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Theorem dfcgrg2 28687
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 28335, already covers that part: see trgcgr 28340. 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 28340. (Contributed by Thierry Arnoux, 18-Jan-2023.)
Hypotheses
Ref Expression
dfcgrg2.p 𝑃 = (Baseβ€˜πΊ)
dfcgrg2.m βˆ’ = (distβ€˜πΊ)
dfcgrg2.g (πœ‘ β†’ 𝐺 ∈ TarskiG)
dfcgrg2.a (πœ‘ β†’ 𝐴 ∈ 𝑃)
dfcgrg2.b (πœ‘ β†’ 𝐡 ∈ 𝑃)
dfcgrg2.c (πœ‘ β†’ 𝐢 ∈ 𝑃)
dfcgrg2.d (πœ‘ β†’ 𝐷 ∈ 𝑃)
dfcgrg2.e (πœ‘ β†’ 𝐸 ∈ 𝑃)
dfcgrg2.f (πœ‘ β†’ 𝐹 ∈ 𝑃)
dfcgrg2.1 (πœ‘ β†’ 𝐴 β‰  𝐡)
dfcgrg2.2 (πœ‘ β†’ 𝐡 β‰  𝐢)
dfcgrg2.3 (πœ‘ β†’ 𝐢 β‰  𝐴)
Assertion
Ref Expression
dfcgrg2 (πœ‘ β†’ (βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrGβ€˜πΊ)βŸ¨β€œπ·πΈπΉβ€βŸ© ↔ (((𝐴 βˆ’ 𝐡) = (𝐷 βˆ’ 𝐸) ∧ (𝐡 βˆ’ 𝐢) = (𝐸 βˆ’ 𝐹) ∧ (𝐢 βˆ’ 𝐴) = (𝐹 βˆ’ 𝐷)) ∧ (βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπ·πΈπΉβ€βŸ© ∧ βŸ¨β€œπΆπ΄π΅β€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπΉπ·πΈβ€βŸ© ∧ βŸ¨β€œπ΅πΆπ΄β€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπΈπΉπ·β€βŸ©))))

Proof of Theorem dfcgrg2
StepHypRef Expression
1 dfcgrg2.p . . . . . 6 𝑃 = (Baseβ€˜πΊ)
2 dfcgrg2.m . . . . . 6 βˆ’ = (distβ€˜πΊ)
3 eqid 2728 . . . . . 6 (Itvβ€˜πΊ) = (Itvβ€˜πΊ)
4 dfcgrg2.g . . . . . . 7 (πœ‘ β†’ 𝐺 ∈ TarskiG)
54adantr 479 . . . . . 6 ((πœ‘ ∧ βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrGβ€˜πΊ)βŸ¨β€œπ·πΈπΉβ€βŸ©) β†’ 𝐺 ∈ TarskiG)
6 dfcgrg2.a . . . . . . 7 (πœ‘ β†’ 𝐴 ∈ 𝑃)
76adantr 479 . . . . . 6 ((πœ‘ ∧ βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrGβ€˜πΊ)βŸ¨β€œπ·πΈπΉβ€βŸ©) β†’ 𝐴 ∈ 𝑃)
8 dfcgrg2.b . . . . . . 7 (πœ‘ β†’ 𝐡 ∈ 𝑃)
98adantr 479 . . . . . 6 ((πœ‘ ∧ βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrGβ€˜πΊ)βŸ¨β€œπ·πΈπΉβ€βŸ©) β†’ 𝐡 ∈ 𝑃)
10 dfcgrg2.c . . . . . . 7 (πœ‘ β†’ 𝐢 ∈ 𝑃)
1110adantr 479 . . . . . 6 ((πœ‘ ∧ βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrGβ€˜πΊ)βŸ¨β€œπ·πΈπΉβ€βŸ©) β†’ 𝐢 ∈ 𝑃)
12 dfcgrg2.d . . . . . . 7 (πœ‘ β†’ 𝐷 ∈ 𝑃)
1312adantr 479 . . . . . 6 ((πœ‘ ∧ βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrGβ€˜πΊ)βŸ¨β€œπ·πΈπΉβ€βŸ©) β†’ 𝐷 ∈ 𝑃)
14 dfcgrg2.e . . . . . . 7 (πœ‘ β†’ 𝐸 ∈ 𝑃)
1514adantr 479 . . . . . 6 ((πœ‘ ∧ βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrGβ€˜πΊ)βŸ¨β€œπ·πΈπΉβ€βŸ©) β†’ 𝐸 ∈ 𝑃)
16 dfcgrg2.f . . . . . . 7 (πœ‘ β†’ 𝐹 ∈ 𝑃)
1716adantr 479 . . . . . 6 ((πœ‘ ∧ βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrGβ€˜πΊ)βŸ¨β€œπ·πΈπΉβ€βŸ©) β†’ 𝐹 ∈ 𝑃)
18 eqid 2728 . . . . . . . . 9 (cgrGβ€˜πΊ) = (cgrGβ€˜πΊ)
191, 2, 18, 4, 6, 8, 10, 12, 14, 16trgcgrg 28339 . . . . . . . 8 (πœ‘ β†’ (βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrGβ€˜πΊ)βŸ¨β€œπ·πΈπΉβ€βŸ© ↔ ((𝐴 βˆ’ 𝐡) = (𝐷 βˆ’ 𝐸) ∧ (𝐡 βˆ’ 𝐢) = (𝐸 βˆ’ 𝐹) ∧ (𝐢 βˆ’ 𝐴) = (𝐹 βˆ’ 𝐷))))
2019biimpa 475 . . . . . . 7 ((πœ‘ ∧ βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrGβ€˜πΊ)βŸ¨β€œπ·πΈπΉβ€βŸ©) β†’ ((𝐴 βˆ’ 𝐡) = (𝐷 βˆ’ 𝐸) ∧ (𝐡 βˆ’ 𝐢) = (𝐸 βˆ’ 𝐹) ∧ (𝐢 βˆ’ 𝐴) = (𝐹 βˆ’ 𝐷)))
2120simp1d 1139 . . . . . 6 ((πœ‘ ∧ βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrGβ€˜πΊ)βŸ¨β€œπ·πΈπΉβ€βŸ©) β†’ (𝐴 βˆ’ 𝐡) = (𝐷 βˆ’ 𝐸))
2220simp2d 1140 . . . . . 6 ((πœ‘ ∧ βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrGβ€˜πΊ)βŸ¨β€œπ·πΈπΉβ€βŸ©) β†’ (𝐡 βˆ’ 𝐢) = (𝐸 βˆ’ 𝐹))
2320simp3d 1141 . . . . . 6 ((πœ‘ ∧ βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrGβ€˜πΊ)βŸ¨β€œπ·πΈπΉβ€βŸ©) β†’ (𝐢 βˆ’ 𝐴) = (𝐹 βˆ’ 𝐷))
24 dfcgrg2.1 . . . . . . 7 (πœ‘ β†’ 𝐴 β‰  𝐡)
2524adantr 479 . . . . . 6 ((πœ‘ ∧ βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrGβ€˜πΊ)βŸ¨β€œπ·πΈπΉβ€βŸ©) β†’ 𝐴 β‰  𝐡)
26 dfcgrg2.2 . . . . . . 7 (πœ‘ β†’ 𝐡 β‰  𝐢)
2726adantr 479 . . . . . 6 ((πœ‘ ∧ βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrGβ€˜πΊ)βŸ¨β€œπ·πΈπΉβ€βŸ©) β†’ 𝐡 β‰  𝐢)
28 dfcgrg2.3 . . . . . . 7 (πœ‘ β†’ 𝐢 β‰  𝐴)
2928adantr 479 . . . . . 6 ((πœ‘ ∧ βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrGβ€˜πΊ)βŸ¨β€œπ·πΈπΉβ€βŸ©) β†’ 𝐢 β‰  𝐴)
301, 2, 3, 5, 7, 9, 11, 13, 15, 17, 21, 22, 23, 25, 27, 29tgsss1 28684 . . . . 5 ((πœ‘ ∧ βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrGβ€˜πΊ)βŸ¨β€œπ·πΈπΉβ€βŸ©) β†’ βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπ·πΈπΉβ€βŸ©)
311, 2, 3, 5, 11, 7, 9, 17, 13, 15, 23, 21, 22, 29, 25, 27tgsss1 28684 . . . . 5 ((πœ‘ ∧ βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrGβ€˜πΊ)βŸ¨β€œπ·πΈπΉβ€βŸ©) β†’ βŸ¨β€œπΆπ΄π΅β€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπΉπ·πΈβ€βŸ©)
321, 2, 3, 5, 9, 11, 7, 15, 17, 13, 22, 23, 21, 27, 29, 25tgsss1 28684 . . . . 5 ((πœ‘ ∧ βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrGβ€˜πΊ)βŸ¨β€œπ·πΈπΉβ€βŸ©) β†’ βŸ¨β€œπ΅πΆπ΄β€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπΈπΉπ·β€βŸ©)
3330, 31, 323jca 1125 . . . 4 ((πœ‘ ∧ βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrGβ€˜πΊ)βŸ¨β€œπ·πΈπΉβ€βŸ©) β†’ (βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπ·πΈπΉβ€βŸ© ∧ βŸ¨β€œπΆπ΄π΅β€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπΉπ·πΈβ€βŸ© ∧ βŸ¨β€œπ΅πΆπ΄β€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπΈπΉπ·β€βŸ©))
3433ex 411 . . 3 (πœ‘ β†’ (βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrGβ€˜πΊ)βŸ¨β€œπ·πΈπΉβ€βŸ© β†’ (βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπ·πΈπΉβ€βŸ© ∧ βŸ¨β€œπΆπ΄π΅β€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπΉπ·πΈβ€βŸ© ∧ βŸ¨β€œπ΅πΆπ΄β€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπΈπΉπ·β€βŸ©)))
3534pm4.71d 560 . 2 (πœ‘ β†’ (βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrGβ€˜πΊ)βŸ¨β€œπ·πΈπΉβ€βŸ© ↔ (βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrGβ€˜πΊ)βŸ¨β€œπ·πΈπΉβ€βŸ© ∧ (βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπ·πΈπΉβ€βŸ© ∧ βŸ¨β€œπΆπ΄π΅β€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπΉπ·πΈβ€βŸ© ∧ βŸ¨β€œπ΅πΆπ΄β€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπΈπΉπ·β€βŸ©))))
3619anbi1d 629 . 2 (πœ‘ β†’ ((βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrGβ€˜πΊ)βŸ¨β€œπ·πΈπΉβ€βŸ© ∧ (βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπ·πΈπΉβ€βŸ© ∧ βŸ¨β€œπΆπ΄π΅β€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπΉπ·πΈβ€βŸ© ∧ βŸ¨β€œπ΅πΆπ΄β€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπΈπΉπ·β€βŸ©)) ↔ (((𝐴 βˆ’ 𝐡) = (𝐷 βˆ’ 𝐸) ∧ (𝐡 βˆ’ 𝐢) = (𝐸 βˆ’ 𝐹) ∧ (𝐢 βˆ’ 𝐴) = (𝐹 βˆ’ 𝐷)) ∧ (βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπ·πΈπΉβ€βŸ© ∧ βŸ¨β€œπΆπ΄π΅β€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπΉπ·πΈβ€βŸ© ∧ βŸ¨β€œπ΅πΆπ΄β€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπΈπΉπ·β€βŸ©))))
3735, 36bitrd 278 1 (πœ‘ β†’ (βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrGβ€˜πΊ)βŸ¨β€œπ·πΈπΉβ€βŸ© ↔ (((𝐴 βˆ’ 𝐡) = (𝐷 βˆ’ 𝐸) ∧ (𝐡 βˆ’ 𝐢) = (𝐸 βˆ’ 𝐹) ∧ (𝐢 βˆ’ 𝐴) = (𝐹 βˆ’ 𝐷)) ∧ (βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπ·πΈπΉβ€βŸ© ∧ βŸ¨β€œπΆπ΄π΅β€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπΉπ·πΈβ€βŸ© ∧ βŸ¨β€œπ΅πΆπ΄β€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπΈπΉπ·β€βŸ©))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2937   class class class wbr 5152  β€˜cfv 6553  (class class class)co 7426  βŸ¨β€œcs3 14833  Basecbs 17187  distcds 17249  TarskiGcstrkg 28251  Itvcitv 28257  cgrGccgrg 28334  cgrAccgra 28631
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11202  ax-resscn 11203  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-addrcl 11207  ax-mulcl 11208  ax-mulrcl 11209  ax-mulcom 11210  ax-addass 11211  ax-mulass 11212  ax-distr 11213  ax-i2m1 11214  ax-1ne0 11215  ax-1rid 11216  ax-rnegex 11217  ax-rrecex 11218  ax-cnre 11219  ax-pre-lttri 11220  ax-pre-lttrn 11221  ax-pre-ltadd 11222  ax-pre-mulgt0 11223
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-tp 4637  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-1st 7999  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-1o 8493  df-er 8731  df-map 8853  df-pm 8854  df-en 8971  df-dom 8972  df-sdom 8973  df-fin 8974  df-card 9970  df-pnf 11288  df-mnf 11289  df-xr 11290  df-ltxr 11291  df-le 11292  df-sub 11484  df-neg 11485  df-nn 12251  df-2 12313  df-3 12314  df-n0 12511  df-z 12597  df-uz 12861  df-fz 13525  df-fzo 13668  df-hash 14330  df-word 14505  df-concat 14561  df-s1 14586  df-s2 14839  df-s3 14840  df-trkgc 28272  df-trkgcb 28274  df-trkg 28277  df-cgrg 28335  df-hlg 28425  df-cgra 28632
This theorem is referenced by: (None)
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