Theorem ragcgr 28670

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 2730	. . . 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 28484	. . . . 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 28445	. . . 4 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → 𝐸 = 𝐹)
25		eqidd 2730	. . . 4 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → 𝐹 = 𝐹)
26	1, 24, 25	s3eqd 14789	. . 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 28665	. . 3 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → ⟨“𝐷𝐹𝐹”⟩ ∈ (∟G‘𝐺))
30	26, 29	eqeltrd 2828	. 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 28660	. . . . 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 28485	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (𝐶 − 𝐴) = (𝐹 − 𝐷))
44	2, 3, 4, 33, 36, 34, 39, 40, 43	tgcgrcomlr 28443	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (𝐴 − 𝐶) = (𝐷 − 𝐹))
45		eqid 2729	. . . . . 6 ⊢ (𝑆‘𝐵) = (𝑆‘𝐵)
46	2, 3, 4, 27, 28, 33, 35, 45, 36	mircl 28624	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → ((𝑆‘𝐵)‘𝐶) ∈ 𝑃)
47		eqid 2729	. . . . . 6 ⊢ (𝑆‘𝐸) = (𝑆‘𝐸)
48	2, 3, 4, 27, 28, 33, 41, 47, 39	mircl 28624	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → ((𝑆‘𝐸)‘𝐹) ∈ 𝑃)
49		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐵 ≠ 𝐶)
50	49	necomd 2980	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐶 ≠ 𝐵)
51	2, 3, 4, 27, 28, 33, 35, 45, 36	mirbtwn 28621	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐵 ∈ (((𝑆‘𝐵)‘𝐶)𝐼𝐶))
52	2, 3, 4, 33, 46, 35, 36, 51	tgbtwncom 28451	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐵 ∈ (𝐶𝐼((𝑆‘𝐵)‘𝐶)))
53	2, 3, 4, 27, 28, 33, 41, 47, 39	mirbtwn 28621	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐸 ∈ (((𝑆‘𝐸)‘𝐹)𝐼𝐹))
54	2, 3, 4, 33, 48, 41, 39, 53	tgbtwncom 28451	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐸 ∈ (𝐹𝐼((𝑆‘𝐸)‘𝐹)))
55	2, 3, 4, 15, 33, 34, 35, 36, 40, 41, 39, 42	cgr3simp2 28484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (𝐵 − 𝐶) = (𝐸 − 𝐹))
56	2, 3, 4, 33, 35, 36, 41, 39, 55	tgcgrcomlr 28443	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (𝐶 − 𝐵) = (𝐹 − 𝐸))
57	2, 3, 4, 27, 28, 33, 35, 45, 36	mircgr 28620	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (𝐵 − ((𝑆‘𝐵)‘𝐶)) = (𝐵 − 𝐶))
58	2, 3, 4, 27, 28, 33, 41, 47, 39	mircgr 28620	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (𝐸 − ((𝑆‘𝐸)‘𝐹)) = (𝐸 − 𝐹))
59	55, 57, 58	3eqtr4d 2774	. . . . . 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 28483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (𝐴 − 𝐵) = (𝐷 − 𝐸))
61	2, 3, 4, 33, 34, 35, 40, 41, 60	tgcgrcomlr 28443	. . . . . 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 28428	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (((𝑆‘𝐵)‘𝐶) − 𝐴) = (((𝑆‘𝐸)‘𝐹) − 𝐷))
63	2, 3, 4, 33, 46, 34, 48, 40, 62	tgcgrcomlr 28443	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (𝐴 − ((𝑆‘𝐵)‘𝐶)) = (𝐷 − ((𝑆‘𝐸)‘𝐹)))
64	38, 44, 63	3eqtr3d 2772	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (𝐷 − 𝐹) = (𝐷 − ((𝑆‘𝐸)‘𝐹)))
65	2, 3, 4, 27, 28, 33, 40, 41, 39	israg 28660	. . 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 3012	1 ⊢ (𝜑 → ⟨“𝐷𝐸𝐹”⟩ ∈ (∟G‘𝐺))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2925   class class class wbr 5095  ‘cfv 6486  (class class class)co 7353  ⟨“cs3 14767  Basecbs 17138  distcds 17188  TarskiGcstrkg 28390  Itvcitv 28396  LineGclng 28397  cgrGccgrg 28473  pInvGcmir 28615  ∟Gcrag 28656

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 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-tp 4584  df-op 4586  df-uni 4862  df-int 4900  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  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 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-er 8632  df-map 8762  df-pm 8763  df-en 8880  df-dom 8881  df-sdom 8882  df-fin 8883  df-card 9854  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11367  df-neg 11368  df-nn 12147  df-2 12209  df-3 12210  df-n0 12403  df-z 12490  df-uz 12754  df-fz 13429  df-fzo 13576  df-hash 14256  df-word 14439  df-concat 14496  df-s1 14521  df-s2 14773  df-s3 14774  df-trkgc 28411  df-trkgb 28412  df-trkgcb 28413  df-trkg 28416  df-cgrg 28474  df-mir 28616  df-rag 28657

This theorem is referenced by:  motrag  28671  footexALT  28681  footexlem1  28682  footexlem2  28683