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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 2740 | . . 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 27171 | . . . 4 ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
14 | 1, 3, 9, 5, 5, 10, 4, 13 | hlid 26968 | . . 3 ⊢ (𝜑 → 𝐴((hlG‘𝐺)‘𝐵)𝐴) |
15 | 1, 3, 9, 4, 5, 10, 6, 7, 11, 8, 12 | cgrane2 27172 | . . . . 5 ⊢ (𝜑 → 𝐵 ≠ 𝐶) |
16 | 15 | necomd 3001 | . . . 4 ⊢ (𝜑 → 𝐶 ≠ 𝐵) |
17 | 1, 3, 9, 6, 5, 10, 4, 16 | hlid 26968 | . . 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 26839 | . . 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 27177 | . 2 ⊢ (𝜑 → (𝐴 − 𝐶) = (𝐷 − 𝐹)) |
22 | 1, 2, 3, 4, 5, 6, 7, 8, 21 | tgcgrcomlr 26839 | 1 ⊢ (𝜑 → (𝐶 − 𝐴) = (𝐹 − 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2110 class class class wbr 5079 ‘cfv 6432 (class class class)co 7271 〈“cs3 14553 Basecbs 16910 distcds 16969 TarskiGcstrkg 26786 Itvcitv 26792 hlGchlg 26959 cgrAccgra 27166 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-om 7707 df-1st 7824 df-2nd 7825 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-1o 8288 df-oadd 8292 df-er 8481 df-map 8600 df-pm 8601 df-en 8717 df-dom 8718 df-sdom 8719 df-fin 8720 df-dju 9660 df-card 9698 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-n0 12234 df-xnn0 12306 df-z 12320 df-uz 12582 df-fz 13239 df-fzo 13382 df-hash 14043 df-word 14216 df-concat 14272 df-s1 14299 df-s2 14559 df-s3 14560 df-trkgc 26807 df-trkgb 26808 df-trkgcb 26809 df-trkg 26812 df-cgrg 26870 df-leg 26942 df-hlg 26960 df-cgra 27167 |
This theorem is referenced by: tgsas 27214 tgsas2 27215 tgsas3 27216 |
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