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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 2728 | . . . 4 β’ (LineGβπΊ) = (LineGβπΊ) | |
10 | eqid 2728 | . . . 4 β’ (pInvGβπΊ) = (pInvGβπΊ) | |
11 | hypcgr.e | . . . . 5 β’ (π β πΈ β π) | |
12 | 1, 2, 3, 4, 5, 7, 11 | midcl 28594 | . . . 4 β’ (π β (π΅(midGβπΊ)πΈ) β π) |
13 | eqid 2728 | . . . 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 28478 | . . 3 β’ (π β (((pInvGβπΊ)β(π΅(midGβπΊ)πΈ))βπ·) β π) |
16 | 1, 2, 3, 9, 10, 4, 12, 13, 11 | mircl 28478 | . . 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 28478 | . . 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 28518 | . . 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 28487 | . . . 4 β’ (π β ((((pInvGβπΊ)β(π΅(midGβπΊ)πΈ))βπ·) β (((pInvGβπΊ)β(π΅(midGβπΊ)πΈ))βπΈ)) = (π· β πΈ)) |
24 | 22, 23 | eqtr4d 2771 | . . 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 28487 | . . . 4 β’ (π β ((((pInvGβπΊ)β(π΅(midGβπΊ)πΈ))βπΈ) β (((pInvGβπΊ)β(π΅(midGβπΊ)πΈ))βπΉ)) = (πΈ β πΉ)) |
27 | 25, 26 | eqtr4d 2771 | . . 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 28599 | . . . 4 β’ (π β (πΈ(midGβπΊ)π΅) = (π΅(midGβπΊ)πΈ)) |
29 | 1, 2, 3, 4, 5, 11, 7, 10, 12 | ismidb 28595 | . . . 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 2728 | . . 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 28617 | . 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 28487 | . 2 β’ (π β ((((pInvGβπΊ)β(π΅(midGβπΊ)πΈ))βπ·) β (((pInvGβπΊ)β(π΅(midGβπΊ)πΈ))βπΉ)) = (π· β πΉ)) |
34 | 32, 33 | eqtrd 2768 | 1 β’ (π β (π΄ β πΆ) = (π· β πΉ)) |
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 2c2 12298 β¨βcs3 14826 Basecbs 17180 distcds 17242 TarskiGcstrkg 28244 DimTarskiGβ₯cstrkgld 28248 Itvcitv 28250 LineGclng 28251 pInvGcmir 28469 βGcrag 28510 midGcmid 28589 lInvGclmi 28590 |
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-rmo 3373 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-trkgld 28269 df-trkg 28270 df-cgrg 28328 df-ismt 28350 df-leg 28400 df-mir 28470 df-rag 28511 df-perpg 28513 df-mid 28591 df-lmi 28592 |
This theorem is referenced by: trgcopy 28621 |
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