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Mirrors > Home > MPE Home > Th. List > mircgr | 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 |
---|---|
mircgr | ⊢ (𝜑 → (𝐴 − (𝑀‘𝐵)) = (𝐴 − 𝐵)) |
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 26442 | . . . 4 ⊢ (𝜑 → (𝑀‘𝐵) = (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵)))) |
11 | 1, 2, 3, 6, 9, 7 | mirreu3 26440 | . . . . 5 ⊢ (𝜑 → ∃!𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))) |
12 | riotacl2 7130 | . . . . 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 2913 | . . 3 ⊢ (𝜑 → (𝑀‘𝐵) ∈ {𝑧 ∈ 𝑃 ∣ ((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))}) |
15 | oveq2 7164 | . . . . . 6 ⊢ (𝑧 = (𝑀‘𝐵) → (𝐴 − 𝑧) = (𝐴 − (𝑀‘𝐵))) | |
16 | 15 | eqeq1d 2823 | . . . . 5 ⊢ (𝑧 = (𝑀‘𝐵) → ((𝐴 − 𝑧) = (𝐴 − 𝐵) ↔ (𝐴 − (𝑀‘𝐵)) = (𝐴 − 𝐵))) |
17 | oveq1 7163 | . . . . . 6 ⊢ (𝑧 = (𝑀‘𝐵) → (𝑧𝐼𝐵) = ((𝑀‘𝐵)𝐼𝐵)) | |
18 | 17 | eleq2d 2898 | . . . . 5 ⊢ (𝑧 = (𝑀‘𝐵) → (𝐴 ∈ (𝑧𝐼𝐵) ↔ 𝐴 ∈ ((𝑀‘𝐵)𝐼𝐵))) |
19 | 16, 18 | anbi12d 632 | . . . 4 ⊢ (𝑧 = (𝑀‘𝐵) → (((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵)) ↔ ((𝐴 − (𝑀‘𝐵)) = (𝐴 − 𝐵) ∧ 𝐴 ∈ ((𝑀‘𝐵)𝐼𝐵)))) |
20 | 19 | elrab 3680 | . . 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 220 | . 2 ⊢ (𝜑 → ((𝑀‘𝐵) ∈ 𝑃 ∧ ((𝐴 − (𝑀‘𝐵)) = (𝐴 − 𝐵) ∧ 𝐴 ∈ ((𝑀‘𝐵)𝐼𝐵)))) |
22 | 21 | simprld 770 | 1 ⊢ (𝜑 → (𝐴 − (𝑀‘𝐵)) = (𝐴 − 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∃!wreu 3140 {crab 3142 ‘cfv 6355 ℩crio 7113 (class class class)co 7156 Basecbs 16483 distcds 16574 TarskiGcstrkg 26216 Itvcitv 26222 LineGclng 26223 pInvGcmir 26438 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-trkgc 26234 df-trkgb 26235 df-trkgcb 26236 df-trkg 26239 df-mir 26439 |
This theorem is referenced by: mirmir 26448 miriso 26456 mirmir2 26460 mircgrextend 26468 mirtrcgr 26469 mirauto 26470 miduniq 26471 krippenlem 26476 ragcol 26485 ragflat 26490 ragcgr 26493 footexALT 26504 footexlem2 26506 colperpexlem1 26516 colperpexlem3 26518 mideulem2 26520 opphllem 26521 midcgr 26566 lmiisolem 26582 |
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