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Theorem dfcgrg2 26212
Description: Congruence for two triangles can also be defined as congruence of sides and angles (6 parts). This is often the actual textbook definition of triangle congruence, see for example https://en.wikipedia.org/wiki/Congruence_(geometry)#Congruence_of_triangles With this definition, the "SSS" congruence theorem has an additional part, namely, that triangle congruence implies congruence of the sides (which means equality of the lengths). Because our development of elementary geometry strives to closely follow Schwabhaeuser's, our original definition of shape congruence, df-cgrg 25862, already covers that part: see trgcgr 25867. This theorem is also named "CPCTC", which stands for "Corresponding Parts of Congruent Triangles are Congruent", see https://en.wikipedia.org/wiki/Congruence_(geometry)#CPCTC (Contributed by Thierry Arnoux, 18-Jan-2023.)
Hypotheses
Ref Expression
dfcgrg2.p 𝑃 = (Base‘𝐺)
dfcgrg2.m = (dist‘𝐺)
dfcgrg2.g (𝜑𝐺 ∈ TarskiG)
dfcgrg2.a (𝜑𝐴𝑃)
dfcgrg2.b (𝜑𝐵𝑃)
dfcgrg2.c (𝜑𝐶𝑃)
dfcgrg2.d (𝜑𝐷𝑃)
dfcgrg2.e (𝜑𝐸𝑃)
dfcgrg2.f (𝜑𝐹𝑃)
dfcgrg2.1 (𝜑𝐴𝐵)
dfcgrg2.2 (𝜑𝐵𝐶)
dfcgrg2.3 (𝜑𝐶𝐴)
Assertion
Ref Expression
dfcgrg2 (𝜑 → (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩ ↔ (((𝐴 𝐵) = (𝐷 𝐸) ∧ (𝐵 𝐶) = (𝐸 𝐹) ∧ (𝐶 𝐴) = (𝐹 𝐷)) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩ ∧ ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝐹𝐷𝐸”⟩ ∧ ⟨“𝐵𝐶𝐴”⟩(cgrA‘𝐺)⟨“𝐸𝐹𝐷”⟩))))

Proof of Theorem dfcgrg2
StepHypRef Expression
1 dfcgrg2.p . . . . . 6 𝑃 = (Base‘𝐺)
2 dfcgrg2.m . . . . . 6 = (dist‘𝐺)
3 eqid 2778 . . . . . 6 (Itv‘𝐺) = (Itv‘𝐺)
4 dfcgrg2.g . . . . . . 7 (𝜑𝐺 ∈ TarskiG)
54adantr 474 . . . . . 6 ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩) → 𝐺 ∈ TarskiG)
6 dfcgrg2.a . . . . . . 7 (𝜑𝐴𝑃)
76adantr 474 . . . . . 6 ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩) → 𝐴𝑃)
8 dfcgrg2.b . . . . . . 7 (𝜑𝐵𝑃)
98adantr 474 . . . . . 6 ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩) → 𝐵𝑃)
10 dfcgrg2.c . . . . . . 7 (𝜑𝐶𝑃)
1110adantr 474 . . . . . 6 ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩) → 𝐶𝑃)
12 dfcgrg2.d . . . . . . 7 (𝜑𝐷𝑃)
1312adantr 474 . . . . . 6 ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩) → 𝐷𝑃)
14 dfcgrg2.e . . . . . . 7 (𝜑𝐸𝑃)
1514adantr 474 . . . . . 6 ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩) → 𝐸𝑃)
16 dfcgrg2.f . . . . . . 7 (𝜑𝐹𝑃)
1716adantr 474 . . . . . 6 ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩) → 𝐹𝑃)
18 eqid 2778 . . . . . . . . 9 (cgrG‘𝐺) = (cgrG‘𝐺)
191, 2, 18, 4, 6, 8, 10, 12, 14, 16trgcgrg 25866 . . . . . . . 8 (𝜑 → (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩ ↔ ((𝐴 𝐵) = (𝐷 𝐸) ∧ (𝐵 𝐶) = (𝐸 𝐹) ∧ (𝐶 𝐴) = (𝐹 𝐷))))
2019biimpa 470 . . . . . . 7 ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩) → ((𝐴 𝐵) = (𝐷 𝐸) ∧ (𝐵 𝐶) = (𝐸 𝐹) ∧ (𝐶 𝐴) = (𝐹 𝐷)))
2120simp1d 1133 . . . . . 6 ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩) → (𝐴 𝐵) = (𝐷 𝐸))
2220simp2d 1134 . . . . . 6 ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩) → (𝐵 𝐶) = (𝐸 𝐹))
2320simp3d 1135 . . . . . 6 ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩) → (𝐶 𝐴) = (𝐹 𝐷))
24 dfcgrg2.1 . . . . . . 7 (𝜑𝐴𝐵)
2524adantr 474 . . . . . 6 ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩) → 𝐴𝐵)
26 dfcgrg2.2 . . . . . . 7 (𝜑𝐵𝐶)
2726adantr 474 . . . . . 6 ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩) → 𝐵𝐶)
28 dfcgrg2.3 . . . . . . 7 (𝜑𝐶𝐴)
2928adantr 474 . . . . . 6 ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩) → 𝐶𝐴)
301, 2, 3, 5, 7, 9, 11, 13, 15, 17, 21, 22, 23, 25, 27, 29tgsss1 26209 . . . . 5 ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩) → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)
311, 2, 3, 5, 11, 7, 9, 17, 13, 15, 23, 21, 22, 29, 25, 27tgsss1 26209 . . . . 5 ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩) → ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝐹𝐷𝐸”⟩)
321, 2, 3, 5, 9, 11, 7, 15, 17, 13, 22, 23, 21, 27, 29, 25tgsss1 26209 . . . . 5 ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩) → ⟨“𝐵𝐶𝐴”⟩(cgrA‘𝐺)⟨“𝐸𝐹𝐷”⟩)
3330, 31, 323jca 1119 . . . 4 ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩) → (⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩ ∧ ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝐹𝐷𝐸”⟩ ∧ ⟨“𝐵𝐶𝐴”⟩(cgrA‘𝐺)⟨“𝐸𝐹𝐷”⟩))
3433ex 403 . . 3 (𝜑 → (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩ → (⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩ ∧ ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝐹𝐷𝐸”⟩ ∧ ⟨“𝐵𝐶𝐴”⟩(cgrA‘𝐺)⟨“𝐸𝐹𝐷”⟩)))
3534pm4.71d 557 . 2 (𝜑 → (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩ ↔ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩ ∧ (⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩ ∧ ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝐹𝐷𝐸”⟩ ∧ ⟨“𝐵𝐶𝐴”⟩(cgrA‘𝐺)⟨“𝐸𝐹𝐷”⟩))))
3619anbi1d 623 . 2 (𝜑 → ((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩ ∧ (⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩ ∧ ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝐹𝐷𝐸”⟩ ∧ ⟨“𝐵𝐶𝐴”⟩(cgrA‘𝐺)⟨“𝐸𝐹𝐷”⟩)) ↔ (((𝐴 𝐵) = (𝐷 𝐸) ∧ (𝐵 𝐶) = (𝐸 𝐹) ∧ (𝐶 𝐴) = (𝐹 𝐷)) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩ ∧ ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝐹𝐷𝐸”⟩ ∧ ⟨“𝐵𝐶𝐴”⟩(cgrA‘𝐺)⟨“𝐸𝐹𝐷”⟩))))
3735, 36bitrd 271 1 (𝜑 → (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩ ↔ (((𝐴 𝐵) = (𝐷 𝐸) ∧ (𝐵 𝐶) = (𝐸 𝐹) ∧ (𝐶 𝐴) = (𝐹 𝐷)) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩ ∧ ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝐹𝐷𝐸”⟩ ∧ ⟨“𝐵𝐶𝐴”⟩(cgrA‘𝐺)⟨“𝐸𝐹𝐷”⟩))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  w3a 1071   = wceq 1601  wcel 2107  wne 2969   class class class wbr 4886  cfv 6135  (class class class)co 6922  ⟨“cs3 13993  Basecbs 16255  distcds 16347  TarskiGcstrkg 25781  Itvcitv 25787  cgrGccgrg 25861  cgrAccgra 26155
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-cnex 10328  ax-resscn 10329  ax-1cn 10330  ax-icn 10331  ax-addcl 10332  ax-addrcl 10333  ax-mulcl 10334  ax-mulrcl 10335  ax-mulcom 10336  ax-addass 10337  ax-mulass 10338  ax-distr 10339  ax-i2m1 10340  ax-1ne0 10341  ax-1rid 10342  ax-rnegex 10343  ax-rrecex 10344  ax-cnre 10345  ax-pre-lttri 10346  ax-pre-lttrn 10347  ax-pre-ltadd 10348  ax-pre-mulgt0 10349
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-int 4711  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-1st 7445  df-2nd 7446  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-1o 7843  df-oadd 7847  df-er 8026  df-map 8142  df-pm 8143  df-en 8242  df-dom 8243  df-sdom 8244  df-fin 8245  df-card 9098  df-pnf 10413  df-mnf 10414  df-xr 10415  df-ltxr 10416  df-le 10417  df-sub 10608  df-neg 10609  df-nn 11375  df-2 11438  df-3 11439  df-n0 11643  df-z 11729  df-uz 11993  df-fz 12644  df-fzo 12785  df-hash 13436  df-word 13600  df-concat 13661  df-s1 13686  df-s2 13999  df-s3 14000  df-trkgc 25799  df-trkgcb 25801  df-trkg 25804  df-cgrg 25862  df-hlg 25952  df-cgra 26156
This theorem is referenced by: (None)
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