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Mirrors > Home > MPE Home > Th. List > tgsas1 | Structured version Visualization version GIF version |
Description: First congruence theorem: SAS (Side-Angle-Side): If two pairs of sides of two triangles are equal in length, and the included angles are equal in measurement, then third sides are equal in length. Theorem 11.49 of [Schwabhauser] p. 107. (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) |
tgsas.1 | ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐷 − 𝐸)) |
tgsas.2 | ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐸𝐹”〉) |
tgsas.3 | ⊢ (𝜑 → (𝐵 − 𝐶) = (𝐸 − 𝐹)) |
Ref | Expression |
---|---|
tgsas1 | ⊢ (𝜑 → (𝐶 − 𝐴) = (𝐹 − 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tgsas.p | . 2 ⊢ 𝑃 = (Base‘𝐺) | |
2 | tgsas.m | . 2 ⊢ − = (dist‘𝐺) | |
3 | tgsas.i | . 2 ⊢ 𝐼 = (Itv‘𝐺) | |
4 | tgsas.g | . 2 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
5 | tgsas.a | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
6 | tgsas.c | . 2 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
7 | tgsas.d | . 2 ⊢ (𝜑 → 𝐷 ∈ 𝑃) | |
8 | tgsas.f | . 2 ⊢ (𝜑 → 𝐹 ∈ 𝑃) | |
9 | eqid 2772 | . . 3 ⊢ (hlG‘𝐺) = (hlG‘𝐺) | |
10 | tgsas.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
11 | tgsas.e | . . 3 ⊢ (𝜑 → 𝐸 ∈ 𝑃) | |
12 | tgsas.2 | . . 3 ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐸𝐹”〉) | |
13 | 1, 3, 9, 4, 5, 10, 6, 7, 11, 8, 12 | cgrane1 26294 | . . . 4 ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
14 | 1, 3, 9, 5, 5, 10, 4, 13 | hlid 26091 | . . 3 ⊢ (𝜑 → 𝐴((hlG‘𝐺)‘𝐵)𝐴) |
15 | 1, 3, 9, 4, 5, 10, 6, 7, 11, 8, 12 | cgrane2 26295 | . . . . 5 ⊢ (𝜑 → 𝐵 ≠ 𝐶) |
16 | 15 | necomd 3016 | . . . 4 ⊢ (𝜑 → 𝐶 ≠ 𝐵) |
17 | 1, 3, 9, 6, 5, 10, 4, 16 | hlid 26091 | . . 3 ⊢ (𝜑 → 𝐶((hlG‘𝐺)‘𝐵)𝐶) |
18 | tgsas.1 | . . . 4 ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐷 − 𝐸)) | |
19 | 1, 2, 3, 4, 5, 10, 7, 11, 18 | tgcgrcomlr 25962 | . . 3 ⊢ (𝜑 → (𝐵 − 𝐴) = (𝐸 − 𝐷)) |
20 | tgsas.3 | . . 3 ⊢ (𝜑 → (𝐵 − 𝐶) = (𝐸 − 𝐹)) | |
21 | 1, 3, 9, 4, 5, 10, 6, 7, 11, 8, 12, 5, 2, 6, 14, 17, 19, 20 | cgracgr 26300 | . 2 ⊢ (𝜑 → (𝐴 − 𝐶) = (𝐷 − 𝐹)) |
22 | 1, 2, 3, 4, 5, 6, 7, 8, 21 | tgcgrcomlr 25962 | 1 ⊢ (𝜑 → (𝐶 − 𝐴) = (𝐹 − 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1507 ∈ wcel 2050 class class class wbr 4923 ‘cfv 6182 (class class class)co 6970 〈“cs3 14060 Basecbs 16333 distcds 16424 TarskiGcstrkg 25912 Itvcitv 25918 hlGchlg 26082 cgrAccgra 26289 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10385 ax-resscn 10386 ax-1cn 10387 ax-icn 10388 ax-addcl 10389 ax-addrcl 10390 ax-mulcl 10391 ax-mulrcl 10392 ax-mulcom 10393 ax-addass 10394 ax-mulass 10395 ax-distr 10396 ax-i2m1 10397 ax-1ne0 10398 ax-1rid 10399 ax-rnegex 10400 ax-rrecex 10401 ax-cnre 10402 ax-pre-lttri 10403 ax-pre-lttrn 10404 ax-pre-ltadd 10405 ax-pre-mulgt0 10406 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-pss 3839 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-int 4744 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5306 df-eprel 5311 df-po 5320 df-so 5321 df-fr 5360 df-we 5362 df-xp 5407 df-rel 5408 df-cnv 5409 df-co 5410 df-dm 5411 df-rn 5412 df-res 5413 df-ima 5414 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-om 7391 df-1st 7495 df-2nd 7496 df-wrecs 7744 df-recs 7806 df-rdg 7844 df-1o 7899 df-oadd 7903 df-er 8083 df-map 8202 df-pm 8203 df-en 8301 df-dom 8302 df-sdom 8303 df-fin 8304 df-dju 9118 df-card 9156 df-pnf 10470 df-mnf 10471 df-xr 10472 df-ltxr 10473 df-le 10474 df-sub 10666 df-neg 10667 df-nn 11434 df-2 11497 df-3 11498 df-n0 11702 df-xnn0 11774 df-z 11788 df-uz 12053 df-fz 12703 df-fzo 12844 df-hash 13500 df-word 13667 df-concat 13728 df-s1 13753 df-s2 14066 df-s3 14067 df-trkgc 25930 df-trkgb 25931 df-trkgcb 25932 df-trkg 25935 df-cgrg 25993 df-leg 26065 df-hlg 26083 df-cgra 26290 |
This theorem is referenced by: tgsas 26338 tgsas2 26339 tgsas3 26340 |
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