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Theorem tgcgrsub2 26308
Description: Removing identical parts from the end of a line segment preserves congruence. In this version the order of points is not known. (Contributed by Thierry Arnoux, 3-Apr-2019.)
Hypotheses
Ref Expression
legval.p 𝑃 = (Base‘𝐺)
legval.d = (dist‘𝐺)
legval.i 𝐼 = (Itv‘𝐺)
legval.l = (≤G‘𝐺)
legval.g (𝜑𝐺 ∈ TarskiG)
legid.a (𝜑𝐴𝑃)
legid.b (𝜑𝐵𝑃)
legtrd.c (𝜑𝐶𝑃)
legtrd.d (𝜑𝐷𝑃)
tgcgrsub2.d (𝜑𝐷𝑃)
tgcgrsub2.e (𝜑𝐸𝑃)
tgcgrsub2.f (𝜑𝐹𝑃)
tgcgrsub2.1 (𝜑 → (𝐵 ∈ (𝐴𝐼𝐶) ∨ 𝐶 ∈ (𝐴𝐼𝐵)))
tgcgrsub2.2 (𝜑 → (𝐸 ∈ (𝐷𝐼𝐹) ∨ 𝐹 ∈ (𝐷𝐼𝐸)))
tgcgrsub2.3 (𝜑 → (𝐴 𝐵) = (𝐷 𝐸))
tgcgrsub2.4 (𝜑 → (𝐴 𝐶) = (𝐷 𝐹))
Assertion
Ref Expression
tgcgrsub2 (𝜑 → (𝐵 𝐶) = (𝐸 𝐹))

Proof of Theorem tgcgrsub2
StepHypRef Expression
1 legval.p . . 3 𝑃 = (Base‘𝐺)
2 legval.d . . 3 = (dist‘𝐺)
3 legval.i . . 3 𝐼 = (Itv‘𝐺)
4 legval.g . . . 4 (𝜑𝐺 ∈ TarskiG)
54adantr 481 . . 3 ((𝜑𝐵 ∈ (𝐴𝐼𝐶)) → 𝐺 ∈ TarskiG)
6 legtrd.c . . . 4 (𝜑𝐶𝑃)
76adantr 481 . . 3 ((𝜑𝐵 ∈ (𝐴𝐼𝐶)) → 𝐶𝑃)
8 legid.b . . . 4 (𝜑𝐵𝑃)
98adantr 481 . . 3 ((𝜑𝐵 ∈ (𝐴𝐼𝐶)) → 𝐵𝑃)
10 tgcgrsub2.f . . . 4 (𝜑𝐹𝑃)
1110adantr 481 . . 3 ((𝜑𝐵 ∈ (𝐴𝐼𝐶)) → 𝐹𝑃)
12 tgcgrsub2.e . . . 4 (𝜑𝐸𝑃)
1312adantr 481 . . 3 ((𝜑𝐵 ∈ (𝐴𝐼𝐶)) → 𝐸𝑃)
14 legid.a . . . . 5 (𝜑𝐴𝑃)
1514adantr 481 . . . 4 ((𝜑𝐵 ∈ (𝐴𝐼𝐶)) → 𝐴𝑃)
16 legtrd.d . . . . 5 (𝜑𝐷𝑃)
1716adantr 481 . . . 4 ((𝜑𝐵 ∈ (𝐴𝐼𝐶)) → 𝐷𝑃)
18 simpr 485 . . . . 5 ((𝜑𝐵 ∈ (𝐴𝐼𝐶)) → 𝐵 ∈ (𝐴𝐼𝐶))
191, 2, 3, 5, 15, 9, 7, 18tgbtwncom 26201 . . . 4 ((𝜑𝐵 ∈ (𝐴𝐼𝐶)) → 𝐵 ∈ (𝐶𝐼𝐴))
20 legval.l . . . . . 6 = (≤G‘𝐺)
21 tgcgrsub2.2 . . . . . . 7 (𝜑 → (𝐸 ∈ (𝐷𝐼𝐹) ∨ 𝐹 ∈ (𝐷𝐼𝐸)))
2221adantr 481 . . . . . 6 ((𝜑𝐵 ∈ (𝐴𝐼𝐶)) → (𝐸 ∈ (𝐷𝐼𝐹) ∨ 𝐹 ∈ (𝐷𝐼𝐸)))
231, 2, 3, 20, 5, 15, 9, 7, 18btwnleg 26301 . . . . . . 7 ((𝜑𝐵 ∈ (𝐴𝐼𝐶)) → (𝐴 𝐵) (𝐴 𝐶))
24 tgcgrsub2.3 . . . . . . . 8 (𝜑 → (𝐴 𝐵) = (𝐷 𝐸))
2524adantr 481 . . . . . . 7 ((𝜑𝐵 ∈ (𝐴𝐼𝐶)) → (𝐴 𝐵) = (𝐷 𝐸))
26 tgcgrsub2.4 . . . . . . . 8 (𝜑 → (𝐴 𝐶) = (𝐷 𝐹))
2726adantr 481 . . . . . . 7 ((𝜑𝐵 ∈ (𝐴𝐼𝐶)) → (𝐴 𝐶) = (𝐷 𝐹))
2823, 25, 273brtr3d 5088 . . . . . 6 ((𝜑𝐵 ∈ (𝐴𝐼𝐶)) → (𝐷 𝐸) (𝐷 𝐹))
291, 2, 3, 20, 5, 13, 11, 17, 17, 22, 28legbtwn 26307 . . . . 5 ((𝜑𝐵 ∈ (𝐴𝐼𝐶)) → 𝐸 ∈ (𝐷𝐼𝐹))
301, 2, 3, 5, 17, 13, 11, 29tgbtwncom 26201 . . . 4 ((𝜑𝐵 ∈ (𝐴𝐼𝐶)) → 𝐸 ∈ (𝐹𝐼𝐷))
311, 2, 3, 4, 14, 6, 16, 10, 26tgcgrcomlr 26193 . . . . 5 (𝜑 → (𝐶 𝐴) = (𝐹 𝐷))
3231adantr 481 . . . 4 ((𝜑𝐵 ∈ (𝐴𝐼𝐶)) → (𝐶 𝐴) = (𝐹 𝐷))
331, 2, 3, 4, 14, 8, 16, 12, 24tgcgrcomlr 26193 . . . . 5 (𝜑 → (𝐵 𝐴) = (𝐸 𝐷))
3433adantr 481 . . . 4 ((𝜑𝐵 ∈ (𝐴𝐼𝐶)) → (𝐵 𝐴) = (𝐸 𝐷))
351, 2, 3, 5, 7, 9, 15, 11, 13, 17, 19, 30, 32, 34tgcgrsub 26222 . . 3 ((𝜑𝐵 ∈ (𝐴𝐼𝐶)) → (𝐶 𝐵) = (𝐹 𝐸))
361, 2, 3, 5, 7, 9, 11, 13, 35tgcgrcomlr 26193 . 2 ((𝜑𝐵 ∈ (𝐴𝐼𝐶)) → (𝐵 𝐶) = (𝐸 𝐹))
374adantr 481 . . 3 ((𝜑𝐶 ∈ (𝐴𝐼𝐵)) → 𝐺 ∈ TarskiG)
388adantr 481 . . 3 ((𝜑𝐶 ∈ (𝐴𝐼𝐵)) → 𝐵𝑃)
396adantr 481 . . 3 ((𝜑𝐶 ∈ (𝐴𝐼𝐵)) → 𝐶𝑃)
4014adantr 481 . . 3 ((𝜑𝐶 ∈ (𝐴𝐼𝐵)) → 𝐴𝑃)
4112adantr 481 . . 3 ((𝜑𝐶 ∈ (𝐴𝐼𝐵)) → 𝐸𝑃)
4210adantr 481 . . 3 ((𝜑𝐶 ∈ (𝐴𝐼𝐵)) → 𝐹𝑃)
4316adantr 481 . . 3 ((𝜑𝐶 ∈ (𝐴𝐼𝐵)) → 𝐷𝑃)
44 simpr 485 . . . 4 ((𝜑𝐶 ∈ (𝐴𝐼𝐵)) → 𝐶 ∈ (𝐴𝐼𝐵))
451, 2, 3, 37, 40, 39, 38, 44tgbtwncom 26201 . . 3 ((𝜑𝐶 ∈ (𝐴𝐼𝐵)) → 𝐶 ∈ (𝐵𝐼𝐴))
4621orcomd 865 . . . . . 6 (𝜑 → (𝐹 ∈ (𝐷𝐼𝐸) ∨ 𝐸 ∈ (𝐷𝐼𝐹)))
4746adantr 481 . . . . 5 ((𝜑𝐶 ∈ (𝐴𝐼𝐵)) → (𝐹 ∈ (𝐷𝐼𝐸) ∨ 𝐸 ∈ (𝐷𝐼𝐹)))
481, 2, 3, 20, 37, 40, 39, 38, 44btwnleg 26301 . . . . . 6 ((𝜑𝐶 ∈ (𝐴𝐼𝐵)) → (𝐴 𝐶) (𝐴 𝐵))
4926adantr 481 . . . . . 6 ((𝜑𝐶 ∈ (𝐴𝐼𝐵)) → (𝐴 𝐶) = (𝐷 𝐹))
5024adantr 481 . . . . . 6 ((𝜑𝐶 ∈ (𝐴𝐼𝐵)) → (𝐴 𝐵) = (𝐷 𝐸))
5148, 49, 503brtr3d 5088 . . . . 5 ((𝜑𝐶 ∈ (𝐴𝐼𝐵)) → (𝐷 𝐹) (𝐷 𝐸))
521, 2, 3, 20, 37, 42, 41, 43, 43, 47, 51legbtwn 26307 . . . 4 ((𝜑𝐶 ∈ (𝐴𝐼𝐵)) → 𝐹 ∈ (𝐷𝐼𝐸))
531, 2, 3, 37, 43, 42, 41, 52tgbtwncom 26201 . . 3 ((𝜑𝐶 ∈ (𝐴𝐼𝐵)) → 𝐹 ∈ (𝐸𝐼𝐷))
5433adantr 481 . . 3 ((𝜑𝐶 ∈ (𝐴𝐼𝐵)) → (𝐵 𝐴) = (𝐸 𝐷))
5531adantr 481 . . 3 ((𝜑𝐶 ∈ (𝐴𝐼𝐵)) → (𝐶 𝐴) = (𝐹 𝐷))
561, 2, 3, 37, 38, 39, 40, 41, 42, 43, 45, 53, 54, 55tgcgrsub 26222 . 2 ((𝜑𝐶 ∈ (𝐴𝐼𝐵)) → (𝐵 𝐶) = (𝐸 𝐹))
57 tgcgrsub2.1 . 2 (𝜑 → (𝐵 ∈ (𝐴𝐼𝐶) ∨ 𝐶 ∈ (𝐴𝐼𝐵)))
5836, 56, 57mpjaodan 952 1 (𝜑 → (𝐵 𝐶) = (𝐸 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wo 841   = wceq 1528  wcel 2105  cfv 6348  (class class class)co 7145  Basecbs 16471  distcds 16562  TarskiGcstrkg 26143  Itvcitv 26149  ≤Gcleg 26295
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  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 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-pm 8398  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-dju 9318  df-card 9356  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-nn 11627  df-2 11688  df-3 11689  df-n0 11886  df-xnn0 11956  df-z 11970  df-uz 12232  df-fz 12881  df-fzo 13022  df-hash 13679  df-word 13850  df-concat 13911  df-s1 13938  df-s2 14198  df-s3 14199  df-trkgc 26161  df-trkgb 26162  df-trkgcb 26163  df-trkg 26166  df-cgrg 26224  df-leg 26296
This theorem is referenced by:  cgracgr  26531  cgraswap  26533
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