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Theorem dfcgrg2 28617
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 28265, already covers that part: see trgcgr 28270. 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 28270. (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 2726 . . . . . 6 (Itvβ€˜πΊ) = (Itvβ€˜πΊ)
4 dfcgrg2.g . . . . . . 7 (πœ‘ β†’ 𝐺 ∈ TarskiG)
54adantr 480 . . . . . 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 480 . . . . . 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 480 . . . . . 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 480 . . . . . 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 480 . . . . . 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 480 . . . . . 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 480 . . . . . 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 2726 . . . . . . . . 9 (cgrGβ€˜πΊ) = (cgrGβ€˜πΊ)
191, 2, 18, 4, 6, 8, 10, 12, 14, 16trgcgrg 28269 . . . . . . . 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 476 . . . . . . 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 480 . . . . . 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 480 . . . . . 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 480 . . . . . 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 28614 . . . . 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 28614 . . . . 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 28614 . . . . 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 412 . . 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 561 . 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 279 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 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2934   class class class wbr 5141  β€˜cfv 6536  (class class class)co 7404  βŸ¨β€œcs3 14796  Basecbs 17150  distcds 17212  TarskiGcstrkg 28181  Itvcitv 28187  cgrGccgrg 28264  cgrAccgra 28561
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6293  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8264  df-wrecs 8295  df-recs 8369  df-rdg 8408  df-1o 8464  df-er 8702  df-map 8821  df-pm 8822  df-en 8939  df-dom 8940  df-sdom 8941  df-fin 8942  df-card 9933  df-pnf 11251  df-mnf 11252  df-xr 11253  df-ltxr 11254  df-le 11255  df-sub 11447  df-neg 11448  df-nn 12214  df-2 12276  df-3 12277  df-n0 12474  df-z 12560  df-uz 12824  df-fz 13488  df-fzo 13631  df-hash 14293  df-word 14468  df-concat 14524  df-s1 14549  df-s2 14802  df-s3 14803  df-trkgc 28202  df-trkgcb 28204  df-trkg 28207  df-cgrg 28265  df-hlg 28355  df-cgra 28562
This theorem is referenced by: (None)
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