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Theorem cgraswap 28579
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 2726 . . . . 5 (distβ€˜πΊ) = (distβ€˜πΊ)
3 eqid 2726 . . . . 5 (cgrGβ€˜πΊ) = (cgrGβ€˜πΊ)
4 cgraid.g . . . . . 6 (πœ‘ β†’ 𝐺 ∈ TarskiG)
54ad3antrrr 727 . . . . 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 727 . . . . 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 727 . . . . 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 727 . . . . 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 773 . . . . 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 766 . . . . 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 777 . . . . . . 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 28239 . . . . . 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 2732 . . . . 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 779 . . . . . 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 2732 . . . . 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 2726 . . . . . . . 8 (LineGβ€˜πΊ) = (LineGβ€˜πΊ)
21 cgraid.k . . . . . . . . . . 11 𝐾 = (hlGβ€˜πΊ)
22 simprll 776 . . . . . . . . . . 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 28366 . . . . . . . . . 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 870 . . . . . . . . 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 28318 . . . . . . . 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 2726 . . . . . . . . . 10 (≀Gβ€˜πΊ) = (≀Gβ€˜πΊ)
271, 14, 21, 12, 11, 9, 5ishlg 28361 . . . . . . . . . . . . 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 868 . . . . . . . . . 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 778 . . . . . . . . . . . 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 28361 . . . . . . . . . . . 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 28354 . . . . . . . . 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 28275 . . . . . . . 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 28221 . . . . . . . 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 727 . . . . . . . 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 28327 . . . . . . 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 28239 . . . . . 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 2732 . . . . 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 28275 . . . 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 2990 . . . . 5 (πœ‘ β†’ 𝐢 β‰  𝐡)
46 cgraid.1 . . . . . 6 (πœ‘ β†’ 𝐴 β‰  𝐡)
4746necomd 2990 . . . . 5 (πœ‘ β†’ 𝐡 β‰  𝐴)
481, 14, 21, 8, 8, 6, 4, 10, 2, 45, 47hlcgrex 28375 . . . 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 28375 . . . 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 582 . . 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 3206 . 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 28568 . 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 395   ∨ wo 844   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2934  βˆƒwrex 3064   class class class wbr 5141  β€˜cfv 6537  (class class class)co 7405  βŸ¨β€œcs3 14799  Basecbs 17153  distcds 17215  TarskiGcstrkg 28186  Itvcitv 28192  LineGclng 28193  cgrGccgrg 28269  β‰€Gcleg 28341  hlGchlg 28359  cgrAccgra 28566
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-1st 7974  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-1o 8467  df-oadd 8471  df-er 8705  df-map 8824  df-pm 8825  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-dju 9898  df-card 9936  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-3 12280  df-n0 12477  df-xnn0 12549  df-z 12563  df-uz 12827  df-fz 13491  df-fzo 13634  df-hash 14296  df-word 14471  df-concat 14527  df-s1 14552  df-s2 14805  df-s3 14806  df-trkgc 28207  df-trkgb 28208  df-trkgcb 28209  df-trkg 28212  df-cgrg 28270  df-leg 28342  df-hlg 28360  df-cgra 28567
This theorem is referenced by:  cgraswaplr  28584  oacgr  28591  tgasa1  28617
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