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Mirrors > Home > MPE Home > Th. List > lmicinv | Structured version Visualization version GIF version |
Description: The mirroring line is an invariant. (Contributed by Thierry Arnoux, 8-Aug-2020.) |
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) |
lmicinv.1 | ⊢ (𝜑 → 𝐴 ∈ 𝐷) |
Ref | Expression |
---|---|
lmicinv | ⊢ (𝜑 → (𝑀‘𝐴) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmicinv.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝐷) | |
2 | ismid.p | . . 3 ⊢ 𝑃 = (Base‘𝐺) | |
3 | ismid.d | . . 3 ⊢ − = (dist‘𝐺) | |
4 | ismid.i | . . 3 ⊢ 𝐼 = (Itv‘𝐺) | |
5 | ismid.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
6 | ismid.1 | . . 3 ⊢ (𝜑 → 𝐺DimTarskiG≥2) | |
7 | lmif.m | . . 3 ⊢ 𝑀 = ((lInvG‘𝐺)‘𝐷) | |
8 | lmif.l | . . 3 ⊢ 𝐿 = (LineG‘𝐺) | |
9 | lmif.d | . . 3 ⊢ (𝜑 → 𝐷 ∈ ran 𝐿) | |
10 | lmicl.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
11 | 2, 3, 4, 5, 6, 7, 8, 9, 10 | lmiinv 26140 | . 2 ⊢ (𝜑 → ((𝑀‘𝐴) = 𝐴 ↔ 𝐴 ∈ 𝐷)) |
12 | 1, 11 | mpbird 249 | 1 ⊢ (𝜑 → (𝑀‘𝐴) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2106 class class class wbr 4886 ran crn 5356 ‘cfv 6135 2c2 11430 Basecbs 16255 distcds 16347 TarskiGcstrkg 25781 DimTarskiG≥cstrkgld 25785 Itvcitv 25787 LineGclng 25788 lInvGclmi 26121 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-er 8026 df-map 8142 df-pm 8143 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-card 9098 df-cda 9325 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-3 11439 df-n0 11643 df-xnn0 11715 df-z 11729 df-uz 11993 df-fz 12644 df-fzo 12785 df-hash 13436 df-word 13600 df-concat 13661 df-s1 13686 df-s2 13999 df-s3 14000 df-trkgc 25799 df-trkgb 25800 df-trkgcb 25801 df-trkgld 25803 df-trkg 25804 df-cgrg 25862 df-leg 25934 df-mir 26004 df-rag 26045 df-perpg 26047 df-mid 26122 df-lmi 26123 |
This theorem is referenced by: trgcopyeulem 26153 |
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