˜πΊ)𝐢)))) β†’ (𝐢 ∈ (𝐡𝐼π‘₯) ∨ π‘₯ ∈ (𝐡𝐼𝐢)))
31 simprrl 779 . . . . . . . . . . . 12 ((((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((π‘₯(πΎβ€˜π΅)𝐢 ∧ (𝐡(distβ€˜πΊ)π‘₯) = (𝐡(distβ€˜πΊ)𝐴)) ∧ (𝑦(πΎβ€˜π΅)𝐴 ∧ (𝐡(distβ€˜πΊ)𝑦) = (𝐡(distβ€˜πΊ)𝐢)))) β†’ 𝑦(πΎβ€˜π΅)𝐴)
321, 14, 21, 13, 7, 9, 5ishlg 28434 . . . . . . . . . . . 12 ((((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((π‘₯(πΎβ€˜π΅)𝐢 ∧ (𝐡(distβ€˜πΊ)π‘₯) = (𝐡(distβ€˜πΊ)𝐴)) ∧ (𝑦(πΎβ€˜π΅)𝐴 ∧ (𝐡(distβ€˜πΊ)𝑦) = (𝐡(distβ€˜πΊ)𝐢)))) β†’ (𝑦(πΎβ€˜π΅)𝐴 ↔ (𝑦 β‰  𝐡 ∧ 𝐴 β‰  𝐡 ∧ (𝑦 ∈ (𝐡𝐼𝐴) ∨ 𝐴 ∈ (𝐡𝐼𝑦)))))
3331, 32mpbid 231 . . . . . . . . . . 11 ((((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((π‘₯(πΎβ€˜π΅)𝐢 ∧ (𝐡(distβ€˜πΊ)π‘₯) = (𝐡(distβ€˜πΊ)𝐴)) ∧ (𝑦(πΎβ€˜π΅)𝐴 ∧ (𝐡(distβ€˜πΊ)𝑦) = (𝐡(distβ€˜πΊ)𝐢)))) β†’ (𝑦 β‰  𝐡 ∧ 𝐴 β‰  𝐡 ∧ (𝑦 ∈ (𝐡𝐼𝐴) ∨ 𝐴 ∈ (𝐡𝐼𝑦))))
3433simp3d 1141 . . . . . . . . . 10 ((((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((π‘₯(πΎβ€˜π΅)𝐢 ∧ (𝐡(distβ€˜πΊ)π‘₯) = (𝐡(distβ€˜πΊ)𝐴)) ∧ (𝑦(πΎβ€˜π΅)𝐴 ∧ (𝐡(distβ€˜πΊ)𝑦) = (𝐡(distβ€˜πΊ)𝐢)))) β†’ (𝑦 ∈ (𝐡𝐼𝐴) ∨ 𝐴 ∈ (𝐡𝐼𝑦)))
351, 2, 14, 26, 5, 9, 11, 12, 9, 9, 13, 7, 30, 34, 19, 15tgcgrsub2 28427 . . . . . . . . 9 ((((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((π‘₯(πΎβ€˜π΅)𝐢 ∧ (𝐡(distβ€˜πΊ)π‘₯) = (𝐡(distβ€˜πΊ)𝐴)) ∧ (𝑦(πΎβ€˜π΅)𝐴 ∧ (𝐡(distβ€˜πΊ)𝑦) = (𝐡(distβ€˜πΊ)𝐢)))) β†’ (𝐢(distβ€˜πΊ)π‘₯) = (𝑦(distβ€˜πΊ)𝐴))
361, 2, 3, 5, 9, 11, 12, 9, 13, 7, 19, 35, 16trgcgr 28348 . . . . . . . 8 ((((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((π‘₯(πΎβ€˜π΅)𝐢 ∧ (𝐡(distβ€˜πΊ)π‘₯) = (𝐡(distβ€˜πΊ)𝐴)) ∧ (𝑦(πΎβ€˜π΅)𝐴 ∧ (𝐡(distβ€˜πΊ)𝑦) = (𝐡(distβ€˜πΊ)𝐢)))) β†’ βŸ¨β€œπ΅πΆπ‘₯β€βŸ©(cgrGβ€˜πΊ)βŸ¨β€œπ΅π‘¦π΄β€βŸ©)
371, 2, 14, 5, 11, 13axtgcgrrflx 28294 . . . . . . . 8 ((((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((π‘₯(πΎβ€˜π΅)𝐢 ∧ (𝐡(distβ€˜πΊ)π‘₯) = (𝐡(distβ€˜πΊ)𝐴)) ∧ (𝑦(πΎβ€˜π΅)𝐴 ∧ (𝐡(distβ€˜πΊ)𝑦) = (𝐡(distβ€˜πΊ)𝐢)))) β†’ (𝐢(distβ€˜πΊ)𝑦) = (𝑦(distβ€˜πΊ)𝐢))
38 cgraid.2 . . . . . . . . 9 (πœ‘ β†’ 𝐡 β‰  𝐢)
3938ad3antrrr 728 . . . . . . . 8 ((((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((π‘₯(πΎβ€˜π΅)𝐢 ∧ (𝐡(distβ€˜πΊ)π‘₯) = (𝐡(distβ€˜πΊ)𝐴)) ∧ (𝑦(πΎβ€˜π΅)𝐴 ∧ (𝐡(distβ€˜πΊ)𝑦) = (𝐡(distβ€˜πΊ)𝐢)))) β†’ 𝐡 β‰  𝐢)
401, 20, 14, 5, 9, 11, 12, 3, 9, 13, 2, 13, 7, 11, 25, 36, 18, 37, 39tgfscgr 28400 . . . . . . 7 ((((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((π‘₯(πΎβ€˜π΅)𝐢 ∧ (𝐡(distβ€˜πΊ)π‘₯) = (𝐡(distβ€˜πΊ)𝐴)) ∧ (𝑦(πΎβ€˜π΅)𝐴 ∧ (𝐡(distβ€˜πΊ)𝑦) = (𝐡(distβ€˜πΊ)𝐢)))) β†’ (π‘₯(distβ€˜πΊ)𝑦) = (𝐴(distβ€˜πΊ)𝐢))
411, 2, 14, 5, 12, 13, 7, 11, 40tgcgrcomlr 28312 . . . . . 6 ((((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((π‘₯(πΎβ€˜π΅)𝐢 ∧ (𝐡(distβ€˜πΊ)π‘₯) = (𝐡(distβ€˜πΊ)𝐴)) ∧ (𝑦(πΎβ€˜π΅)𝐴 ∧ (𝐡(distβ€˜πΊ)𝑦) = (𝐡(distβ€˜πΊ)𝐢)))) β†’ (𝑦(distβ€˜πΊ)π‘₯) = (𝐢(distβ€˜πΊ)𝐴))
4241eqcomd 2734 . . . . 5 ((((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((π‘₯(πΎβ€˜π΅)𝐢 ∧ (𝐡(distβ€˜πΊ)π‘₯) = (𝐡(distβ€˜πΊ)𝐴)) ∧ (𝑦(πΎβ€˜π΅)𝐴 ∧ (𝐡(distβ€˜πΊ)𝑦) = (𝐡(distβ€˜πΊ)𝐢)))) β†’ (𝐢(distβ€˜πΊ)𝐴) = (𝑦(distβ€˜πΊ)π‘₯))
431, 2, 3, 5, 7, 9, 11, 12, 9, 13, 17, 19, 42trgcgr 28348 . . . 4 ((((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((π‘₯(πΎβ€˜π΅)𝐢 ∧ (𝐡(distβ€˜πΊ)π‘₯) = (𝐡(distβ€˜πΊ)𝐴)) ∧ (𝑦(πΎβ€˜π΅)𝐴 ∧ (𝐡(distβ€˜πΊ)𝑦) = (𝐡(distβ€˜πΊ)𝐢)))) β†’ βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrGβ€˜πΊ)βŸ¨β€œπ‘₯π΅π‘¦β€βŸ©)
4443, 22, 313jca 1125 . . 3 ((((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((π‘₯(πΎβ€˜π΅)𝐢 ∧ (𝐡(distβ€˜πΊ)π‘₯) = (𝐡(distβ€˜πΊ)𝐴)) ∧ (𝑦(πΎβ€˜π΅)𝐴 ∧ (𝐡(distβ€˜πΊ)𝑦) = (𝐡(distβ€˜πΊ)𝐢)))) β†’ (βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrGβ€˜πΊ)βŸ¨β€œπ‘₯π΅π‘¦β€βŸ© ∧ π‘₯(πΎβ€˜π΅)𝐢 ∧ 𝑦(πΎβ€˜π΅)𝐴))
4538necomd 2993 . . . . 5 (πœ‘ β†’ 𝐢 β‰  𝐡)
46 cgraid.1 . . . . . 6 (πœ‘ β†’ 𝐴 β‰  𝐡)
4746necomd 2993 . . . . 5 (πœ‘ β†’ 𝐡 β‰  𝐴)
481, 14, 21, 8, 8, 6, 4, 10, 2, 45, 47hlcgrex 28448 . . . 4 (πœ‘ β†’ βˆƒπ‘₯ ∈ 𝑃 (π‘₯(πΎβ€˜π΅)𝐢 ∧ (𝐡(distβ€˜πΊ)π‘₯) = (𝐡(distβ€˜πΊ)𝐴)))
491, 14, 21, 8, 8, 10, 4, 6, 2, 46, 38hlcgrex 28448 . . . 4 (πœ‘ β†’ βˆƒπ‘¦ ∈ 𝑃 (𝑦(πΎβ€˜π΅)𝐴 ∧ (𝐡(distβ€˜πΊ)𝑦) = (𝐡(distβ€˜πΊ)𝐢)))
50 reeanv 3224 . . . 4 (βˆƒπ‘₯ ∈ 𝑃 βˆƒπ‘¦ ∈ 𝑃 ((π‘₯(πΎβ€˜π΅)𝐢 ∧ (𝐡(distβ€˜πΊ)π‘₯) = (𝐡(distβ€˜πΊ)𝐴)) ∧ (𝑦(πΎβ€˜π΅)𝐴 ∧ (𝐡(distβ€˜πΊ)𝑦) = (𝐡(distβ€˜πΊ)𝐢))) ↔ (βˆƒπ‘₯ ∈ 𝑃 (π‘₯(πΎβ€˜π΅)𝐢 ∧ (𝐡(distβ€˜πΊ)π‘₯) = (𝐡(distβ€˜πΊ)𝐴)) ∧ βˆƒπ‘¦ ∈ 𝑃 (𝑦(πΎβ€˜π΅)𝐴 ∧ (𝐡(distβ€˜πΊ)𝑦) = (𝐡(distβ€˜πΊ)𝐢))))
5148, 49, 50sylanbrc 581 . . 3 (πœ‘ β†’ βˆƒπ‘₯ ∈ 𝑃 βˆƒπ‘¦ ∈ 𝑃 ((π‘₯(πΎβ€˜π΅)𝐢 ∧ (𝐡(distβ€˜πΊ)π‘₯) = (𝐡(distβ€˜πΊ)𝐴)) ∧ (𝑦(πΎβ€˜π΅)𝐴 ∧ (𝐡(distβ€˜πΊ)𝑦) = (𝐡(distβ€˜πΊ)𝐢))))
5244, 51reximddv2 3210 . 2 (πœ‘ β†’ βˆƒπ‘₯ ∈ 𝑃 βˆƒπ‘¦ ∈ 𝑃 (βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrGβ€˜πΊ)βŸ¨β€œπ‘₯π΅π‘¦β€βŸ© ∧ π‘₯(πΎβ€˜π΅)𝐢 ∧ 𝑦(πΎβ€˜π΅)𝐴))
531, 14, 21, 4, 6, 8, 10, 10, 8, 6iscgra 28641 . 2 (πœ‘ β†’ (βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπΆπ΅π΄β€βŸ© ↔ βˆƒπ‘₯ ∈ 𝑃 βˆƒπ‘¦ ∈ 𝑃 (βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrGβ€˜πΊ)βŸ¨β€œπ‘₯π΅π‘¦β€βŸ© ∧ π‘₯(πΎβ€˜π΅)𝐢 ∧ 𝑦(πΎβ€˜π΅)𝐴)))
5452, 53mpbird 256 1 (πœ‘ β†’ βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπΆπ΅π΄β€βŸ©)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   ∨ wo 845   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2937  βˆƒwrex 3067   class class class wbr 5152  β€˜cfv 6553  (class class class)co 7426  βŸ¨β€œcs3 14835  Basecbs 17189  distcds 17251  TarskiGcstrkg 28259  Itvcitv 28265  LineGclng 28266  cgrGccgrg 28342  β‰€Gcleg 28414  hlGchlg 28432  cgrAccgra 28639
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7748  ax-cnex 11204  ax-resscn 11205  ax-1cn 11206  ax-icn 11207  ax-addcl 11208  ax-addrcl 11209  ax-mulcl 11210  ax-mulrcl 11211  ax-mulcom 11212  ax-addass 11213  ax-mulass 11214  ax-distr 11215  ax-i2m1 11216  ax-1ne0 11217  ax-1rid 11218  ax-rnegex 11219  ax-rrecex 11220  ax-cnre 11221  ax-pre-lttri 11222  ax-pre-lttrn 11223  ax-pre-ltadd 11224  ax-pre-mulgt0 11225
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-tp 4637  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7879  df-1st 8001  df-2nd 8002  df-frecs 8295  df-wrecs 8326  df-recs 8400  df-rdg 8439  df-1o 8495  df-oadd 8499  df-er 8733  df-map 8855  df-pm 8856  df-en 8973  df-dom 8974  df-sdom 8975  df-fin 8976  df-dju 9934  df-card 9972  df-pnf 11290  df-mnf 11291  df-xr 11292  df-ltxr 11293  df-le 11294  df-sub 11486  df-neg 11487  df-nn 12253  df-2 12315  df-3 12316  df-n0 12513  df-xnn0 12585  df-z 12599  df-uz 12863  df-fz 13527  df-fzo 13670  df-hash 14332  df-word 14507  df-concat 14563  df-s1 14588  df-s2 14841  df-s3 14842  df-trkgc 28280  df-trkgb 28281  df-trkgcb 28282  df-trkg 28285  df-cgrg 28343  df-leg 28415  df-hlg 28433  df-cgra 28640
This theorem is referenced by:  cgraswaplr  28657  oacgr  28664  tgasa1  28690
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