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Mirrors > Home > MPE Home > Th. List > hypcgr | Structured version Visualization version GIF version |
Description: If the catheti of two right-angled triangles are congruent, so is their hypothenuse. Theorem 10.12 of [Schwabhauser] p. 91. (Contributed by Thierry Arnoux, 16-Dec-2019.) |
Ref | Expression |
---|---|
hypcgr.p | β’ π = (BaseβπΊ) |
hypcgr.m | β’ β = (distβπΊ) |
hypcgr.i | β’ πΌ = (ItvβπΊ) |
hypcgr.g | β’ (π β πΊ β TarskiG) |
hypcgr.h | β’ (π β πΊDimTarskiGβ₯2) |
hypcgr.a | β’ (π β π΄ β π) |
hypcgr.b | β’ (π β π΅ β π) |
hypcgr.c | β’ (π β πΆ β π) |
hypcgr.d | β’ (π β π· β π) |
hypcgr.e | β’ (π β πΈ β π) |
hypcgr.f | β’ (π β πΉ β π) |
hypcgr.1 | β’ (π β β¨βπ΄π΅πΆββ© β (βGβπΊ)) |
hypcgr.2 | β’ (π β β¨βπ·πΈπΉββ© β (βGβπΊ)) |
hypcgr.3 | β’ (π β (π΄ β π΅) = (π· β πΈ)) |
hypcgr.4 | β’ (π β (π΅ β πΆ) = (πΈ β πΉ)) |
Ref | Expression |
---|---|
hypcgr | β’ (π β (π΄ β πΆ) = (π· β πΉ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hypcgr.p | . . 3 β’ π = (BaseβπΊ) | |
2 | hypcgr.m | . . 3 β’ β = (distβπΊ) | |
3 | hypcgr.i | . . 3 β’ πΌ = (ItvβπΊ) | |
4 | hypcgr.g | . . 3 β’ (π β πΊ β TarskiG) | |
5 | hypcgr.h | . . 3 β’ (π β πΊDimTarskiGβ₯2) | |
6 | hypcgr.a | . . 3 β’ (π β π΄ β π) | |
7 | hypcgr.b | . . 3 β’ (π β π΅ β π) | |
8 | hypcgr.c | . . 3 β’ (π β πΆ β π) | |
9 | eqid 2726 | . . . 4 β’ (LineGβπΊ) = (LineGβπΊ) | |
10 | eqid 2726 | . . . 4 β’ (pInvGβπΊ) = (pInvGβπΊ) | |
11 | hypcgr.e | . . . . 5 β’ (π β πΈ β π) | |
12 | 1, 2, 3, 4, 5, 7, 11 | midcl 28532 | . . . 4 β’ (π β (π΅(midGβπΊ)πΈ) β π) |
13 | eqid 2726 | . . . 4 β’ ((pInvGβπΊ)β(π΅(midGβπΊ)πΈ)) = ((pInvGβπΊ)β(π΅(midGβπΊ)πΈ)) | |
14 | hypcgr.d | . . . 4 β’ (π β π· β π) | |
15 | 1, 2, 3, 9, 10, 4, 12, 13, 14 | mircl 28416 | . . 3 β’ (π β (((pInvGβπΊ)β(π΅(midGβπΊ)πΈ))βπ·) β π) |
16 | 1, 2, 3, 9, 10, 4, 12, 13, 11 | mircl 28416 | . . 3 β’ (π β (((pInvGβπΊ)β(π΅(midGβπΊ)πΈ))βπΈ) β π) |
17 | hypcgr.f | . . . 4 β’ (π β πΉ β π) | |
18 | 1, 2, 3, 9, 10, 4, 12, 13, 17 | mircl 28416 | . . 3 β’ (π β (((pInvGβπΊ)β(π΅(midGβπΊ)πΈ))βπΉ) β π) |
19 | hypcgr.1 | . . 3 β’ (π β β¨βπ΄π΅πΆββ© β (βGβπΊ)) | |
20 | hypcgr.2 | . . . 4 β’ (π β β¨βπ·πΈπΉββ© β (βGβπΊ)) | |
21 | 1, 2, 3, 9, 10, 4, 14, 11, 17, 20, 13, 12 | mirrag 28456 | . . 3 β’ (π β β¨β(((pInvGβπΊ)β(π΅(midGβπΊ)πΈ))βπ·)(((pInvGβπΊ)β(π΅(midGβπΊ)πΈ))βπΈ)(((pInvGβπΊ)β(π΅(midGβπΊ)πΈ))βπΉ)ββ© β (βGβπΊ)) |
22 | hypcgr.3 | . . . 4 β’ (π β (π΄ β π΅) = (π· β πΈ)) | |
23 | 1, 2, 3, 9, 10, 4, 12, 13, 14, 11 | miriso 28425 | . . . 4 β’ (π β ((((pInvGβπΊ)β(π΅(midGβπΊ)πΈ))βπ·) β (((pInvGβπΊ)β(π΅(midGβπΊ)πΈ))βπΈ)) = (π· β πΈ)) |
24 | 22, 23 | eqtr4d 2769 | . . 3 β’ (π β (π΄ β π΅) = ((((pInvGβπΊ)β(π΅(midGβπΊ)πΈ))βπ·) β (((pInvGβπΊ)β(π΅(midGβπΊ)πΈ))βπΈ))) |
25 | hypcgr.4 | . . . 4 β’ (π β (π΅ β πΆ) = (πΈ β πΉ)) | |
26 | 1, 2, 3, 9, 10, 4, 12, 13, 11, 17 | miriso 28425 | . . . 4 β’ (π β ((((pInvGβπΊ)β(π΅(midGβπΊ)πΈ))βπΈ) β (((pInvGβπΊ)β(π΅(midGβπΊ)πΈ))βπΉ)) = (πΈ β πΉ)) |
27 | 25, 26 | eqtr4d 2769 | . . 3 β’ (π β (π΅ β πΆ) = ((((pInvGβπΊ)β(π΅(midGβπΊ)πΈ))βπΈ) β (((pInvGβπΊ)β(π΅(midGβπΊ)πΈ))βπΉ))) |
28 | 1, 2, 3, 4, 5, 11, 7 | midcom 28537 | . . . 4 β’ (π β (πΈ(midGβπΊ)π΅) = (π΅(midGβπΊ)πΈ)) |
29 | 1, 2, 3, 4, 5, 11, 7, 10, 12 | ismidb 28533 | . . . 4 β’ (π β (π΅ = (((pInvGβπΊ)β(π΅(midGβπΊ)πΈ))βπΈ) β (πΈ(midGβπΊ)π΅) = (π΅(midGβπΊ)πΈ))) |
30 | 28, 29 | mpbird 257 | . . 3 β’ (π β π΅ = (((pInvGβπΊ)β(π΅(midGβπΊ)πΈ))βπΈ)) |
31 | eqid 2726 | . . 3 β’ ((lInvGβπΊ)β((πΆ(midGβπΊ)(((pInvGβπΊ)β(π΅(midGβπΊ)πΈ))βπΉ))(LineGβπΊ)π΅)) = ((lInvGβπΊ)β((πΆ(midGβπΊ)(((pInvGβπΊ)β(π΅(midGβπΊ)πΈ))βπΉ))(LineGβπΊ)π΅)) | |
32 | 1, 2, 3, 4, 5, 6, 7, 8, 15, 16, 18, 19, 21, 24, 27, 30, 31 | hypcgrlem2 28555 | . 2 β’ (π β (π΄ β πΆ) = ((((pInvGβπΊ)β(π΅(midGβπΊ)πΈ))βπ·) β (((pInvGβπΊ)β(π΅(midGβπΊ)πΈ))βπΉ))) |
33 | 1, 2, 3, 9, 10, 4, 12, 13, 14, 17 | miriso 28425 | . 2 β’ (π β ((((pInvGβπΊ)β(π΅(midGβπΊ)πΈ))βπ·) β (((pInvGβπΊ)β(π΅(midGβπΊ)πΈ))βπΉ)) = (π· β πΉ)) |
34 | 32, 33 | eqtrd 2766 | 1 β’ (π β (π΄ β πΆ) = (π· β πΉ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 class class class wbr 5141 βcfv 6536 (class class class)co 7404 2c2 12268 β¨βcs3 14797 Basecbs 17151 distcds 17213 TarskiGcstrkg 28182 DimTarskiGβ₯cstrkgld 28186 Itvcitv 28188 LineGclng 28189 pInvGcmir 28407 βGcrag 28448 midGcmid 28527 lInvGclmi 28528 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-oadd 8468 df-er 8702 df-map 8821 df-pm 8822 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-dju 9895 df-card 9933 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-2 12276 df-3 12277 df-n0 12474 df-xnn0 12546 df-z 12560 df-uz 12824 df-fz 13488 df-fzo 13631 df-hash 14294 df-word 14469 df-concat 14525 df-s1 14550 df-s2 14803 df-s3 14804 df-trkgc 28203 df-trkgb 28204 df-trkgcb 28205 df-trkgld 28207 df-trkg 28208 df-cgrg 28266 df-ismt 28288 df-leg 28338 df-mir 28408 df-rag 28449 df-perpg 28451 df-mid 28529 df-lmi 28530 |
This theorem is referenced by: trgcopy 28559 |
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