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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 | ⊢ 𝑃 = (Base‘𝐺) |
ismid.d | ⊢ − = (dist‘𝐺) |
ismid.i | ⊢ 𝐼 = (Itv‘𝐺) |
ismid.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
ismid.1 | ⊢ (𝜑 → 𝐺DimTarskiG≥2) |
lmif.m | ⊢ 𝑀 = ((lInvG‘𝐺)‘𝐷) |
lmif.l | ⊢ 𝐿 = (LineG‘𝐺) |
lmif.d | ⊢ (𝜑 → 𝐷 ∈ ran 𝐿) |
lmicl.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
Ref | Expression |
---|---|
lmilmi | ⊢ (𝜑 → (𝑀‘(𝑀‘𝐴)) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ismid.p | . 2 ⊢ 𝑃 = (Base‘𝐺) | |
2 | ismid.d | . 2 ⊢ − = (dist‘𝐺) | |
3 | ismid.i | . 2 ⊢ 𝐼 = (Itv‘𝐺) | |
4 | ismid.g | . 2 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
5 | ismid.1 | . 2 ⊢ (𝜑 → 𝐺DimTarskiG≥2) | |
6 | lmif.m | . 2 ⊢ 𝑀 = ((lInvG‘𝐺)‘𝐷) | |
7 | lmif.l | . 2 ⊢ 𝐿 = (LineG‘𝐺) | |
8 | lmif.d | . 2 ⊢ (𝜑 → 𝐷 ∈ ran 𝐿) | |
9 | lmicl.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
10 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | lmicl 28808 | . 2 ⊢ (𝜑 → (𝑀‘𝐴) ∈ 𝑃) |
11 | eqidd 2735 | . 2 ⊢ (𝜑 → (𝑀‘𝐴) = (𝑀‘𝐴)) | |
12 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 | lmicom 28810 | 1 ⊢ (𝜑 → (𝑀‘(𝑀‘𝐴)) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2105 class class class wbr 5147 ran crn 5689 ‘cfv 6562 2c2 12318 Basecbs 17244 distcds 17306 TarskiGcstrkg 28449 DimTarskiG≥cstrkgld 28453 Itvcitv 28455 LineGclng 28456 lInvGclmi 28795 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-oadd 8508 df-er 8743 df-map 8866 df-pm 8867 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-dju 9938 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-2 12326 df-3 12327 df-n0 12524 df-xnn0 12597 df-z 12611 df-uz 12876 df-fz 13544 df-fzo 13691 df-hash 14366 df-word 14549 df-concat 14605 df-s1 14630 df-s2 14883 df-s3 14884 df-trkgc 28470 df-trkgb 28471 df-trkgcb 28472 df-trkgld 28474 df-trkg 28475 df-cgrg 28533 df-leg 28605 df-mir 28675 df-rag 28716 df-perpg 28718 df-mid 28796 df-lmi 28797 |
This theorem is referenced by: lmireu 28812 lmif1o 28817 lmiopp 28824 |
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