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Theorem cgraswap 27804
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 2737 . . . . 5 (distβ€˜πΊ) = (distβ€˜πΊ)
3 eqid 2737 . . . . 5 (cgrGβ€˜πΊ) = (cgrGβ€˜πΊ)
4 cgraid.g . . . . . 6 (πœ‘ β†’ 𝐺 ∈ TarskiG)
54ad3antrrr 729 . . . . 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 729 . . . . 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 729 . . . . 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 729 . . . . 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 775 . . . . 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 768 . . . . 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 779 . . . . . . 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 27464 . . . . . 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 2743 . . . . 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 781 . . . . . 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 2743 . . . . 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 2737 . . . . . . . 8 (LineGβ€˜πΊ) = (LineGβ€˜πΊ)
21 cgraid.k . . . . . . . . . . 11 𝐾 = (hlGβ€˜πΊ)
22 simprll 778 . . . . . . . . . . 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 27591 . . . . . . . . . 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 872 . . . . . . . . 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 27543 . . . . . . . 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 2737 . . . . . . . . . 10 (≀Gβ€˜πΊ) = (≀Gβ€˜πΊ)
271, 14, 21, 12, 11, 9, 5ishlg 27586 . . . . . . . . . . . . 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 1145 . . . . . . . . . . 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 870 . . . . . . . . . 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 780 . . . . . . . . . . . 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 27586 . . . . . . . . . . . 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 1145 . . . . . . . . . 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 27579 . . . . . . . . 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 27500 . . . . . . . 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 27446 . . . . . . . 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 729 . . . . . . . 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 27552 . . . . . . 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 27464 . . . . . 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 2743 . . . . 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 27500 . . . 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 1129 . . 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 3000 . . . . 5 (πœ‘ β†’ 𝐢 β‰  𝐡)
46 cgraid.1 . . . . . 6 (πœ‘ β†’ 𝐴 β‰  𝐡)
4746necomd 3000 . . . . 5 (πœ‘ β†’ 𝐡 β‰  𝐴)
481, 14, 21, 8, 8, 6, 4, 10, 2, 45, 47hlcgrex 27600 . . . 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 27600 . . . 4 (πœ‘ β†’ βˆƒπ‘¦ ∈ 𝑃 (𝑦(πΎβ€˜π΅)𝐴 ∧ (𝐡(distβ€˜πΊ)𝑦) = (𝐡(distβ€˜πΊ)𝐢)))
50 reeanv 3220 . . . 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 584 . . 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 3207 . 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 27793 . 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 257 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 397   ∨ wo 846   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2944  βˆƒwrex 3074   class class class wbr 5110  β€˜cfv 6501  (class class class)co 7362  βŸ¨β€œcs3 14738  Basecbs 17090  distcds 17149  TarskiGcstrkg 27411  Itvcitv 27417  LineGclng 27418  cgrGccgrg 27494  β‰€Gcleg 27566  hlGchlg 27584  cgrAccgra 27791
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11114  ax-resscn 11115  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-addrcl 11119  ax-mulcl 11120  ax-mulrcl 11121  ax-mulcom 11122  ax-addass 11123  ax-mulass 11124  ax-distr 11125  ax-i2m1 11126  ax-1ne0 11127  ax-1rid 11128  ax-rnegex 11129  ax-rrecex 11130  ax-cnre 11131  ax-pre-lttri 11132  ax-pre-lttrn 11133  ax-pre-ltadd 11134  ax-pre-mulgt0 11135
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-oadd 8421  df-er 8655  df-map 8774  df-pm 8775  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-dju 9844  df-card 9882  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-sub 11394  df-neg 11395  df-nn 12161  df-2 12223  df-3 12224  df-n0 12421  df-xnn0 12493  df-z 12507  df-uz 12771  df-fz 13432  df-fzo 13575  df-hash 14238  df-word 14410  df-concat 14466  df-s1 14491  df-s2 14744  df-s3 14745  df-trkgc 27432  df-trkgb 27433  df-trkgcb 27434  df-trkg 27437  df-cgrg 27495  df-leg 27567  df-hlg 27585  df-cgra 27792
This theorem is referenced by:  cgraswaplr  27809  oacgr  27816  tgasa1  27842
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