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Theorem tgcgrsub2 25290
 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 480 . . 3 ((𝜑𝐵 ∈ (𝐴𝐼𝐶)) → 𝐺 ∈ TarskiG)
6 legtrd.c . . . 4 (𝜑𝐶𝑃)
76adantr 480 . . 3 ((𝜑𝐵 ∈ (𝐴𝐼𝐶)) → 𝐶𝑃)
8 legid.b . . . 4 (𝜑𝐵𝑃)
98adantr 480 . . 3 ((𝜑𝐵 ∈ (𝐴𝐼𝐶)) → 𝐵𝑃)
10 tgcgrsub2.f . . . 4 (𝜑𝐹𝑃)
1110adantr 480 . . 3 ((𝜑𝐵 ∈ (𝐴𝐼𝐶)) → 𝐹𝑃)
12 tgcgrsub2.e . . . 4 (𝜑𝐸𝑃)
1312adantr 480 . . 3 ((𝜑𝐵 ∈ (𝐴𝐼𝐶)) → 𝐸𝑃)
14 legid.a . . . . 5 (𝜑𝐴𝑃)
1514adantr 480 . . . 4 ((𝜑𝐵 ∈ (𝐴𝐼𝐶)) → 𝐴𝑃)
16 legtrd.d . . . . 5 (𝜑𝐷𝑃)
1716adantr 480 . . . 4 ((𝜑𝐵 ∈ (𝐴𝐼𝐶)) → 𝐷𝑃)
18 simpr 476 . . . . 5 ((𝜑𝐵 ∈ (𝐴𝐼𝐶)) → 𝐵 ∈ (𝐴𝐼𝐶))
191, 2, 3, 5, 15, 9, 7, 18tgbtwncom 25183 . . . 4 ((𝜑𝐵 ∈ (𝐴𝐼𝐶)) → 𝐵 ∈ (𝐶𝐼𝐴))
20 legval.l . . . . . 6 = (≤G‘𝐺)
21 tgcgrsub2.2 . . . . . . 7 (𝜑 → (𝐸 ∈ (𝐷𝐼𝐹) ∨ 𝐹 ∈ (𝐷𝐼𝐸)))
2221adantr 480 . . . . . 6 ((𝜑𝐵 ∈ (𝐴𝐼𝐶)) → (𝐸 ∈ (𝐷𝐼𝐹) ∨ 𝐹 ∈ (𝐷𝐼𝐸)))
231, 2, 3, 20, 5, 15, 9, 7, 18btwnleg 25283 . . . . . . 7 ((𝜑𝐵 ∈ (𝐴𝐼𝐶)) → (𝐴 𝐵) (𝐴 𝐶))
24 tgcgrsub2.3 . . . . . . . 8 (𝜑 → (𝐴 𝐵) = (𝐷 𝐸))
2524adantr 480 . . . . . . 7 ((𝜑𝐵 ∈ (𝐴𝐼𝐶)) → (𝐴 𝐵) = (𝐷 𝐸))
26 tgcgrsub2.4 . . . . . . . 8 (𝜑 → (𝐴 𝐶) = (𝐷 𝐹))
2726adantr 480 . . . . . . 7 ((𝜑𝐵 ∈ (𝐴𝐼𝐶)) → (𝐴 𝐶) = (𝐷 𝐹))
2823, 25, 273brtr3d 4614 . . . . . 6 ((𝜑𝐵 ∈ (𝐴𝐼𝐶)) → (𝐷 𝐸) (𝐷 𝐹))
291, 2, 3, 20, 5, 13, 11, 17, 17, 22, 28legbtwn 25289 . . . . 5 ((𝜑𝐵 ∈ (𝐴𝐼𝐶)) → 𝐸 ∈ (𝐷𝐼𝐹))
301, 2, 3, 5, 17, 13, 11, 29tgbtwncom 25183 . . . 4 ((𝜑𝐵 ∈ (𝐴𝐼𝐶)) → 𝐸 ∈ (𝐹𝐼𝐷))
311, 2, 3, 4, 14, 6, 16, 10, 26tgcgrcomlr 25175 . . . . 5 (𝜑 → (𝐶 𝐴) = (𝐹 𝐷))
3231adantr 480 . . . 4 ((𝜑𝐵 ∈ (𝐴𝐼𝐶)) → (𝐶 𝐴) = (𝐹 𝐷))
331, 2, 3, 4, 14, 8, 16, 12, 24tgcgrcomlr 25175 . . . . 5 (𝜑 → (𝐵 𝐴) = (𝐸 𝐷))
3433adantr 480 . . . 4 ((𝜑𝐵 ∈ (𝐴𝐼𝐶)) → (𝐵 𝐴) = (𝐸 𝐷))
351, 2, 3, 5, 7, 9, 15, 11, 13, 17, 19, 30, 32, 34tgcgrsub 25204 . . 3 ((𝜑𝐵 ∈ (𝐴𝐼𝐶)) → (𝐶 𝐵) = (𝐹 𝐸))
361, 2, 3, 5, 7, 9, 11, 13, 35tgcgrcomlr 25175 . 2 ((𝜑𝐵 ∈ (𝐴𝐼𝐶)) → (𝐵 𝐶) = (𝐸 𝐹))
374adantr 480 . . 3 ((𝜑𝐶 ∈ (𝐴𝐼𝐵)) → 𝐺 ∈ TarskiG)
388adantr 480 . . 3 ((𝜑𝐶 ∈ (𝐴𝐼𝐵)) → 𝐵𝑃)
396adantr 480 . . 3 ((𝜑𝐶 ∈ (𝐴𝐼𝐵)) → 𝐶𝑃)
4014adantr 480 . . 3 ((𝜑𝐶 ∈ (𝐴𝐼𝐵)) → 𝐴𝑃)
4112adantr 480 . . 3 ((𝜑𝐶 ∈ (𝐴𝐼𝐵)) → 𝐸𝑃)
4210adantr 480 . . 3 ((𝜑𝐶 ∈ (𝐴𝐼𝐵)) → 𝐹𝑃)
4316adantr 480 . . 3 ((𝜑𝐶 ∈ (𝐴𝐼𝐵)) → 𝐷𝑃)
44 simpr 476 . . . 4 ((𝜑𝐶 ∈ (𝐴𝐼𝐵)) → 𝐶 ∈ (𝐴𝐼𝐵))
451, 2, 3, 37, 40, 39, 38, 44tgbtwncom 25183 . . 3 ((𝜑𝐶 ∈ (𝐴𝐼𝐵)) → 𝐶 ∈ (𝐵𝐼𝐴))
4621orcomd 402 . . . . . 6 (𝜑 → (𝐹 ∈ (𝐷𝐼𝐸) ∨ 𝐸 ∈ (𝐷𝐼𝐹)))
4746adantr 480 . . . . 5 ((𝜑𝐶 ∈ (𝐴𝐼𝐵)) → (𝐹 ∈ (𝐷𝐼𝐸) ∨ 𝐸 ∈ (𝐷𝐼𝐹)))
481, 2, 3, 20, 37, 40, 39, 38, 44btwnleg 25283 . . . . . 6 ((𝜑𝐶 ∈ (𝐴𝐼𝐵)) → (𝐴 𝐶) (𝐴 𝐵))
4926adantr 480 . . . . . 6 ((𝜑𝐶 ∈ (𝐴𝐼𝐵)) → (𝐴 𝐶) = (𝐷 𝐹))
5024adantr 480 . . . . . 6 ((𝜑𝐶 ∈ (𝐴𝐼𝐵)) → (𝐴 𝐵) = (𝐷 𝐸))
5148, 49, 503brtr3d 4614 . . . . 5 ((𝜑𝐶 ∈ (𝐴𝐼𝐵)) → (𝐷 𝐹) (𝐷 𝐸))
521, 2, 3, 20, 37, 42, 41, 43, 43, 47, 51legbtwn 25289 . . . 4 ((𝜑𝐶 ∈ (𝐴𝐼𝐵)) → 𝐹 ∈ (𝐷𝐼𝐸))
531, 2, 3, 37, 43, 42, 41, 52tgbtwncom 25183 . . 3 ((𝜑𝐶 ∈ (𝐴𝐼𝐵)) → 𝐹 ∈ (𝐸𝐼𝐷))
5433adantr 480 . . 3 ((𝜑𝐶 ∈ (𝐴𝐼𝐵)) → (𝐵 𝐴) = (𝐸 𝐷))
5531adantr 480 . . 3 ((𝜑𝐶 ∈ (𝐴𝐼𝐵)) → (𝐶 𝐴) = (𝐹 𝐷))
561, 2, 3, 37, 38, 39, 40, 41, 42, 43, 45, 53, 54, 55tgcgrsub 25204 . 2 ((𝜑𝐶 ∈ (𝐴𝐼𝐵)) → (𝐵 𝐶) = (𝐸 𝐹))
57 tgcgrsub2.1 . 2 (𝜑 → (𝐵 ∈ (𝐴𝐼𝐶) ∨ 𝐶 ∈ (𝐴𝐼𝐵)))
5836, 56, 57mpjaodan 823 1 (𝜑 → (𝐵 𝐶) = (𝐸 𝐹))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  distcds 15777  TarskiGcstrkg 25129  Itvcitv 25135  ≤Gcleg 25277 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-concat 13156  df-s1 13157  df-s2 13444  df-s3 13445  df-trkgc 25147  df-trkgb 25148  df-trkgcb 25149  df-trkg 25152  df-cgrg 25206  df-leg 25278 This theorem is referenced by:  cgracgr  25510  cgraswap  25512
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