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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 28664 | . . . 4 ⊢ (𝜑 → (𝑀‘𝐵) = (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵)))) |
| 11 | 1, 2, 3, 6, 9, 7 | mirreu3 28662 | . . . . 5 ⊢ (𝜑 → ∃!𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))) |
| 12 | riotacl2 7404 | . . . . 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 2841 | . . 3 ⊢ (𝜑 → (𝑀‘𝐵) ∈ {𝑧 ∈ 𝑃 ∣ ((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))}) |
| 15 | oveq2 7439 | . . . . . 6 ⊢ (𝑧 = (𝑀‘𝐵) → (𝐴 − 𝑧) = (𝐴 − (𝑀‘𝐵))) | |
| 16 | 15 | eqeq1d 2739 | . . . . 5 ⊢ (𝑧 = (𝑀‘𝐵) → ((𝐴 − 𝑧) = (𝐴 − 𝐵) ↔ (𝐴 − (𝑀‘𝐵)) = (𝐴 − 𝐵))) |
| 17 | oveq1 7438 | . . . . . 6 ⊢ (𝑧 = (𝑀‘𝐵) → (𝑧𝐼𝐵) = ((𝑀‘𝐵)𝐼𝐵)) | |
| 18 | 17 | eleq2d 2827 | . . . . 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 3692 | . . 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 218 | . 2 ⊢ (𝜑 → ((𝑀‘𝐵) ∈ 𝑃 ∧ ((𝐴 − (𝑀‘𝐵)) = (𝐴 − 𝐵) ∧ 𝐴 ∈ ((𝑀‘𝐵)𝐼𝐵)))) |
| 22 | 21 | simprld 772 | 1 ⊢ (𝜑 → (𝐴 − (𝑀‘𝐵)) = (𝐴 − 𝐵)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∃!wreu 3378 {crab 3436 ‘cfv 6561 ℩crio 7387 (class class class)co 7431 Basecbs 17247 distcds 17306 TarskiGcstrkg 28435 Itvcitv 28441 LineGclng 28442 pInvGcmir 28660 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-trkgc 28456 df-trkgb 28457 df-trkgcb 28458 df-trkg 28461 df-mir 28661 |
| This theorem is referenced by: mirmir 28670 miriso 28678 mirmir2 28682 mircgrextend 28690 mirtrcgr 28691 mirauto 28692 miduniq 28693 krippenlem 28698 ragcol 28707 ragflat 28712 ragcgr 28715 footexALT 28726 footexlem2 28728 colperpexlem1 28738 colperpexlem3 28740 mideulem2 28742 opphllem 28743 midcgr 28788 lmiisolem 28804 |
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