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Theorem colperpexlem1 28573
Description: Lemma for colperp 28572. First part of lemma 8.20 of [Schwabhauser] p. 62. (Contributed by Thierry Arnoux, 27-Oct-2019.)
Hypotheses
Ref Expression
colperpex.p 𝑃 = (Baseβ€˜πΊ)
colperpex.d βˆ’ = (distβ€˜πΊ)
colperpex.i 𝐼 = (Itvβ€˜πΊ)
colperpex.l 𝐿 = (LineGβ€˜πΊ)
colperpex.g (πœ‘ β†’ 𝐺 ∈ TarskiG)
colperpexlem.s 𝑆 = (pInvGβ€˜πΊ)
colperpexlem.m 𝑀 = (π‘†β€˜π΄)
colperpexlem.n 𝑁 = (π‘†β€˜π΅)
colperpexlem.k 𝐾 = (π‘†β€˜π‘„)
colperpexlem.a (πœ‘ β†’ 𝐴 ∈ 𝑃)
colperpexlem.b (πœ‘ β†’ 𝐡 ∈ 𝑃)
colperpexlem.c (πœ‘ β†’ 𝐢 ∈ 𝑃)
colperpexlem.q (πœ‘ β†’ 𝑄 ∈ 𝑃)
colperpexlem.1 (πœ‘ β†’ βŸ¨β€œπ΄π΅πΆβ€βŸ© ∈ (∟Gβ€˜πΊ))
colperpexlem.2 (πœ‘ β†’ (πΎβ€˜(π‘€β€˜πΆ)) = (π‘β€˜πΆ))
Assertion
Ref Expression
colperpexlem1 (πœ‘ β†’ βŸ¨β€œπ΅π΄π‘„β€βŸ© ∈ (∟Gβ€˜πΊ))

Proof of Theorem colperpexlem1
StepHypRef Expression
1 colperpex.p . . . 4 𝑃 = (Baseβ€˜πΊ)
2 colperpex.d . . . 4 βˆ’ = (distβ€˜πΊ)
3 colperpex.i . . . 4 𝐼 = (Itvβ€˜πΊ)
4 colperpex.g . . . 4 (πœ‘ β†’ 𝐺 ∈ TarskiG)
5 colperpexlem.q . . . 4 (πœ‘ β†’ 𝑄 ∈ 𝑃)
6 colperpexlem.b . . . 4 (πœ‘ β†’ 𝐡 ∈ 𝑃)
7 colperpex.l . . . . 5 𝐿 = (LineGβ€˜πΊ)
8 colperpexlem.s . . . . 5 𝑆 = (pInvGβ€˜πΊ)
9 colperpexlem.a . . . . 5 (πœ‘ β†’ 𝐴 ∈ 𝑃)
10 colperpexlem.m . . . . 5 𝑀 = (π‘†β€˜π΄)
111, 2, 3, 7, 8, 4, 9, 10, 5mircl 28504 . . . 4 (πœ‘ β†’ (π‘€β€˜π‘„) ∈ 𝑃)
12 colperpexlem.c . . . . . 6 (πœ‘ β†’ 𝐢 ∈ 𝑃)
131, 2, 3, 7, 8, 4, 9, 10, 12mircl 28504 . . . . 5 (πœ‘ β†’ (π‘€β€˜πΆ) ∈ 𝑃)
14 eqid 2725 . . . . . 6 (π‘†β€˜π΅) = (π‘†β€˜π΅)
151, 2, 3, 7, 8, 4, 6, 14, 12mircl 28504 . . . . 5 (πœ‘ β†’ ((π‘†β€˜π΅)β€˜πΆ) ∈ 𝑃)
161, 2, 3, 7, 8, 4, 9, 10, 15mircl 28504 . . . . 5 (πœ‘ β†’ (π‘€β€˜((π‘†β€˜π΅)β€˜πΆ)) ∈ 𝑃)
17 colperpexlem.2 . . . . . . . 8 (πœ‘ β†’ (πΎβ€˜(π‘€β€˜πΆ)) = (π‘β€˜πΆ))
18 colperpexlem.n . . . . . . . . 9 𝑁 = (π‘†β€˜π΅)
191, 2, 3, 7, 8, 4, 6, 18, 12mircl 28504 . . . . . . . 8 (πœ‘ β†’ (π‘β€˜πΆ) ∈ 𝑃)
2017, 19eqeltrd 2825 . . . . . . 7 (πœ‘ β†’ (πΎβ€˜(π‘€β€˜πΆ)) ∈ 𝑃)
21 colperpexlem.k . . . . . . . 8 𝐾 = (π‘†β€˜π‘„)
221, 2, 3, 7, 8, 4, 5, 21, 13mirbtwn 28501 . . . . . . 7 (πœ‘ β†’ 𝑄 ∈ ((πΎβ€˜(π‘€β€˜πΆ))𝐼(π‘€β€˜πΆ)))
231, 2, 3, 4, 20, 5, 13, 22tgbtwncom 28331 . . . . . 6 (πœ‘ β†’ 𝑄 ∈ ((π‘€β€˜πΆ)𝐼(πΎβ€˜(π‘€β€˜πΆ))))
2418fveq1i 6891 . . . . . . . 8 (π‘β€˜πΆ) = ((π‘†β€˜π΅)β€˜πΆ)
2517, 24eqtrdi 2781 . . . . . . 7 (πœ‘ β†’ (πΎβ€˜(π‘€β€˜πΆ)) = ((π‘†β€˜π΅)β€˜πΆ))
2625oveq2d 7429 . . . . . 6 (πœ‘ β†’ ((π‘€β€˜πΆ)𝐼(πΎβ€˜(π‘€β€˜πΆ))) = ((π‘€β€˜πΆ)𝐼((π‘†β€˜π΅)β€˜πΆ)))
2723, 26eleqtrd 2827 . . . . 5 (πœ‘ β†’ 𝑄 ∈ ((π‘€β€˜πΆ)𝐼((π‘†β€˜π΅)β€˜πΆ)))
281, 2, 3, 4, 13, 5, 15, 27tgbtwncom 28331 . . . . . . 7 (πœ‘ β†’ 𝑄 ∈ (((π‘†β€˜π΅)β€˜πΆ)𝐼(π‘€β€˜πΆ)))
291, 2, 3, 7, 8, 4, 9, 10, 15, 5, 13, 28mirbtwni 28514 . . . . . 6 (πœ‘ β†’ (π‘€β€˜π‘„) ∈ ((π‘€β€˜((π‘†β€˜π΅)β€˜πΆ))𝐼(π‘€β€˜(π‘€β€˜πΆ))))
301, 2, 3, 7, 8, 4, 9, 10, 12mirmir 28505 . . . . . . 7 (πœ‘ β†’ (π‘€β€˜(π‘€β€˜πΆ)) = 𝐢)
3130oveq2d 7429 . . . . . 6 (πœ‘ β†’ ((π‘€β€˜((π‘†β€˜π΅)β€˜πΆ))𝐼(π‘€β€˜(π‘€β€˜πΆ))) = ((π‘€β€˜((π‘†β€˜π΅)β€˜πΆ))𝐼𝐢))
3229, 31eleqtrd 2827 . . . . 5 (πœ‘ β†’ (π‘€β€˜π‘„) ∈ ((π‘€β€˜((π‘†β€˜π΅)β€˜πΆ))𝐼𝐢))
331, 2, 3, 4, 13, 15axtgcgrrflx 28305 . . . . . 6 (πœ‘ β†’ ((π‘€β€˜πΆ) βˆ’ ((π‘†β€˜π΅)β€˜πΆ)) = (((π‘†β€˜π΅)β€˜πΆ) βˆ’ (π‘€β€˜πΆ)))
341, 2, 3, 7, 8, 4, 9, 10, 15, 13miriso 28513 . . . . . 6 (πœ‘ β†’ ((π‘€β€˜((π‘†β€˜π΅)β€˜πΆ)) βˆ’ (π‘€β€˜(π‘€β€˜πΆ))) = (((π‘†β€˜π΅)β€˜πΆ) βˆ’ (π‘€β€˜πΆ)))
3530oveq2d 7429 . . . . . 6 (πœ‘ β†’ ((π‘€β€˜((π‘†β€˜π΅)β€˜πΆ)) βˆ’ (π‘€β€˜(π‘€β€˜πΆ))) = ((π‘€β€˜((π‘†β€˜π΅)β€˜πΆ)) βˆ’ 𝐢))
3633, 34, 353eqtr2d 2771 . . . . 5 (πœ‘ β†’ ((π‘€β€˜πΆ) βˆ’ ((π‘†β€˜π΅)β€˜πΆ)) = ((π‘€β€˜((π‘†β€˜π΅)β€˜πΆ)) βˆ’ 𝐢))
3725oveq2d 7429 . . . . . . 7 (πœ‘ β†’ (𝑄 βˆ’ (πΎβ€˜(π‘€β€˜πΆ))) = (𝑄 βˆ’ ((π‘†β€˜π΅)β€˜πΆ)))
381, 2, 3, 7, 8, 4, 5, 21, 13mircgr 28500 . . . . . . 7 (πœ‘ β†’ (𝑄 βˆ’ (πΎβ€˜(π‘€β€˜πΆ))) = (𝑄 βˆ’ (π‘€β€˜πΆ)))
3937, 38eqtr3d 2767 . . . . . 6 (πœ‘ β†’ (𝑄 βˆ’ ((π‘†β€˜π΅)β€˜πΆ)) = (𝑄 βˆ’ (π‘€β€˜πΆ)))
401, 2, 3, 7, 8, 4, 9, 10, 5, 13miriso 28513 . . . . . 6 (πœ‘ β†’ ((π‘€β€˜π‘„) βˆ’ (π‘€β€˜(π‘€β€˜πΆ))) = (𝑄 βˆ’ (π‘€β€˜πΆ)))
4130oveq2d 7429 . . . . . 6 (πœ‘ β†’ ((π‘€β€˜π‘„) βˆ’ (π‘€β€˜(π‘€β€˜πΆ))) = ((π‘€β€˜π‘„) βˆ’ 𝐢))
4239, 40, 413eqtr2d 2771 . . . . 5 (πœ‘ β†’ (𝑄 βˆ’ ((π‘†β€˜π΅)β€˜πΆ)) = ((π‘€β€˜π‘„) βˆ’ 𝐢))
431, 2, 3, 7, 8, 4, 9, 10, 6mirmir 28505 . . . . . . . . . 10 (πœ‘ β†’ (π‘€β€˜(π‘€β€˜π΅)) = 𝐡)
44 eqidd 2726 . . . . . . . . . 10 (πœ‘ β†’ (π‘€β€˜π΅) = (π‘€β€˜π΅))
45 eqidd 2726 . . . . . . . . . 10 (πœ‘ β†’ (π‘€β€˜πΆ) = (π‘€β€˜πΆ))
4643, 44, 45s3eqd 14842 . . . . . . . . 9 (πœ‘ β†’ βŸ¨β€œ(π‘€β€˜(π‘€β€˜π΅))(π‘€β€˜π΅)(π‘€β€˜πΆ)β€βŸ© = βŸ¨β€œπ΅(π‘€β€˜π΅)(π‘€β€˜πΆ)β€βŸ©)
471, 2, 3, 7, 8, 4, 9, 10, 6mircl 28504 . . . . . . . . . 10 (πœ‘ β†’ (π‘€β€˜π΅) ∈ 𝑃)
48 simpr 483 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ 𝐴 = 𝐡) β†’ 𝐴 = 𝐡)
4948fveq2d 6894 . . . . . . . . . . . . . 14 ((πœ‘ ∧ 𝐴 = 𝐡) β†’ (π‘€β€˜π΄) = (π‘€β€˜π΅))
504adantr 479 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ 𝐴 = 𝐡) β†’ 𝐺 ∈ TarskiG)
519adantr 479 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ 𝐴 = 𝐡) β†’ 𝐴 ∈ 𝑃)
521, 2, 3, 7, 8, 50, 51, 10mircinv 28511 . . . . . . . . . . . . . 14 ((πœ‘ ∧ 𝐴 = 𝐡) β†’ (π‘€β€˜π΄) = 𝐴)
5349, 52eqtr3d 2767 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝐴 = 𝐡) β†’ (π‘€β€˜π΅) = 𝐴)
54 eqidd 2726 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝐴 = 𝐡) β†’ 𝐡 = 𝐡)
55 eqidd 2726 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝐴 = 𝐡) β†’ 𝐢 = 𝐢)
5653, 54, 55s3eqd 14842 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝐴 = 𝐡) β†’ βŸ¨β€œ(π‘€β€˜π΅)π΅πΆβ€βŸ© = βŸ¨β€œπ΄π΅πΆβ€βŸ©)
57 colperpexlem.1 . . . . . . . . . . . . 13 (πœ‘ β†’ βŸ¨β€œπ΄π΅πΆβ€βŸ© ∈ (∟Gβ€˜πΊ))
5857adantr 479 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝐴 = 𝐡) β†’ βŸ¨β€œπ΄π΅πΆβ€βŸ© ∈ (∟Gβ€˜πΊ))
5956, 58eqeltrd 2825 . . . . . . . . . . 11 ((πœ‘ ∧ 𝐴 = 𝐡) β†’ βŸ¨β€œ(π‘€β€˜π΅)π΅πΆβ€βŸ© ∈ (∟Gβ€˜πΊ))
604adantr 479 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝐴 β‰  𝐡) β†’ 𝐺 ∈ TarskiG)
619adantr 479 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝐴 β‰  𝐡) β†’ 𝐴 ∈ 𝑃)
626adantr 479 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝐴 β‰  𝐡) β†’ 𝐡 ∈ 𝑃)
6312adantr 479 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝐴 β‰  𝐡) β†’ 𝐢 ∈ 𝑃)
641, 2, 3, 7, 8, 60, 61, 10, 62mircl 28504 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝐴 β‰  𝐡) β†’ (π‘€β€˜π΅) ∈ 𝑃)
6557adantr 479 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝐴 β‰  𝐡) β†’ βŸ¨β€œπ΄π΅πΆβ€βŸ© ∈ (∟Gβ€˜πΊ))
66 simpr 483 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝐴 β‰  𝐡) β†’ 𝐴 β‰  𝐡)
671, 2, 3, 7, 8, 60, 61, 10, 62mirbtwn 28501 . . . . . . . . . . . . . 14 ((πœ‘ ∧ 𝐴 β‰  𝐡) β†’ 𝐴 ∈ ((π‘€β€˜π΅)𝐼𝐡))
681, 7, 3, 60, 64, 62, 61, 67btwncolg1 28398 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝐴 β‰  𝐡) β†’ (𝐴 ∈ ((π‘€β€˜π΅)𝐿𝐡) ∨ (π‘€β€˜π΅) = 𝐡))
691, 7, 3, 60, 64, 62, 61, 68colcom 28401 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝐴 β‰  𝐡) β†’ (𝐴 ∈ (𝐡𝐿(π‘€β€˜π΅)) ∨ 𝐡 = (π‘€β€˜π΅)))
701, 2, 3, 7, 8, 60, 61, 62, 63, 64, 65, 66, 69ragcol 28542 . . . . . . . . . . 11 ((πœ‘ ∧ 𝐴 β‰  𝐡) β†’ βŸ¨β€œ(π‘€β€˜π΅)π΅πΆβ€βŸ© ∈ (∟Gβ€˜πΊ))
7159, 70pm2.61dane 3019 . . . . . . . . . 10 (πœ‘ β†’ βŸ¨β€œ(π‘€β€˜π΅)π΅πΆβ€βŸ© ∈ (∟Gβ€˜πΊ))
721, 2, 3, 7, 8, 4, 47, 6, 12, 71, 10, 9mirrag 28544 . . . . . . . . 9 (πœ‘ β†’ βŸ¨β€œ(π‘€β€˜(π‘€β€˜π΅))(π‘€β€˜π΅)(π‘€β€˜πΆ)β€βŸ© ∈ (∟Gβ€˜πΊ))
7346, 72eqeltrrd 2826 . . . . . . . 8 (πœ‘ β†’ βŸ¨β€œπ΅(π‘€β€˜π΅)(π‘€β€˜πΆ)β€βŸ© ∈ (∟Gβ€˜πΊ))
741, 2, 3, 7, 8, 4, 6, 47, 13israg 28540 . . . . . . . 8 (πœ‘ β†’ (βŸ¨β€œπ΅(π‘€β€˜π΅)(π‘€β€˜πΆ)β€βŸ© ∈ (∟Gβ€˜πΊ) ↔ (𝐡 βˆ’ (π‘€β€˜πΆ)) = (𝐡 βˆ’ ((π‘†β€˜(π‘€β€˜π΅))β€˜(π‘€β€˜πΆ)))))
7573, 74mpbid 231 . . . . . . 7 (πœ‘ β†’ (𝐡 βˆ’ (π‘€β€˜πΆ)) = (𝐡 βˆ’ ((π‘†β€˜(π‘€β€˜π΅))β€˜(π‘€β€˜πΆ))))
761, 2, 3, 7, 8, 4, 9, 10, 12, 6mirmir2 28517 . . . . . . . 8 (πœ‘ β†’ (π‘€β€˜((π‘†β€˜π΅)β€˜πΆ)) = ((π‘†β€˜(π‘€β€˜π΅))β€˜(π‘€β€˜πΆ)))
7776oveq2d 7429 . . . . . . 7 (πœ‘ β†’ (𝐡 βˆ’ (π‘€β€˜((π‘†β€˜π΅)β€˜πΆ))) = (𝐡 βˆ’ ((π‘†β€˜(π‘€β€˜π΅))β€˜(π‘€β€˜πΆ))))
7875, 77eqtr4d 2768 . . . . . 6 (πœ‘ β†’ (𝐡 βˆ’ (π‘€β€˜πΆ)) = (𝐡 βˆ’ (π‘€β€˜((π‘†β€˜π΅)β€˜πΆ))))
791, 2, 3, 4, 6, 13, 6, 16, 78tgcgrcomlr 28323 . . . . 5 (πœ‘ β†’ ((π‘€β€˜πΆ) βˆ’ 𝐡) = ((π‘€β€˜((π‘†β€˜π΅)β€˜πΆ)) βˆ’ 𝐡))
801, 2, 3, 7, 8, 4, 6, 14, 12mircgr 28500 . . . . . 6 (πœ‘ β†’ (𝐡 βˆ’ ((π‘†β€˜π΅)β€˜πΆ)) = (𝐡 βˆ’ 𝐢))
811, 2, 3, 4, 6, 15, 6, 12, 80tgcgrcomlr 28323 . . . . 5 (πœ‘ β†’ (((π‘†β€˜π΅)β€˜πΆ) βˆ’ 𝐡) = (𝐢 βˆ’ 𝐡))
821, 2, 3, 4, 13, 5, 15, 6, 16, 11, 12, 6, 27, 32, 36, 42, 79, 81tgifscgr 28351 . . . 4 (πœ‘ β†’ (𝑄 βˆ’ 𝐡) = ((π‘€β€˜π‘„) βˆ’ 𝐡))
831, 2, 3, 4, 5, 6, 11, 6, 82tgcgrcomlr 28323 . . 3 (πœ‘ β†’ (𝐡 βˆ’ 𝑄) = (𝐡 βˆ’ (π‘€β€˜π‘„)))
8410fveq1i 6891 . . . 4 (π‘€β€˜π‘„) = ((π‘†β€˜π΄)β€˜π‘„)
8584oveq2i 7424 . . 3 (𝐡 βˆ’ (π‘€β€˜π‘„)) = (𝐡 βˆ’ ((π‘†β€˜π΄)β€˜π‘„))
8683, 85eqtrdi 2781 . 2 (πœ‘ β†’ (𝐡 βˆ’ 𝑄) = (𝐡 βˆ’ ((π‘†β€˜π΄)β€˜π‘„)))
871, 2, 3, 7, 8, 4, 6, 9, 5israg 28540 . 2 (πœ‘ β†’ (βŸ¨β€œπ΅π΄π‘„β€βŸ© ∈ (∟Gβ€˜πΊ) ↔ (𝐡 βˆ’ 𝑄) = (𝐡 βˆ’ ((π‘†β€˜π΄)β€˜π‘„))))
8886, 87mpbird 256 1 (πœ‘ β†’ βŸ¨β€œπ΅π΄π‘„β€βŸ© ∈ (∟Gβ€˜πΊ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098   β‰  wne 2930  β€˜cfv 6543  (class class class)co 7413  βŸ¨β€œcs3 14820  Basecbs 17174  distcds 17236  TarskiGcstrkg 28270  Itvcitv 28276  LineGclng 28277  pInvGcmir 28495  βˆŸGcrag 28536
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 2696  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7735  ax-cnex 11189  ax-resscn 11190  ax-1cn 11191  ax-icn 11192  ax-addcl 11193  ax-addrcl 11194  ax-mulcl 11195  ax-mulrcl 11196  ax-mulcom 11197  ax-addass 11198  ax-mulass 11199  ax-distr 11200  ax-i2m1 11201  ax-1ne0 11202  ax-1rid 11203  ax-rnegex 11204  ax-rrecex 11205  ax-cnre 11206  ax-pre-lttri 11207  ax-pre-lttrn 11208  ax-pre-ltadd 11209  ax-pre-mulgt0 11210
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 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3961  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4905  df-int 4946  df-iun 4994  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7369  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7866  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-oadd 8484  df-er 8718  df-map 8840  df-pm 8841  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-dju 9919  df-card 9957  df-pnf 11275  df-mnf 11276  df-xr 11277  df-ltxr 11278  df-le 11279  df-sub 11471  df-neg 11472  df-nn 12238  df-2 12300  df-3 12301  df-n0 12498  df-xnn0 12570  df-z 12584  df-uz 12848  df-fz 13512  df-fzo 13655  df-hash 14317  df-word 14492  df-concat 14548  df-s1 14573  df-s2 14826  df-s3 14827  df-trkgc 28291  df-trkgb 28292  df-trkgcb 28293  df-trkg 28296  df-cgrg 28354  df-mir 28496  df-rag 28537
This theorem is referenced by:  colperpexlem3  28575
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