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Theorem cgracom 26619
Description: Angle congruence commutes. Theorem 11.7 of [Schwabhauser] p. 97. (Contributed by Thierry Arnoux, 5-Mar-2020.)
Hypotheses
Ref Expression
cgraid.p 𝑃 = (Base‘𝐺)
cgraid.i 𝐼 = (Itv‘𝐺)
cgraid.g (𝜑𝐺 ∈ TarskiG)
cgraid.k 𝐾 = (hlG‘𝐺)
cgraid.a (𝜑𝐴𝑃)
cgraid.b (𝜑𝐵𝑃)
cgraid.c (𝜑𝐶𝑃)
cgracom.d (𝜑𝐷𝑃)
cgracom.e (𝜑𝐸𝑃)
cgracom.f (𝜑𝐹𝑃)
cgracom.1 (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)
Assertion
Ref Expression
cgracom (𝜑 → ⟨“𝐷𝐸𝐹”⟩(cgrA‘𝐺)⟨“𝐴𝐵𝐶”⟩)

Proof of Theorem cgracom
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cgraid.p . . . . 5 𝑃 = (Base‘𝐺)
2 eqid 2824 . . . . 5 (dist‘𝐺) = (dist‘𝐺)
3 eqid 2824 . . . . 5 (cgrG‘𝐺) = (cgrG‘𝐺)
4 cgraid.g . . . . . 6 (𝜑𝐺 ∈ TarskiG)
54ad3antrrr 729 . . . . 5 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐸(dist‘𝐺)𝐷)) ∧ (𝑦(𝐾𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐸(dist‘𝐺)𝐹)))) → 𝐺 ∈ TarskiG)
6 cgracom.d . . . . . 6 (𝜑𝐷𝑃)
76ad3antrrr 729 . . . . 5 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐸(dist‘𝐺)𝐷)) ∧ (𝑦(𝐾𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐸(dist‘𝐺)𝐹)))) → 𝐷𝑃)
8 cgracom.e . . . . . 6 (𝜑𝐸𝑃)
98ad3antrrr 729 . . . . 5 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐸(dist‘𝐺)𝐷)) ∧ (𝑦(𝐾𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐸(dist‘𝐺)𝐹)))) → 𝐸𝑃)
10 cgracom.f . . . . . 6 (𝜑𝐹𝑃)
1110ad3antrrr 729 . . . . 5 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐸(dist‘𝐺)𝐷)) ∧ (𝑦(𝐾𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐸(dist‘𝐺)𝐹)))) → 𝐹𝑃)
12 simpllr 775 . . . . 5 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐸(dist‘𝐺)𝐷)) ∧ (𝑦(𝐾𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐸(dist‘𝐺)𝐹)))) → 𝑥𝑃)
13 cgraid.b . . . . . 6 (𝜑𝐵𝑃)
1413ad3antrrr 729 . . . . 5 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐸(dist‘𝐺)𝐷)) ∧ (𝑦(𝐾𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐸(dist‘𝐺)𝐹)))) → 𝐵𝑃)
15 simplr 768 . . . . 5 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐸(dist‘𝐺)𝐷)) ∧ (𝑦(𝐾𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐸(dist‘𝐺)𝐹)))) → 𝑦𝑃)
16 cgraid.i . . . . . 6 𝐼 = (Itv‘𝐺)
17 simprlr 779 . . . . . . 7 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐸(dist‘𝐺)𝐷)) ∧ (𝑦(𝐾𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐸(dist‘𝐺)𝐹)))) → (𝐵(dist‘𝐺)𝑥) = (𝐸(dist‘𝐺)𝐷))
1817eqcomd 2830 . . . . . 6 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐸(dist‘𝐺)𝐷)) ∧ (𝑦(𝐾𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐸(dist‘𝐺)𝐹)))) → (𝐸(dist‘𝐺)𝐷) = (𝐵(dist‘𝐺)𝑥))
191, 2, 16, 5, 9, 7, 14, 12, 18tgcgrcomlr 26277 . . . . 5 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐸(dist‘𝐺)𝐷)) ∧ (𝑦(𝐾𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐸(dist‘𝐺)𝐹)))) → (𝐷(dist‘𝐺)𝐸) = (𝑥(dist‘𝐺)𝐵))
20 simprrr 781 . . . . . 6 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐸(dist‘𝐺)𝐷)) ∧ (𝑦(𝐾𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐸(dist‘𝐺)𝐹)))) → (𝐵(dist‘𝐺)𝑦) = (𝐸(dist‘𝐺)𝐹))
2120eqcomd 2830 . . . . 5 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐸(dist‘𝐺)𝐷)) ∧ (𝑦(𝐾𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐸(dist‘𝐺)𝐹)))) → (𝐸(dist‘𝐺)𝐹) = (𝐵(dist‘𝐺)𝑦))
22 cgraid.k . . . . . . . 8 𝐾 = (hlG‘𝐺)
23 cgraid.a . . . . . . . . 9 (𝜑𝐴𝑃)
2423ad3antrrr 729 . . . . . . . 8 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐸(dist‘𝐺)𝐷)) ∧ (𝑦(𝐾𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐸(dist‘𝐺)𝐹)))) → 𝐴𝑃)
25 cgraid.c . . . . . . . . 9 (𝜑𝐶𝑃)
2625ad3antrrr 729 . . . . . . . 8 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐸(dist‘𝐺)𝐷)) ∧ (𝑦(𝐾𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐸(dist‘𝐺)𝐹)))) → 𝐶𝑃)
27 cgracom.1 . . . . . . . . 9 (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)
2827ad3antrrr 729 . . . . . . . 8 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐸(dist‘𝐺)𝐷)) ∧ (𝑦(𝐾𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐸(dist‘𝐺)𝐹)))) → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)
29 simprll 778 . . . . . . . 8 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐸(dist‘𝐺)𝐷)) ∧ (𝑦(𝐾𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐸(dist‘𝐺)𝐹)))) → 𝑥(𝐾𝐵)𝐴)
30 simprrl 780 . . . . . . . 8 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐸(dist‘𝐺)𝐷)) ∧ (𝑦(𝐾𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐸(dist‘𝐺)𝐹)))) → 𝑦(𝐾𝐵)𝐶)
311, 16, 22, 5, 24, 14, 26, 7, 9, 11, 28, 12, 2, 15, 29, 30, 17, 20cgracgr 26615 . . . . . . 7 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐸(dist‘𝐺)𝐷)) ∧ (𝑦(𝐾𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐸(dist‘𝐺)𝐹)))) → (𝑥(dist‘𝐺)𝑦) = (𝐷(dist‘𝐺)𝐹))
3231eqcomd 2830 . . . . . 6 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐸(dist‘𝐺)𝐷)) ∧ (𝑦(𝐾𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐸(dist‘𝐺)𝐹)))) → (𝐷(dist‘𝐺)𝐹) = (𝑥(dist‘𝐺)𝑦))
331, 2, 16, 5, 7, 11, 12, 15, 32tgcgrcomlr 26277 . . . . 5 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐸(dist‘𝐺)𝐷)) ∧ (𝑦(𝐾𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐸(dist‘𝐺)𝐹)))) → (𝐹(dist‘𝐺)𝐷) = (𝑦(dist‘𝐺)𝑥))
341, 2, 3, 5, 7, 9, 11, 12, 14, 15, 19, 21, 33trgcgr 26313 . . . 4 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐸(dist‘𝐺)𝐷)) ∧ (𝑦(𝐾𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐸(dist‘𝐺)𝐹)))) → ⟨“𝐷𝐸𝐹”⟩(cgrG‘𝐺)⟨“𝑥𝐵𝑦”⟩)
3534, 29, 303jca 1125 . . 3 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐸(dist‘𝐺)𝐷)) ∧ (𝑦(𝐾𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐸(dist‘𝐺)𝐹)))) → (⟨“𝐷𝐸𝐹”⟩(cgrG‘𝐺)⟨“𝑥𝐵𝑦”⟩ ∧ 𝑥(𝐾𝐵)𝐴𝑦(𝐾𝐵)𝐶))
361, 16, 22, 4, 23, 13, 25, 6, 8, 10, 27cgrane1 26609 . . . . 5 (𝜑𝐴𝐵)
371, 16, 22, 4, 23, 13, 25, 6, 8, 10, 27cgrane3 26611 . . . . 5 (𝜑𝐸𝐷)
381, 16, 22, 13, 8, 6, 4, 23, 2, 36, 37hlcgrex 26413 . . . 4 (𝜑 → ∃𝑥𝑃 (𝑥(𝐾𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐸(dist‘𝐺)𝐷)))
391, 16, 22, 4, 23, 13, 25, 6, 8, 10, 27cgrane2 26610 . . . . . 6 (𝜑𝐵𝐶)
4039necomd 3069 . . . . 5 (𝜑𝐶𝐵)
411, 16, 22, 4, 23, 13, 25, 6, 8, 10, 27cgrane4 26612 . . . . 5 (𝜑𝐸𝐹)
421, 16, 22, 13, 8, 10, 4, 25, 2, 40, 41hlcgrex 26413 . . . 4 (𝜑 → ∃𝑦𝑃 (𝑦(𝐾𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐸(dist‘𝐺)𝐹)))
43 reeanv 3358 . . . 4 (∃𝑥𝑃𝑦𝑃 ((𝑥(𝐾𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐸(dist‘𝐺)𝐷)) ∧ (𝑦(𝐾𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐸(dist‘𝐺)𝐹))) ↔ (∃𝑥𝑃 (𝑥(𝐾𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐸(dist‘𝐺)𝐷)) ∧ ∃𝑦𝑃 (𝑦(𝐾𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐸(dist‘𝐺)𝐹))))
4438, 42, 43sylanbrc 586 . . 3 (𝜑 → ∃𝑥𝑃𝑦𝑃 ((𝑥(𝐾𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐸(dist‘𝐺)𝐷)) ∧ (𝑦(𝐾𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐸(dist‘𝐺)𝐹))))
4535, 44reximddv2 3270 . 2 (𝜑 → ∃𝑥𝑃𝑦𝑃 (⟨“𝐷𝐸𝐹”⟩(cgrG‘𝐺)⟨“𝑥𝐵𝑦”⟩ ∧ 𝑥(𝐾𝐵)𝐴𝑦(𝐾𝐵)𝐶))
461, 16, 22, 4, 6, 8, 10, 23, 13, 25iscgra 26606 . 2 (𝜑 → (⟨“𝐷𝐸𝐹”⟩(cgrA‘𝐺)⟨“𝐴𝐵𝐶”⟩ ↔ ∃𝑥𝑃𝑦𝑃 (⟨“𝐷𝐸𝐹”⟩(cgrG‘𝐺)⟨“𝑥𝐵𝑦”⟩ ∧ 𝑥(𝐾𝐵)𝐴𝑦(𝐾𝐵)𝐶)))
4745, 46mpbird 260 1 (𝜑 → ⟨“𝐷𝐸𝐹”⟩(cgrA‘𝐺)⟨“𝐴𝐵𝐶”⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2115  wrex 3134   class class class wbr 5052  cfv 6343  (class class class)co 7149  ⟨“cs3 14204  Basecbs 16483  distcds 16574  TarskiGcstrkg 26227  Itvcitv 26233  cgrGccgrg 26307  hlGchlg 26397  cgrAccgra 26604
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-cnex 10591  ax-resscn 10592  ax-1cn 10593  ax-icn 10594  ax-addcl 10595  ax-addrcl 10596  ax-mulcl 10597  ax-mulrcl 10598  ax-mulcom 10599  ax-addass 10600  ax-mulass 10601  ax-distr 10602  ax-i2m1 10603  ax-1ne0 10604  ax-1rid 10605  ax-rnegex 10606  ax-rrecex 10607  ax-cnre 10608  ax-pre-lttri 10609  ax-pre-lttrn 10610  ax-pre-ltadd 10611  ax-pre-mulgt0 10612
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-om 7575  df-1st 7684  df-2nd 7685  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-oadd 8102  df-er 8285  df-map 8404  df-pm 8405  df-en 8506  df-dom 8507  df-sdom 8508  df-fin 8509  df-dju 9327  df-card 9365  df-pnf 10675  df-mnf 10676  df-xr 10677  df-ltxr 10678  df-le 10679  df-sub 10870  df-neg 10871  df-nn 11635  df-2 11697  df-3 11698  df-n0 11895  df-xnn0 11965  df-z 11979  df-uz 12241  df-fz 12895  df-fzo 13038  df-hash 13696  df-word 13867  df-concat 13923  df-s1 13950  df-s2 14210  df-s3 14211  df-trkgc 26245  df-trkgb 26246  df-trkgcb 26247  df-trkg 26250  df-cgrg 26308  df-leg 26380  df-hlg 26398  df-cgra 26605
This theorem is referenced by:  cgracol  26625  cgrancol  26626  dfcgra2  26627  tgasa1  26655  isoas  26661
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