dfcgrg2 - Metamath Proof Explorer

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Theorem dfcgrg2 28842

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 28490, already covers that part: see trgcgr 28495. 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 28495. (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 2735	. . . . . 6 ⊢ (Itv‘𝐺) = (Itv‘𝐺)
4		dfcgrg2.g	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5	4	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩) → 𝐺 ∈ TarskiG)
6		dfcgrg2.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
7	6	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩) → 𝐴 ∈ 𝑃)
8		dfcgrg2.b	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
9	8	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩) → 𝐵 ∈ 𝑃)
10		dfcgrg2.c	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
11	10	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩) → 𝐶 ∈ 𝑃)
12		dfcgrg2.d	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ 𝑃)
13	12	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩) → 𝐷 ∈ 𝑃)
14		dfcgrg2.e	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ 𝑃)
15	14	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩) → 𝐸 ∈ 𝑃)
16		dfcgrg2.f	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ 𝑃)
17	16	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩) → 𝐹 ∈ 𝑃)
18		eqid 2735	. . . . . . . . 9 ⊢ (cgrG‘𝐺) = (cgrG‘𝐺)
19	1, 2, 18, 4, 6, 8, 10, 12, 14, 16	trgcgrg 28494	. . . . . . . 8 ⊢ (𝜑 → (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩ ↔ ((𝐴 − 𝐵) = (𝐷 − 𝐸) ∧ (𝐵 − 𝐶) = (𝐸 − 𝐹) ∧ (𝐶 − 𝐴) = (𝐹 − 𝐷))))
20	19	biimpa 476	. . . . . . 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 480	. . . . . 6 ⊢ ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩) → 𝐴 ≠ 𝐵)
26		dfcgrg2.2	. . . . . . 7 ⊢ (𝜑 → 𝐵 ≠ 𝐶)
27	26	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩) → 𝐵 ≠ 𝐶)
28		dfcgrg2.3	. . . . . . 7 ⊢ (𝜑 → 𝐶 ≠ 𝐴)
29	28	adantr 480	. . . . . 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 28839	. . . . 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 28839	. . . . 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 28839	. . . . 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 412	. . 3 ⊢ (𝜑 → (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩ → (⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩ ∧ ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝐹𝐷𝐸”⟩ ∧ ⟨“𝐵𝐶𝐴”⟩(cgrA‘𝐺)⟨“𝐸𝐹𝐷”⟩)))
35	34	pm4.71d 561	. 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 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2108 ≠ wne 2932 class class class wbr 5119 ‘cfv 6531 (class class class)co 7405 ⟨“cs3 14861 Basecbs 17228 distcds 17280 TarskiGcstrkg 28406 Itvcitv 28412 cgrGccgrg 28489 cgrAccgra 28786

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-tp 4606 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7862 df-1st 7988 df-2nd 7989 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8719 df-map 8842 df-pm 8843 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-card 9953 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-nn 12241 df-2 12303 df-3 12304 df-n0 12502 df-z 12589 df-uz 12853 df-fz 13525 df-fzo 13672 df-hash 14349 df-word 14532 df-concat 14589 df-s1 14614 df-s2 14867 df-s3 14868 df-trkgc 28427 df-trkgcb 28429 df-trkg 28432 df-cgrg 28490 df-hlg 28580 df-cgra 28787

This theorem is referenced by: (None)

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