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Mirrors > Home > MPE Home > Th. List > mirbtwn | Structured version Visualization version GIF version |
Description: Property of the image by the point inversion function. Definition 7.5 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) |
mirfv.b | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
Ref | Expression |
---|---|
mirbtwn | ⊢ (𝜑 → 𝐴 ∈ ((𝑀‘𝐵)𝐼𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mirval.p | . . . . 5 ⊢ 𝑃 = (Base‘𝐺) | |
2 | mirval.d | . . . . 5 ⊢ − = (dist‘𝐺) | |
3 | mirval.i | . . . . 5 ⊢ 𝐼 = (Itv‘𝐺) | |
4 | mirval.l | . . . . 5 ⊢ 𝐿 = (LineG‘𝐺) | |
5 | mirval.s | . . . . 5 ⊢ 𝑆 = (pInvG‘𝐺) | |
6 | mirval.g | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
7 | mirval.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
8 | mirfv.m | . . . . 5 ⊢ 𝑀 = (𝑆‘𝐴) | |
9 | mirfv.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
10 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | mirfv 26369 | . . . 4 ⊢ (𝜑 → (𝑀‘𝐵) = (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵)))) |
11 | 1, 2, 3, 6, 9, 7 | mirreu3 26367 | . . . . 5 ⊢ (𝜑 → ∃!𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))) |
12 | riotacl2 7119 | . . . . 5 ⊢ (∃!𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵)) → (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))) ∈ {𝑧 ∈ 𝑃 ∣ ((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))}) | |
13 | 11, 12 | syl 17 | . . . 4 ⊢ (𝜑 → (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))) ∈ {𝑧 ∈ 𝑃 ∣ ((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))}) |
14 | 10, 13 | eqeltrd 2910 | . . 3 ⊢ (𝜑 → (𝑀‘𝐵) ∈ {𝑧 ∈ 𝑃 ∣ ((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))}) |
15 | oveq2 7153 | . . . . . 6 ⊢ (𝑧 = (𝑀‘𝐵) → (𝐴 − 𝑧) = (𝐴 − (𝑀‘𝐵))) | |
16 | 15 | eqeq1d 2820 | . . . . 5 ⊢ (𝑧 = (𝑀‘𝐵) → ((𝐴 − 𝑧) = (𝐴 − 𝐵) ↔ (𝐴 − (𝑀‘𝐵)) = (𝐴 − 𝐵))) |
17 | oveq1 7152 | . . . . . 6 ⊢ (𝑧 = (𝑀‘𝐵) → (𝑧𝐼𝐵) = ((𝑀‘𝐵)𝐼𝐵)) | |
18 | 17 | eleq2d 2895 | . . . . 5 ⊢ (𝑧 = (𝑀‘𝐵) → (𝐴 ∈ (𝑧𝐼𝐵) ↔ 𝐴 ∈ ((𝑀‘𝐵)𝐼𝐵))) |
19 | 16, 18 | anbi12d 630 | . . . 4 ⊢ (𝑧 = (𝑀‘𝐵) → (((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵)) ↔ ((𝐴 − (𝑀‘𝐵)) = (𝐴 − 𝐵) ∧ 𝐴 ∈ ((𝑀‘𝐵)𝐼𝐵)))) |
20 | 19 | elrab 3677 | . . 3 ⊢ ((𝑀‘𝐵) ∈ {𝑧 ∈ 𝑃 ∣ ((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))} ↔ ((𝑀‘𝐵) ∈ 𝑃 ∧ ((𝐴 − (𝑀‘𝐵)) = (𝐴 − 𝐵) ∧ 𝐴 ∈ ((𝑀‘𝐵)𝐼𝐵)))) |
21 | 14, 20 | sylib 219 | . 2 ⊢ (𝜑 → ((𝑀‘𝐵) ∈ 𝑃 ∧ ((𝐴 − (𝑀‘𝐵)) = (𝐴 − 𝐵) ∧ 𝐴 ∈ ((𝑀‘𝐵)𝐼𝐵)))) |
22 | 21 | simprrd 770 | 1 ⊢ (𝜑 → 𝐴 ∈ ((𝑀‘𝐵)𝐼𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∃!wreu 3137 {crab 3139 ‘cfv 6348 ℩crio 7102 (class class class)co 7145 Basecbs 16471 distcds 16562 TarskiGcstrkg 26143 Itvcitv 26149 LineGclng 26150 pInvGcmir 26365 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-trkgc 26161 df-trkgb 26162 df-trkgcb 26163 df-trkg 26166 df-mir 26366 |
This theorem is referenced by: mirmir 26375 mirinv 26379 miriso 26383 mirmir2 26387 mirln 26389 mirln2 26390 mirconn 26391 mirhl2 26394 mircgrextend 26395 mirtrcgr 26396 mirauto 26397 miduniq 26398 krippenlem 26403 ragflat 26417 ragcgr 26420 footexALT 26431 footexlem1 26432 footexlem2 26433 colperpexlem1 26443 colperpexlem3 26445 mideulem2 26447 opphllem 26448 opphllem1 26460 opphllem2 26461 opphllem4 26463 colhp 26483 midbtwn 26492 lmieu 26497 lmiisolem 26509 |
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