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Mirrors > Home > MPE Home > Th. List > mirmir | Structured version Visualization version GIF version |
Description: The point inversion function is an involution. Theorem 7.7 of [Schwabhauser] p. 49. (Contributed by Thierry Arnoux, 3-Jun-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) |
mirmir.b | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
Ref | Expression |
---|---|
mirmir | ⊢ (𝜑 → (𝑀‘(𝑀‘𝐵)) = 𝐵) |
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 | mirmir.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
10 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | mircl 26752 | . . 3 ⊢ (𝜑 → (𝑀‘𝐵) ∈ 𝑃) |
11 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | mircgr 26748 | . . . 4 ⊢ (𝜑 → (𝐴 − (𝑀‘𝐵)) = (𝐴 − 𝐵)) |
12 | 11 | eqcomd 2743 | . . 3 ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐴 − (𝑀‘𝐵))) |
13 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | mirbtwn 26749 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ((𝑀‘𝐵)𝐼𝐵)) |
14 | 1, 2, 3, 6, 10, 7, 9, 13 | tgbtwncom 26579 | . . 3 ⊢ (𝜑 → 𝐴 ∈ (𝐵𝐼(𝑀‘𝐵))) |
15 | 1, 2, 3, 4, 5, 6, 7, 8, 10, 9, 12, 14 | ismir 26750 | . 2 ⊢ (𝜑 → 𝐵 = (𝑀‘(𝑀‘𝐵))) |
16 | 15 | eqcomd 2743 | 1 ⊢ (𝜑 → (𝑀‘(𝑀‘𝐵)) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2110 ‘cfv 6380 (class class class)co 7213 Basecbs 16760 distcds 16811 TarskiGcstrkg 26521 Itvcitv 26527 LineGclng 26528 pInvGcmir 26743 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pr 5322 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-trkgc 26539 df-trkgb 26540 df-trkgcb 26541 df-trkg 26544 df-mir 26744 |
This theorem is referenced by: mircom 26754 mirreu 26755 mireq 26756 mirne 26758 mirf1o 26760 mirbtwnb 26763 miduniq2 26778 ragcom 26789 ragmir 26791 colperpexlem1 26821 colperpexlem2 26822 opphllem2 26839 opphllem3 26840 opphllem4 26841 opphllem6 26843 opphl 26845 colhp 26861 sacgr 26922 |
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