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Mirrors > Home > MPE Home > Th. List > ismir | 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) |
ismir.1 | ⊢ (𝜑 → 𝐶 ∈ 𝑃) |
ismir.2 | ⊢ (𝜑 → (𝐴 − 𝐶) = (𝐴 − 𝐵)) |
ismir.3 | ⊢ (𝜑 → 𝐴 ∈ (𝐶𝐼𝐵)) |
Ref | Expression |
---|---|
ismir | ⊢ (𝜑 → 𝐶 = (𝑀‘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 | mirfv.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
10 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | mirfv 28473 | . 2 ⊢ (𝜑 → (𝑀‘𝐵) = (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵)))) |
11 | ismir.2 | . . 3 ⊢ (𝜑 → (𝐴 − 𝐶) = (𝐴 − 𝐵)) | |
12 | ismir.3 | . . 3 ⊢ (𝜑 → 𝐴 ∈ (𝐶𝐼𝐵)) | |
13 | ismir.1 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
14 | 1, 2, 3, 6, 9, 7 | mirreu3 28471 | . . . 4 ⊢ (𝜑 → ∃!𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))) |
15 | oveq2 7428 | . . . . . . 7 ⊢ (𝑧 = 𝐶 → (𝐴 − 𝑧) = (𝐴 − 𝐶)) | |
16 | 15 | eqeq1d 2730 | . . . . . 6 ⊢ (𝑧 = 𝐶 → ((𝐴 − 𝑧) = (𝐴 − 𝐵) ↔ (𝐴 − 𝐶) = (𝐴 − 𝐵))) |
17 | oveq1 7427 | . . . . . . 7 ⊢ (𝑧 = 𝐶 → (𝑧𝐼𝐵) = (𝐶𝐼𝐵)) | |
18 | 17 | eleq2d 2815 | . . . . . 6 ⊢ (𝑧 = 𝐶 → (𝐴 ∈ (𝑧𝐼𝐵) ↔ 𝐴 ∈ (𝐶𝐼𝐵))) |
19 | 16, 18 | anbi12d 631 | . . . . 5 ⊢ (𝑧 = 𝐶 → (((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵)) ↔ ((𝐴 − 𝐶) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝐶𝐼𝐵)))) |
20 | 19 | riota2 7402 | . . . 4 ⊢ ((𝐶 ∈ 𝑃 ∧ ∃!𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))) → (((𝐴 − 𝐶) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝐶𝐼𝐵)) ↔ (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))) = 𝐶)) |
21 | 13, 14, 20 | syl2anc 583 | . . 3 ⊢ (𝜑 → (((𝐴 − 𝐶) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝐶𝐼𝐵)) ↔ (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))) = 𝐶)) |
22 | 11, 12, 21 | mpbi2and 711 | . 2 ⊢ (𝜑 → (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))) = 𝐶) |
23 | 10, 22 | eqtr2d 2769 | 1 ⊢ (𝜑 → 𝐶 = (𝑀‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∃!wreu 3371 ‘cfv 6548 ℩crio 7375 (class class class)co 7420 Basecbs 17180 distcds 17242 TarskiGcstrkg 28244 Itvcitv 28250 LineGclng 28251 pInvGcmir 28469 |
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 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pr 5429 |
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 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-trkgc 28265 df-trkgb 28266 df-trkgcb 28267 df-trkg 28270 df-mir 28470 |
This theorem is referenced by: mirmir 28479 mireq 28482 mirinv 28483 miriso 28487 mirmir2 28491 mirauto 28501 colmid 28505 krippenlem 28507 midexlem 28509 mideulem2 28551 opphllem 28552 midcom 28599 trgcopyeulem 28622 |
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