Theorem ragcgr 28686

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 2734	. . . 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 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → 𝐺 ∈ TarskiG)
7		israg.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
8	7	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → 𝐵 ∈ 𝑃)
9		israg.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
10	9	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → 𝐶 ∈ 𝑃)
11		ragcgr.e	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ 𝑃)
12	11	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → 𝐸 ∈ 𝑃)
13		ragcgr.f	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ 𝑃)
14	13	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → 𝐹 ∈ 𝑃)
15		ragcgr.c	. . . . . 6 ⊢ ∼ = (cgrG‘𝐺)
16		israg.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
17	16	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → 𝐴 ∈ 𝑃)
18		ragcgr.d	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ 𝑃)
19	18	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → 𝐷 ∈ 𝑃)
20		ragcgr.2	. . . . . . 7 ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∼ ⟨“𝐷𝐸𝐹”⟩)
21	20	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → ⟨“𝐴𝐵𝐶”⟩ ∼ ⟨“𝐷𝐸𝐹”⟩)
22	2, 3, 4, 15, 6, 17, 8, 10, 19, 12, 14, 21	cgr3simp2 28500	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → (𝐵 − 𝐶) = (𝐸 − 𝐹))
23		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → 𝐵 = 𝐶)
24	2, 3, 4, 6, 8, 10, 12, 14, 22, 23	tgcgreq 28461	. . . 4 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → 𝐸 = 𝐹)
25		eqidd 2734	. . . 4 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → 𝐹 = 𝐹)
26	1, 24, 25	s3eqd 14773	. . 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 28681	. . 3 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → ⟨“𝐷𝐹𝐹”⟩ ∈ (∟G‘𝐺))
30	26, 29	eqeltrd 2833	. 2 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → ⟨“𝐷𝐸𝐹”⟩ ∈ (∟G‘𝐺))
31		ragcgr.1	. . . . . 6 ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
32	31	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
33	5	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐺 ∈ TarskiG)
34	16	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐴 ∈ 𝑃)
35	7	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐵 ∈ 𝑃)
36	9	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐶 ∈ 𝑃)
37	2, 3, 4, 27, 28, 33, 34, 35, 36	israg 28676	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺) ↔ (𝐴 − 𝐶) = (𝐴 − ((𝑆‘𝐵)‘𝐶))))
38	32, 37	mpbid 232	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (𝐴 − 𝐶) = (𝐴 − ((𝑆‘𝐵)‘𝐶)))
39	13	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐹 ∈ 𝑃)
40	18	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐷 ∈ 𝑃)
41	11	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐸 ∈ 𝑃)
42	20	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → ⟨“𝐴𝐵𝐶”⟩ ∼ ⟨“𝐷𝐸𝐹”⟩)
43	2, 3, 4, 15, 33, 34, 35, 36, 40, 41, 39, 42	cgr3simp3 28501	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (𝐶 − 𝐴) = (𝐹 − 𝐷))
44	2, 3, 4, 33, 36, 34, 39, 40, 43	tgcgrcomlr 28459	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (𝐴 − 𝐶) = (𝐷 − 𝐹))
45		eqid 2733	. . . . . 6 ⊢ (𝑆‘𝐵) = (𝑆‘𝐵)
46	2, 3, 4, 27, 28, 33, 35, 45, 36	mircl 28640	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → ((𝑆‘𝐵)‘𝐶) ∈ 𝑃)
47		eqid 2733	. . . . . 6 ⊢ (𝑆‘𝐸) = (𝑆‘𝐸)
48	2, 3, 4, 27, 28, 33, 41, 47, 39	mircl 28640	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → ((𝑆‘𝐸)‘𝐹) ∈ 𝑃)
49		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐵 ≠ 𝐶)
50	49	necomd 2984	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐶 ≠ 𝐵)
51	2, 3, 4, 27, 28, 33, 35, 45, 36	mirbtwn 28637	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐵 ∈ (((𝑆‘𝐵)‘𝐶)𝐼𝐶))
52	2, 3, 4, 33, 46, 35, 36, 51	tgbtwncom 28467	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐵 ∈ (𝐶𝐼((𝑆‘𝐵)‘𝐶)))
53	2, 3, 4, 27, 28, 33, 41, 47, 39	mirbtwn 28637	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐸 ∈ (((𝑆‘𝐸)‘𝐹)𝐼𝐹))
54	2, 3, 4, 33, 48, 41, 39, 53	tgbtwncom 28467	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐸 ∈ (𝐹𝐼((𝑆‘𝐸)‘𝐹)))
55	2, 3, 4, 15, 33, 34, 35, 36, 40, 41, 39, 42	cgr3simp2 28500	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (𝐵 − 𝐶) = (𝐸 − 𝐹))
56	2, 3, 4, 33, 35, 36, 41, 39, 55	tgcgrcomlr 28459	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (𝐶 − 𝐵) = (𝐹 − 𝐸))
57	2, 3, 4, 27, 28, 33, 35, 45, 36	mircgr 28636	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (𝐵 − ((𝑆‘𝐵)‘𝐶)) = (𝐵 − 𝐶))
58	2, 3, 4, 27, 28, 33, 41, 47, 39	mircgr 28636	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (𝐸 − ((𝑆‘𝐸)‘𝐹)) = (𝐸 − 𝐹))
59	55, 57, 58	3eqtr4d 2778	. . . . . 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 28499	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (𝐴 − 𝐵) = (𝐷 − 𝐸))
61	2, 3, 4, 33, 34, 35, 40, 41, 60	tgcgrcomlr 28459	. . . . . 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 28444	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (((𝑆‘𝐵)‘𝐶) − 𝐴) = (((𝑆‘𝐸)‘𝐹) − 𝐷))
63	2, 3, 4, 33, 46, 34, 48, 40, 62	tgcgrcomlr 28459	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (𝐴 − ((𝑆‘𝐵)‘𝐶)) = (𝐷 − ((𝑆‘𝐸)‘𝐹)))
64	38, 44, 63	3eqtr3d 2776	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (𝐷 − 𝐹) = (𝐷 − ((𝑆‘𝐸)‘𝐹)))
65	2, 3, 4, 27, 28, 33, 40, 41, 39	israg 28676	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (⟨“𝐷𝐸𝐹”⟩ ∈ (∟G‘𝐺) ↔ (𝐷 − 𝐹) = (𝐷 − ((𝑆‘𝐸)‘𝐹))))
66	64, 65	mpbird 257	. 2 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → ⟨“𝐷𝐸𝐹”⟩ ∈ (∟G‘𝐺))
67	30, 66	pm2.61dane 3016	1 ⊢ (𝜑 → ⟨“𝐷𝐸𝐹”⟩ ∈ (∟G‘𝐺))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113   ≠ wne 2929   class class class wbr 5093  ‘cfv 6486  (class class class)co 7352  ⟨“cs3 14751  Basecbs 17122  distcds 17172  TarskiGcstrkg 28406  Itvcitv 28412  LineGclng 28413  cgrGccgrg 28489  pInvGcmir 28631  ∟Gcrag 28672

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674  ax-cnex 11069  ax-resscn 11070  ax-1cn 11071  ax-icn 11072  ax-addcl 11073  ax-addrcl 11074  ax-mulcl 11075  ax-mulrcl 11076  ax-mulcom 11077  ax-addass 11078  ax-mulass 11079  ax-distr 11080  ax-i2m1 11081  ax-1ne0 11082  ax-1rid 11083  ax-rnegex 11084  ax-rrecex 11085  ax-cnre 11086  ax-pre-lttri 11087  ax-pre-lttrn 11088  ax-pre-ltadd 11089  ax-pre-mulgt0 11090

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-nel 3034  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-tp 4580  df-op 4582  df-uni 4859  df-int 4898  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7309  df-ov 7355  df-oprab 7356  df-mpo 7357  df-om 7803  df-1st 7927  df-2nd 7928  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-rdg 8335  df-1o 8391  df-er 8628  df-map 8758  df-pm 8759  df-en 8876  df-dom 8877  df-sdom 8878  df-fin 8879  df-card 9839  df-pnf 11155  df-mnf 11156  df-xr 11157  df-ltxr 11158  df-le 11159  df-sub 11353  df-neg 11354  df-nn 12133  df-2 12195  df-3 12196  df-n0 12389  df-z 12476  df-uz 12739  df-fz 13410  df-fzo 13557  df-hash 14240  df-word 14423  df-concat 14480  df-s1 14506  df-s2 14757  df-s3 14758  df-trkgc 28427  df-trkgb 28428  df-trkgcb 28429  df-trkg 28432  df-cgrg 28490  df-mir 28632  df-rag 28673

This theorem is referenced by:  motrag  28687  footexALT  28697  footexlem1  28698  footexlem2  28699