Theorem ragcgr 26491

Description: Right angle and colinearity. Theorem 8.10 of [Schwabhauser] p. 58. (Contributed by Thierry Arnoux, 4-Sep-2019.)

Hypotheses

Ref	Expression
israg.p	⊢ 𝑃 = (Base‘𝐺)
israg.d	⊢ − = (dist‘𝐺)
israg.i	⊢ 𝐼 = (Itv‘𝐺)
israg.l	⊢ 𝐿 = (LineG‘𝐺)
israg.s	⊢ 𝑆 = (pInvG‘𝐺)
israg.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
israg.a	⊢ (𝜑 → 𝐴 ∈ 𝑃)
israg.b	⊢ (𝜑 → 𝐵 ∈ 𝑃)
israg.c	⊢ (𝜑 → 𝐶 ∈ 𝑃)
ragcgr.c	⊢ ∼ = (cgrG‘𝐺)
ragcgr.d	⊢ (𝜑 → 𝐷 ∈ 𝑃)
ragcgr.e	⊢ (𝜑 → 𝐸 ∈ 𝑃)
ragcgr.f	⊢ (𝜑 → 𝐹 ∈ 𝑃)
ragcgr.1	⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
ragcgr.2	⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∼ ⟨“𝐷𝐸𝐹”⟩)

Assertion

Ref	Expression
ragcgr	⊢ (𝜑 → ⟨“𝐷𝐸𝐹”⟩ ∈ (∟G‘𝐺))

Proof of Theorem ragcgr

Step	Hyp	Ref	Expression
1		eqidd 2821	. . . 4 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → 𝐷 = 𝐷)
2		israg.p	. . . . 5 ⊢ 𝑃 = (Base‘𝐺)
3		israg.d	. . . . 5 ⊢ − = (dist‘𝐺)
4		israg.i	. . . . 5 ⊢ 𝐼 = (Itv‘𝐺)
5		israg.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
6	5	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → 𝐺 ∈ TarskiG)
7		israg.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
8	7	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → 𝐵 ∈ 𝑃)
9		israg.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
10	9	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → 𝐶 ∈ 𝑃)
11		ragcgr.e	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ 𝑃)
12	11	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → 𝐸 ∈ 𝑃)
13		ragcgr.f	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ 𝑃)
14	13	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → 𝐹 ∈ 𝑃)
15		ragcgr.c	. . . . . 6 ⊢ ∼ = (cgrG‘𝐺)
16		israg.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
17	16	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → 𝐴 ∈ 𝑃)
18		ragcgr.d	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ 𝑃)
19	18	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → 𝐷 ∈ 𝑃)
20		ragcgr.2	. . . . . . 7 ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∼ ⟨“𝐷𝐸𝐹”⟩)
21	20	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → ⟨“𝐴𝐵𝐶”⟩ ∼ ⟨“𝐷𝐸𝐹”⟩)
22	2, 3, 4, 15, 6, 17, 8, 10, 19, 12, 14, 21	cgr3simp2 26305	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → (𝐵 − 𝐶) = (𝐸 − 𝐹))
23		simpr 487	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → 𝐵 = 𝐶)
24	2, 3, 4, 6, 8, 10, 12, 14, 22, 23	tgcgreq 26266	. . . 4 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → 𝐸 = 𝐹)
25		eqidd 2821	. . . 4 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → 𝐹 = 𝐹)
26	1, 24, 25	s3eqd 14221	. . 3 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → ⟨“𝐷𝐸𝐹”⟩ = ⟨“𝐷𝐹𝐹”⟩)
27		israg.l	. . . 4 ⊢ 𝐿 = (LineG‘𝐺)
28		israg.s	. . . 4 ⊢ 𝑆 = (pInvG‘𝐺)
29	2, 3, 4, 27, 28, 6, 19, 14, 12	ragtrivb 26486	. . 3 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → ⟨“𝐷𝐹𝐹”⟩ ∈ (∟G‘𝐺))
30	26, 29	eqeltrd 2912	. 2 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → ⟨“𝐷𝐸𝐹”⟩ ∈ (∟G‘𝐺))
31		ragcgr.1	. . . . . 6 ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
32	31	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
33	5	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐺 ∈ TarskiG)
34	16	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐴 ∈ 𝑃)
35	7	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐵 ∈ 𝑃)
36	9	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐶 ∈ 𝑃)
37	2, 3, 4, 27, 28, 33, 34, 35, 36	israg 26481	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺) ↔ (𝐴 − 𝐶) = (𝐴 − ((𝑆‘𝐵)‘𝐶))))
38	32, 37	mpbid 234	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (𝐴 − 𝐶) = (𝐴 − ((𝑆‘𝐵)‘𝐶)))
39	13	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐹 ∈ 𝑃)
40	18	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐷 ∈ 𝑃)
41	11	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐸 ∈ 𝑃)
42	20	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → ⟨“𝐴𝐵𝐶”⟩ ∼ ⟨“𝐷𝐸𝐹”⟩)
43	2, 3, 4, 15, 33, 34, 35, 36, 40, 41, 39, 42	cgr3simp3 26306	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (𝐶 − 𝐴) = (𝐹 − 𝐷))
44	2, 3, 4, 33, 36, 34, 39, 40, 43	tgcgrcomlr 26264	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (𝐴 − 𝐶) = (𝐷 − 𝐹))
45		eqid 2820	. . . . . 6 ⊢ (𝑆‘𝐵) = (𝑆‘𝐵)
46	2, 3, 4, 27, 28, 33, 35, 45, 36	mircl 26445	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → ((𝑆‘𝐵)‘𝐶) ∈ 𝑃)
47		eqid 2820	. . . . . 6 ⊢ (𝑆‘𝐸) = (𝑆‘𝐸)
48	2, 3, 4, 27, 28, 33, 41, 47, 39	mircl 26445	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → ((𝑆‘𝐸)‘𝐹) ∈ 𝑃)
49		simpr 487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐵 ≠ 𝐶)
50	49	necomd 3070	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐶 ≠ 𝐵)
51	2, 3, 4, 27, 28, 33, 35, 45, 36	mirbtwn 26442	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐵 ∈ (((𝑆‘𝐵)‘𝐶)𝐼𝐶))
52	2, 3, 4, 33, 46, 35, 36, 51	tgbtwncom 26272	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐵 ∈ (𝐶𝐼((𝑆‘𝐵)‘𝐶)))
53	2, 3, 4, 27, 28, 33, 41, 47, 39	mirbtwn 26442	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐸 ∈ (((𝑆‘𝐸)‘𝐹)𝐼𝐹))
54	2, 3, 4, 33, 48, 41, 39, 53	tgbtwncom 26272	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐸 ∈ (𝐹𝐼((𝑆‘𝐸)‘𝐹)))
55	2, 3, 4, 15, 33, 34, 35, 36, 40, 41, 39, 42	cgr3simp2 26305	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (𝐵 − 𝐶) = (𝐸 − 𝐹))
56	2, 3, 4, 33, 35, 36, 41, 39, 55	tgcgrcomlr 26264	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (𝐶 − 𝐵) = (𝐹 − 𝐸))
57	2, 3, 4, 27, 28, 33, 35, 45, 36	mircgr 26441	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (𝐵 − ((𝑆‘𝐵)‘𝐶)) = (𝐵 − 𝐶))
58	2, 3, 4, 27, 28, 33, 41, 47, 39	mircgr 26441	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (𝐸 − ((𝑆‘𝐸)‘𝐹)) = (𝐸 − 𝐹))
59	55, 57, 58	3eqtr4d 2865	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (𝐵 − ((𝑆‘𝐵)‘𝐶)) = (𝐸 − ((𝑆‘𝐸)‘𝐹)))
60	2, 3, 4, 15, 33, 34, 35, 36, 40, 41, 39, 42	cgr3simp1 26304	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (𝐴 − 𝐵) = (𝐷 − 𝐸))
61	2, 3, 4, 33, 34, 35, 40, 41, 60	tgcgrcomlr 26264	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (𝐵 − 𝐴) = (𝐸 − 𝐷))
62	2, 3, 4, 33, 36, 35, 46, 39, 41, 48, 34, 40, 50, 52, 54, 56, 59, 43, 61	axtg5seg 26249	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (((𝑆‘𝐵)‘𝐶) − 𝐴) = (((𝑆‘𝐸)‘𝐹) − 𝐷))
63	2, 3, 4, 33, 46, 34, 48, 40, 62	tgcgrcomlr 26264	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (𝐴 − ((𝑆‘𝐵)‘𝐶)) = (𝐷 − ((𝑆‘𝐸)‘𝐹)))
64	38, 44, 63	3eqtr3d 2863	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (𝐷 − 𝐹) = (𝐷 − ((𝑆‘𝐸)‘𝐹)))
65	2, 3, 4, 27, 28, 33, 40, 41, 39	israg 26481	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (⟨“𝐷𝐸𝐹”⟩ ∈ (∟G‘𝐺) ↔ (𝐷 − 𝐹) = (𝐷 − ((𝑆‘𝐸)‘𝐹))))
66	64, 65	mpbird 259	. 2 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → ⟨“𝐷𝐸𝐹”⟩ ∈ (∟G‘𝐺))
67	30, 66	pm2.61dane 3103	1 ⊢ (𝜑 → ⟨“𝐷𝐸𝐹”⟩ ∈ (∟G‘𝐺))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1536   ∈ wcel 2113   ≠ wne 3015   class class class wbr 5059  ‘cfv 6348  (class class class)co 7149  ⟨“cs3 14199  Basecbs 16478  distcds 16569  TarskiGcstrkg 26214  Itvcitv 26220  LineGclng 26221  cgrGccgrg 26294  pInvGcmir 26436  ∟Gcrag 26477

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5323  ax-un 7454  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-ne 3016  df-nel 3123  df-ral 3142  df-rex 3143  df-reu 3144  df-rmo 3145  df-rab 3146  df-v 3493  df-sbc 3769  df-csb 3877  df-dif 3932  df-un 3934  df-in 3936  df-ss 3945  df-pss 3947  df-nul 4285  df-if 4461  df-pw 4534  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-int 4870  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-tr 5166  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-om 7574  df-1st 7682  df-2nd 7683  df-wrecs 7940  df-recs 8001  df-rdg 8039  df-1o 8095  df-oadd 8099  df-er 8282  df-map 8401  df-pm 8402  df-en 8503  df-dom 8504  df-sdom 8505  df-fin 8506  df-card 9361  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-nn 11632  df-2 11694  df-3 11695  df-n0 11892  df-z 11976  df-uz 12238  df-fz 12890  df-fzo 13031  df-hash 13688  df-word 13859  df-concat 13918  df-s1 13945  df-s2 14205  df-s3 14206  df-trkgc 26232  df-trkgb 26233  df-trkgcb 26234  df-trkg 26237  df-cgrg 26295  df-mir 26437  df-rag 26478

This theorem is referenced by:  motrag  26492  footexALT  26502  footexlem1  26503  footexlem2  26504