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Mirrors > Home > MPE Home > Th. List > lmilmi | Structured version Visualization version GIF version |
Description: Line mirroring is an involution. Theorem 10.5 of [Schwabhauser] p. 89. (Contributed by Thierry Arnoux, 11-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) |
lmicl.1 | β’ (π β π΄ β π) |
Ref | Expression |
---|---|
lmilmi | β’ (π β (πβ(πβπ΄)) = π΄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ismid.p | . 2 β’ π = (BaseβπΊ) | |
2 | ismid.d | . 2 β’ β = (distβπΊ) | |
3 | ismid.i | . 2 β’ πΌ = (ItvβπΊ) | |
4 | ismid.g | . 2 β’ (π β πΊ β TarskiG) | |
5 | ismid.1 | . 2 β’ (π β πΊDimTarskiGβ₯2) | |
6 | lmif.m | . 2 β’ π = ((lInvGβπΊ)βπ·) | |
7 | lmif.l | . 2 β’ πΏ = (LineGβπΊ) | |
8 | lmif.d | . 2 β’ (π β π· β ran πΏ) | |
9 | lmicl.1 | . 2 β’ (π β π΄ β π) | |
10 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | lmicl 28034 | . 2 β’ (π β (πβπ΄) β π) |
11 | eqidd 2733 | . 2 β’ (π β (πβπ΄) = (πβπ΄)) | |
12 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 | lmicom 28036 | 1 β’ (π β (πβ(πβπ΄)) = π΄) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1541 β wcel 2106 class class class wbr 5148 ran crn 5677 βcfv 6543 2c2 12266 Basecbs 17143 distcds 17205 TarskiGcstrkg 27675 DimTarskiGβ₯cstrkgld 27679 Itvcitv 27681 LineGclng 27682 lInvGclmi 28021 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 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 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-1st 7974 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-oadd 8469 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 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-3 12275 df-n0 12472 df-xnn0 12544 df-z 12558 df-uz 12822 df-fz 13484 df-fzo 13627 df-hash 14290 df-word 14464 df-concat 14520 df-s1 14545 df-s2 14798 df-s3 14799 df-trkgc 27696 df-trkgb 27697 df-trkgcb 27698 df-trkgld 27700 df-trkg 27701 df-cgrg 27759 df-leg 27831 df-mir 27901 df-rag 27942 df-perpg 27944 df-mid 28022 df-lmi 28023 |
This theorem is referenced by: lmireu 28038 lmif1o 28043 lmiopp 28050 |
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