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Mirrors > Home > MPE Home > Th. List > lmimot | Structured version Visualization version GIF version |
Description: Line mirroring is a motion of the geometric space. Theorem 10.11 of [Schwabhauser] p. 90. (Contributed by Thierry Arnoux, 15-Dec-2019.) |
Ref | Expression |
---|---|
ismid.p | β’ π = (BaseβπΊ) |
ismid.d | β’ β = (distβπΊ) |
ismid.i | β’ πΌ = (ItvβπΊ) |
ismid.g | β’ (π β πΊ β TarskiG) |
ismid.1 | β’ (π β πΊDimTarskiGβ₯2) |
lmif.m | β’ π = ((lInvGβπΊ)βπ·) |
lmif.l | β’ πΏ = (LineGβπΊ) |
lmif.d | β’ (π β π· β ran πΏ) |
Ref | Expression |
---|---|
lmimot | β’ (π β π β (πΊIsmtπΊ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ismid.p | . . 3 β’ π = (BaseβπΊ) | |
2 | ismid.d | . . 3 β’ β = (distβπΊ) | |
3 | ismid.i | . . 3 β’ πΌ = (ItvβπΊ) | |
4 | ismid.g | . . 3 β’ (π β πΊ β TarskiG) | |
5 | ismid.1 | . . 3 β’ (π β πΊDimTarskiGβ₯2) | |
6 | lmif.m | . . 3 β’ π = ((lInvGβπΊ)βπ·) | |
7 | lmif.l | . . 3 β’ πΏ = (LineGβπΊ) | |
8 | lmif.d | . . 3 β’ (π β π· β ran πΏ) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | lmif1o 28619 | . 2 β’ (π β π:πβ1-1-ontoβπ) |
10 | 4 | adantr 479 | . . . 4 β’ ((π β§ (π β π β§ π β π)) β πΊ β TarskiG) |
11 | 5 | adantr 479 | . . . 4 β’ ((π β§ (π β π β§ π β π)) β πΊDimTarskiGβ₯2) |
12 | 8 | adantr 479 | . . . 4 β’ ((π β§ (π β π β§ π β π)) β π· β ran πΏ) |
13 | simprl 769 | . . . 4 β’ ((π β§ (π β π β§ π β π)) β π β π) | |
14 | simprr 771 | . . . 4 β’ ((π β§ (π β π β§ π β π)) β π β π) | |
15 | 1, 2, 3, 10, 11, 6, 7, 12, 13, 14 | lmiiso 28621 | . . 3 β’ ((π β§ (π β π β§ π β π)) β ((πβπ) β (πβπ)) = (π β π)) |
16 | 15 | ralrimivva 3198 | . 2 β’ (π β βπ β π βπ β π ((πβπ) β (πβπ)) = (π β π)) |
17 | 1, 2 | ismot 28359 | . . 3 β’ (πΊ β TarskiG β (π β (πΊIsmtπΊ) β (π:πβ1-1-ontoβπ β§ βπ β π βπ β π ((πβπ) β (πβπ)) = (π β π)))) |
18 | 4, 17 | syl 17 | . 2 β’ (π β (π β (πΊIsmtπΊ) β (π:πβ1-1-ontoβπ β§ βπ β π βπ β π ((πβπ) β (πβπ)) = (π β π)))) |
19 | 9, 16, 18 | mpbir2and 711 | 1 β’ (π β π β (πΊIsmtπΊ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 394 = wceq 1533 β wcel 2098 βwral 3058 class class class wbr 5152 ran crn 5683 β1-1-ontoβwf1o 6552 βcfv 6553 (class class class)co 7426 2c2 12305 Basecbs 17187 distcds 17249 TarskiGcstrkg 28251 DimTarskiGβ₯cstrkgld 28255 Itvcitv 28257 LineGclng 28258 Ismtcismt 28356 lInvGclmi 28597 |
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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 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 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-oadd 8497 df-er 8731 df-map 8853 df-pm 8854 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-dju 9932 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-3 12314 df-n0 12511 df-xnn0 12583 df-z 12597 df-uz 12861 df-fz 13525 df-fzo 13668 df-hash 14330 df-word 14505 df-concat 14561 df-s1 14586 df-s2 14839 df-s3 14840 df-trkgc 28272 df-trkgb 28273 df-trkgcb 28274 df-trkgld 28276 df-trkg 28277 df-cgrg 28335 df-ismt 28357 df-leg 28407 df-mir 28477 df-rag 28518 df-perpg 28520 df-mid 28598 df-lmi 28599 |
This theorem is referenced by: hypcgrlem2 28624 |
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