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Mirrors > Home > MPE Home > Th. List > mircl | Structured version Visualization version GIF version |
Description: Closure of the point inversion function. (Contributed by Thierry Arnoux, 20-Oct-2019.) |
Ref | Expression |
---|---|
mirval.p | ⊢ 𝑃 = (Base‘𝐺) |
mirval.d | ⊢ − = (dist‘𝐺) |
mirval.i | ⊢ 𝐼 = (Itv‘𝐺) |
mirval.l | ⊢ 𝐿 = (LineG‘𝐺) |
mirval.s | ⊢ 𝑆 = (pInvG‘𝐺) |
mirval.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
mirval.a | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
mirfv.m | ⊢ 𝑀 = (𝑆‘𝐴) |
mircl.x | ⊢ (𝜑 → 𝑋 ∈ 𝑃) |
Ref | Expression |
---|---|
mircl | ⊢ (𝜑 → (𝑀‘𝑋) ∈ 𝑃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mirval.p | . . 3 ⊢ 𝑃 = (Base‘𝐺) | |
2 | mirval.d | . . 3 ⊢ − = (dist‘𝐺) | |
3 | mirval.i | . . 3 ⊢ 𝐼 = (Itv‘𝐺) | |
4 | mirval.l | . . 3 ⊢ 𝐿 = (LineG‘𝐺) | |
5 | mirval.s | . . 3 ⊢ 𝑆 = (pInvG‘𝐺) | |
6 | mirval.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
7 | mirval.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
8 | mirfv.m | . . 3 ⊢ 𝑀 = (𝑆‘𝐴) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | mirf 25971 | . 2 ⊢ (𝜑 → 𝑀:𝑃⟶𝑃) |
10 | mircl.x | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝑃) | |
11 | 9, 10 | ffvelrnd 6608 | 1 ⊢ (𝜑 → (𝑀‘𝑋) ∈ 𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1658 ∈ wcel 2166 ‘cfv 6122 Basecbs 16221 distcds 16313 TarskiGcstrkg 25741 Itvcitv 25747 LineGclng 25748 pInvGcmir 25963 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 ax-rep 4993 ax-sep 5004 ax-nul 5012 ax-pr 5126 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2604 df-eu 2639 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-ne 2999 df-ral 3121 df-rex 3122 df-reu 3123 df-rmo 3124 df-rab 3125 df-v 3415 df-sbc 3662 df-csb 3757 df-dif 3800 df-un 3802 df-in 3804 df-ss 3811 df-nul 4144 df-if 4306 df-pw 4379 df-sn 4397 df-pr 4399 df-op 4403 df-uni 4658 df-iun 4741 df-br 4873 df-opab 4935 df-mpt 4952 df-id 5249 df-xp 5347 df-rel 5348 df-cnv 5349 df-co 5350 df-dm 5351 df-rn 5352 df-res 5353 df-ima 5354 df-iota 6085 df-fun 6124 df-fn 6125 df-f 6126 df-f1 6127 df-fo 6128 df-f1o 6129 df-fv 6130 df-riota 6865 df-ov 6907 df-trkgc 25759 df-trkgb 25760 df-trkgcb 25761 df-trkg 25764 df-mir 25964 |
This theorem is referenced by: mirmir 25973 mirreu 25975 mireq 25976 miriso 25981 mirmir2 25985 mirln 25987 mirconn 25989 mirhl 25990 mirbtwnhl 25991 mirhl2 25992 mircgrextend 25993 mirtrcgr 25994 miduniq 25996 miduniq1 25997 miduniq2 25998 ragcom 26009 ragcol 26010 ragmir 26011 mirrag 26012 ragflat2 26014 ragflat 26015 ragcgr 26018 footex 26029 colperpexlem1 26038 colperpexlem3 26040 mideulem2 26042 opphllem 26043 opphllem2 26056 opphllem3 26057 opphllem4 26058 opphllem6 26060 opphl 26062 colhp 26078 mirmid 26091 lmieu 26092 lmimid 26102 lmiisolem 26104 hypcgrlem1 26107 hypcgrlem2 26108 hypcgr 26109 sacgr 26138 sacgrOLD 26139 |
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