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Mirrors > Home > MPE Home > Th. List > mircinv | Structured version Visualization version GIF version |
Description: The center point is invariant of a point inversion. (Contributed by Thierry Arnoux, 25-Aug-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) |
Ref | Expression |
---|---|
mircinv | ⊢ (𝜑 → (𝑀‘𝐴) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2825 | . 2 ⊢ 𝐴 = 𝐴 | |
2 | mirval.p | . . 3 ⊢ 𝑃 = (Base‘𝐺) | |
3 | mirval.d | . . 3 ⊢ − = (dist‘𝐺) | |
4 | mirval.i | . . 3 ⊢ 𝐼 = (Itv‘𝐺) | |
5 | mirval.l | . . 3 ⊢ 𝐿 = (LineG‘𝐺) | |
6 | mirval.s | . . 3 ⊢ 𝑆 = (pInvG‘𝐺) | |
7 | mirval.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
8 | mirval.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
9 | mirfv.m | . . 3 ⊢ 𝑀 = (𝑆‘𝐴) | |
10 | 2, 3, 4, 5, 6, 7, 8, 9, 8 | mirinv 25985 | . 2 ⊢ (𝜑 → ((𝑀‘𝐴) = 𝐴 ↔ 𝐴 = 𝐴)) |
11 | 1, 10 | mpbiri 250 | 1 ⊢ (𝜑 → (𝑀‘𝐴) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1656 ∈ wcel 2164 ‘cfv 6127 Basecbs 16229 distcds 16321 TarskiGcstrkg 25749 Itvcitv 25755 LineGclng 25756 pInvGcmir 25971 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pr 5129 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-trkgc 25767 df-trkgb 25768 df-trkgcb 25769 df-trkg 25772 df-mir 25972 |
This theorem is referenced by: mirln 25995 mirconn 25997 mirbtwnhl 25999 mirhl2 26000 midexlem 26011 ragtrivb 26021 colperpexlem1 26046 colperpexlem3 26048 midex 26053 lmieu 26100 |
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