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Mirrors > Home > MPE Home > Th. List > tgsss1 | Structured version Visualization version GIF version |
Description: Third congruence theorem: SSS (Side-Side-Side): If the three pairs of sides of two triangles are equal in length, then the triangles are congruent. Theorem 11.51 of [Schwabhauser] p. 109. (Contributed by Thierry Arnoux, 1-Aug-2020.) |
Ref | Expression |
---|---|
tgsas.p | ⊢ 𝑃 = (Base‘𝐺) |
tgsas.m | ⊢ − = (dist‘𝐺) |
tgsas.i | ⊢ 𝐼 = (Itv‘𝐺) |
tgsas.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
tgsas.a | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
tgsas.b | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
tgsas.c | ⊢ (𝜑 → 𝐶 ∈ 𝑃) |
tgsas.d | ⊢ (𝜑 → 𝐷 ∈ 𝑃) |
tgsas.e | ⊢ (𝜑 → 𝐸 ∈ 𝑃) |
tgsas.f | ⊢ (𝜑 → 𝐹 ∈ 𝑃) |
tgsss.1 | ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐷 − 𝐸)) |
tgsss.2 | ⊢ (𝜑 → (𝐵 − 𝐶) = (𝐸 − 𝐹)) |
tgsss.3 | ⊢ (𝜑 → (𝐶 − 𝐴) = (𝐹 − 𝐷)) |
tgsss.4 | ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
tgsss.5 | ⊢ (𝜑 → 𝐵 ≠ 𝐶) |
tgsss.6 | ⊢ (𝜑 → 𝐶 ≠ 𝐴) |
Ref | Expression |
---|---|
tgsss1 | ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐸𝐹”〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tgsas.p | . 2 ⊢ 𝑃 = (Base‘𝐺) | |
2 | tgsas.i | . 2 ⊢ 𝐼 = (Itv‘𝐺) | |
3 | tgsas.g | . 2 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
4 | eqid 2821 | . 2 ⊢ (hlG‘𝐺) = (hlG‘𝐺) | |
5 | tgsas.a | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
6 | tgsas.b | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
7 | tgsas.c | . 2 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
8 | tgsas.d | . 2 ⊢ (𝜑 → 𝐷 ∈ 𝑃) | |
9 | tgsas.e | . 2 ⊢ (𝜑 → 𝐸 ∈ 𝑃) | |
10 | tgsas.f | . 2 ⊢ (𝜑 → 𝐹 ∈ 𝑃) | |
11 | tgsss.4 | . 2 ⊢ (𝜑 → 𝐴 ≠ 𝐵) | |
12 | tgsss.5 | . 2 ⊢ (𝜑 → 𝐵 ≠ 𝐶) | |
13 | tgsas.m | . . 3 ⊢ − = (dist‘𝐺) | |
14 | eqid 2821 | . . 3 ⊢ (cgrG‘𝐺) = (cgrG‘𝐺) | |
15 | tgsss.1 | . . 3 ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐷 − 𝐸)) | |
16 | tgsss.2 | . . 3 ⊢ (𝜑 → (𝐵 − 𝐶) = (𝐸 − 𝐹)) | |
17 | tgsss.3 | . . 3 ⊢ (𝜑 → (𝐶 − 𝐴) = (𝐹 − 𝐷)) | |
18 | 1, 13, 14, 3, 5, 6, 7, 8, 9, 10, 15, 16, 17 | trgcgr 26296 | . 2 ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝐷𝐸𝐹”〉) |
19 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 18 | cgrcgra 26601 | 1 ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐸𝐹”〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 class class class wbr 5059 ‘cfv 6350 (class class class)co 7150 〈“cs3 14198 Basecbs 16477 distcds 16568 TarskiGcstrkg 26210 Itvcitv 26216 cgrGccgrg 26290 hlGchlg 26380 cgrAccgra 26587 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-map 8402 df-pm 8403 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 df-fzo 13028 df-hash 13685 df-word 13856 df-concat 13917 df-s1 13944 df-s2 14204 df-s3 14205 df-trkgc 26228 df-trkgcb 26230 df-trkg 26233 df-cgrg 26291 df-hlg 26381 df-cgra 26588 |
This theorem is referenced by: tgsss2 26641 tgsss3 26642 dfcgrg2 26643 |
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