dfcgrg2 - Metamath Proof Explorer

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Theorem dfcgrg2 27513

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 27161, already covers that part: see trgcgr 27166. 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 27166. (Contributed by Thierry Arnoux, 18-Jan-2023.)

**Hypotheses**
Ref	Expression
dfcgrg2.p	⊢ 𝑃 = (Base‘𝐺)
dfcgrg2.m	⊢ − = (dist‘𝐺)
dfcgrg2.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
dfcgrg2.a	⊢ (𝜑 → 𝐴 ∈ 𝑃)
dfcgrg2.b	⊢ (𝜑 → 𝐵 ∈ 𝑃)
dfcgrg2.c	⊢ (𝜑 → 𝐶 ∈ 𝑃)
dfcgrg2.d	⊢ (𝜑 → 𝐷 ∈ 𝑃)
dfcgrg2.e	⊢ (𝜑 → 𝐸 ∈ 𝑃)
dfcgrg2.f	⊢ (𝜑 → 𝐹 ∈ 𝑃)
dfcgrg2.1	⊢ (𝜑 → 𝐴 ≠ 𝐵)
dfcgrg2.2	⊢ (𝜑 → 𝐵 ≠ 𝐶)
dfcgrg2.3	⊢ (𝜑 → 𝐶 ≠ 𝐴)

**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**
Step	Hyp	Ref	Expression
1		dfcgrg2.p	. . . . . 6 ⊢ 𝑃 = (Base‘𝐺)
2		dfcgrg2.m	. . . . . 6 ⊢ − = (dist‘𝐺)
3		eqid 2737	. . . . . 6 ⊢ (Itv‘𝐺) = (Itv‘𝐺)
4		dfcgrg2.g	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5	4	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩) → 𝐺 ∈ TarskiG)
6		dfcgrg2.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
7	6	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩) → 𝐴 ∈ 𝑃)
8		dfcgrg2.b	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
9	8	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩) → 𝐵 ∈ 𝑃)
10		dfcgrg2.c	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
11	10	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩) → 𝐶 ∈ 𝑃)
12		dfcgrg2.d	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ 𝑃)
13	12	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩) → 𝐷 ∈ 𝑃)
14		dfcgrg2.e	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ 𝑃)
15	14	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩) → 𝐸 ∈ 𝑃)
16		dfcgrg2.f	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ 𝑃)
17	16	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩) → 𝐹 ∈ 𝑃)
18		eqid 2737	. . . . . . . . 9 ⊢ (cgrG‘𝐺) = (cgrG‘𝐺)
19	1, 2, 18, 4, 6, 8, 10, 12, 14, 16	trgcgrg 27165	. . . . . . . 8 ⊢ (𝜑 → (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩ ↔ ((𝐴 − 𝐵) = (𝐷 − 𝐸) ∧ (𝐵 − 𝐶) = (𝐸 − 𝐹) ∧ (𝐶 − 𝐴) = (𝐹 − 𝐷))))
20	19	biimpa 478	. . . . . . 7 ⊢ ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩) → ((𝐴 − 𝐵) = (𝐷 − 𝐸) ∧ (𝐵 − 𝐶) = (𝐸 − 𝐹) ∧ (𝐶 − 𝐴) = (𝐹 − 𝐷)))
21	20	simp1d 1142	. . . . . 6 ⊢ ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩) → (𝐴 − 𝐵) = (𝐷 − 𝐸))
22	20	simp2d 1143	. . . . . 6 ⊢ ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩) → (𝐵 − 𝐶) = (𝐸 − 𝐹))
23	20	simp3d 1144	. . . . . 6 ⊢ ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩) → (𝐶 − 𝐴) = (𝐹 − 𝐷))
24		dfcgrg2.1	. . . . . . 7 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
25	24	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩) → 𝐴 ≠ 𝐵)
26		dfcgrg2.2	. . . . . . 7 ⊢ (𝜑 → 𝐵 ≠ 𝐶)
27	26	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩) → 𝐵 ≠ 𝐶)
28		dfcgrg2.3	. . . . . . 7 ⊢ (𝜑 → 𝐶 ≠ 𝐴)
29	28	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩) → 𝐶 ≠ 𝐴)
30	1, 2, 3, 5, 7, 9, 11, 13, 15, 17, 21, 22, 23, 25, 27, 29	tgsss1 27510	. . . . 5 ⊢ ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩) → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)
31	1, 2, 3, 5, 11, 7, 9, 17, 13, 15, 23, 21, 22, 29, 25, 27	tgsss1 27510	. . . . 5 ⊢ ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩) → ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝐹𝐷𝐸”⟩)
32	1, 2, 3, 5, 9, 11, 7, 15, 17, 13, 22, 23, 21, 27, 29, 25	tgsss1 27510	. . . . 5 ⊢ ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩) → ⟨“𝐵𝐶𝐴”⟩(cgrA‘𝐺)⟨“𝐸𝐹𝐷”⟩)
33	30, 31, 32	3jca 1128	. . . 4 ⊢ ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩) → (⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩ ∧ ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝐹𝐷𝐸”⟩ ∧ ⟨“𝐵𝐶𝐴”⟩(cgrA‘𝐺)⟨“𝐸𝐹𝐷”⟩))
34	33	ex 414	. . 3 ⊢ (𝜑 → (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩ → (⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩ ∧ ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝐹𝐷𝐸”⟩ ∧ ⟨“𝐵𝐶𝐴”⟩(cgrA‘𝐺)⟨“𝐸𝐹𝐷”⟩)))
35	34	pm4.71d 563	. 2 ⊢ (𝜑 → (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩ ↔ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩ ∧ (⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩ ∧ ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝐹𝐷𝐸”⟩ ∧ ⟨“𝐵𝐶𝐴”⟩(cgrA‘𝐺)⟨“𝐸𝐹𝐷”⟩))))
36	19	anbi1d 631	. 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”⟩))))
37	35, 36	bitrd 279	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 205 ∧ wa 397 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2941 class class class wbr 5097 ‘cfv 6484 (class class class)co 7342 ⟨“cs3 14655 Basecbs 17010 distcds 17069 TarskiGcstrkg 27077 Itvcitv 27083 cgrGccgrg 27160 cgrAccgra 27457

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-rep 5234 ax-sep 5248 ax-nul 5255 ax-pow 5313 ax-pr 5377 ax-un 7655 ax-cnex 11033 ax-resscn 11034 ax-1cn 11035 ax-icn 11036 ax-addcl 11037 ax-addrcl 11038 ax-mulcl 11039 ax-mulrcl 11040 ax-mulcom 11041 ax-addass 11042 ax-mulass 11043 ax-distr 11044 ax-i2m1 11045 ax-1ne0 11046 ax-1rid 11047 ax-rnegex 11048 ax-rrecex 11049 ax-cnre 11050 ax-pre-lttri 11051 ax-pre-lttrn 11052 ax-pre-ltadd 11053 ax-pre-mulgt0 11054

This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3351 df-rab 3405 df-v 3444 df-sbc 3732 df-csb 3848 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3921 df-nul 4275 df-if 4479 df-pw 4554 df-sn 4579 df-pr 4581 df-tp 4583 df-op 4585 df-uni 4858 df-int 4900 df-iun 4948 df-br 5098 df-opab 5160 df-mpt 5181 df-tr 5215 df-id 5523 df-eprel 5529 df-po 5537 df-so 5538 df-fr 5580 df-we 5582 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6243 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6436 df-fun 6486 df-fn 6487 df-f 6488 df-f1 6489 df-fo 6490 df-f1o 6491 df-fv 6492 df-riota 7298 df-ov 7345 df-oprab 7346 df-mpo 7347 df-om 7786 df-1st 7904 df-2nd 7905 df-frecs 8172 df-wrecs 8203 df-recs 8277 df-rdg 8316 df-1o 8372 df-er 8574 df-map 8693 df-pm 8694 df-en 8810 df-dom 8811 df-sdom 8812 df-fin 8813 df-card 9801 df-pnf 11117 df-mnf 11118 df-xr 11119 df-ltxr 11120 df-le 11121 df-sub 11313 df-neg 11314 df-nn 12080 df-2 12142 df-3 12143 df-n0 12340 df-z 12426 df-uz 12689 df-fz 13346 df-fzo 13489 df-hash 14151 df-word 14323 df-concat 14379 df-s1 14404 df-s2 14661 df-s3 14662 df-trkgc 27098 df-trkgcb 27100 df-trkg 27103 df-cgrg 27161 df-hlg 27251 df-cgra 27458

This theorem is referenced by: (None)

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