![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 | β’ π = (BaseβπΊ) |
tgsas.m | β’ β = (distβπΊ) |
tgsas.i | β’ πΌ = (ItvβπΊ) |
tgsas.g | β’ (π β πΊ β TarskiG) |
tgsas.a | β’ (π β π΄ β π) |
tgsas.b | β’ (π β π΅ β π) |
tgsas.c | β’ (π β πΆ β π) |
tgsas.d | β’ (π β π· β π) |
tgsas.e | β’ (π β πΈ β π) |
tgsas.f | β’ (π β πΉ β π) |
tgsas.1 | β’ (π β (π΄ β π΅) = (π· β πΈ)) |
tgsas.2 | β’ (π β β¨βπ΄π΅πΆββ©(cgrAβπΊ)β¨βπ·πΈπΉββ©) |
tgsas.3 | β’ (π β (π΅ β πΆ) = (πΈ β πΉ)) |
Ref | Expression |
---|---|
tgsas1 | β’ (π β (πΆ β π΄) = (πΉ β π·)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tgsas.p | . 2 β’ π = (BaseβπΊ) | |
2 | tgsas.m | . 2 β’ β = (distβπΊ) | |
3 | tgsas.i | . 2 β’ πΌ = (ItvβπΊ) | |
4 | tgsas.g | . 2 β’ (π β πΊ β TarskiG) | |
5 | tgsas.a | . 2 β’ (π β π΄ β π) | |
6 | tgsas.c | . 2 β’ (π β πΆ β π) | |
7 | tgsas.d | . 2 β’ (π β π· β π) | |
8 | tgsas.f | . 2 β’ (π β πΉ β π) | |
9 | eqid 2728 | . . 3 β’ (hlGβπΊ) = (hlGβπΊ) | |
10 | tgsas.b | . . 3 β’ (π β π΅ β π) | |
11 | tgsas.e | . . 3 β’ (π β πΈ β π) | |
12 | tgsas.2 | . . 3 β’ (π β β¨βπ΄π΅πΆββ©(cgrAβπΊ)β¨βπ·πΈπΉββ©) | |
13 | 1, 3, 9, 4, 5, 10, 6, 7, 11, 8, 12 | cgrane1 28629 | . . . 4 β’ (π β π΄ β π΅) |
14 | 1, 3, 9, 5, 5, 10, 4, 13 | hlid 28426 | . . 3 β’ (π β π΄((hlGβπΊ)βπ΅)π΄) |
15 | 1, 3, 9, 4, 5, 10, 6, 7, 11, 8, 12 | cgrane2 28630 | . . . . 5 β’ (π β π΅ β πΆ) |
16 | 15 | necomd 2993 | . . . 4 β’ (π β πΆ β π΅) |
17 | 1, 3, 9, 6, 5, 10, 4, 16 | hlid 28426 | . . 3 β’ (π β πΆ((hlGβπΊ)βπ΅)πΆ) |
18 | tgsas.1 | . . . 4 β’ (π β (π΄ β π΅) = (π· β πΈ)) | |
19 | 1, 2, 3, 4, 5, 10, 7, 11, 18 | tgcgrcomlr 28297 | . . 3 β’ (π β (π΅ β π΄) = (πΈ β π·)) |
20 | tgsas.3 | . . 3 β’ (π β (π΅ β πΆ) = (πΈ β πΉ)) | |
21 | 1, 3, 9, 4, 5, 10, 6, 7, 11, 8, 12, 5, 2, 6, 14, 17, 19, 20 | cgracgr 28635 | . 2 β’ (π β (π΄ β πΆ) = (π· β πΉ)) |
22 | 1, 2, 3, 4, 5, 6, 7, 8, 21 | tgcgrcomlr 28297 | 1 β’ (π β (πΆ β π΄) = (πΉ β π·)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1534 β wcel 2099 class class class wbr 5148 βcfv 6548 (class class class)co 7420 β¨βcs3 14826 Basecbs 17180 distcds 17242 TarskiGcstrkg 28244 Itvcitv 28250 hlGchlg 28417 cgrAccgra 28624 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-oadd 8491 df-er 8725 df-map 8847 df-pm 8848 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-dju 9925 df-card 9963 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-2 12306 df-3 12307 df-n0 12504 df-xnn0 12576 df-z 12590 df-uz 12854 df-fz 13518 df-fzo 13661 df-hash 14323 df-word 14498 df-concat 14554 df-s1 14579 df-s2 14832 df-s3 14833 df-trkgc 28265 df-trkgb 28266 df-trkgcb 28267 df-trkg 28270 df-cgrg 28328 df-leg 28400 df-hlg 28418 df-cgra 28625 |
This theorem is referenced by: tgsas 28672 tgsas2 28673 tgsas3 28674 |
Copyright terms: Public domain | W3C validator |