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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 2727 | . 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 28463 | . 2 ⊢ (𝜑 → ((𝑀‘𝐴) = 𝐴 ↔ 𝐴 = 𝐴)) |
11 | 1, 10 | mpbiri 258 | 1 ⊢ (𝜑 → (𝑀‘𝐴) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ‘cfv 6542 Basecbs 17173 distcds 17235 TarskiGcstrkg 28224 Itvcitv 28230 LineGclng 28231 pInvGcmir 28449 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-trkgc 28245 df-trkgb 28246 df-trkgcb 28247 df-trkg 28250 df-mir 28450 |
This theorem is referenced by: mirln 28473 mirconn 28475 mirbtwnhl 28477 midexlem 28489 ragtrivb 28499 colperpexlem1 28527 colperpexlem3 28529 midex 28534 lmieu 28581 |
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