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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 26701 | . 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 26699 | . . . 4 ⊢ (𝜑 → ∃!𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))) |
15 | oveq2 7199 | . . . . . . 7 ⊢ (𝑧 = 𝐶 → (𝐴 − 𝑧) = (𝐴 − 𝐶)) | |
16 | 15 | eqeq1d 2738 | . . . . . 6 ⊢ (𝑧 = 𝐶 → ((𝐴 − 𝑧) = (𝐴 − 𝐵) ↔ (𝐴 − 𝐶) = (𝐴 − 𝐵))) |
17 | oveq1 7198 | . . . . . . 7 ⊢ (𝑧 = 𝐶 → (𝑧𝐼𝐵) = (𝐶𝐼𝐵)) | |
18 | 17 | eleq2d 2816 | . . . . . 6 ⊢ (𝑧 = 𝐶 → (𝐴 ∈ (𝑧𝐼𝐵) ↔ 𝐴 ∈ (𝐶𝐼𝐵))) |
19 | 16, 18 | anbi12d 634 | . . . . 5 ⊢ (𝑧 = 𝐶 → (((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵)) ↔ ((𝐴 − 𝐶) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝐶𝐼𝐵)))) |
20 | 19 | riota2 7174 | . . . 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 587 | . . 3 ⊢ (𝜑 → (((𝐴 − 𝐶) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝐶𝐼𝐵)) ↔ (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))) = 𝐶)) |
22 | 11, 12, 21 | mpbi2and 712 | . 2 ⊢ (𝜑 → (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))) = 𝐶) |
23 | 10, 22 | eqtr2d 2772 | 1 ⊢ (𝜑 → 𝐶 = (𝑀‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ∃!wreu 3053 ‘cfv 6358 ℩crio 7147 (class class class)co 7191 Basecbs 16666 distcds 16758 TarskiGcstrkg 26475 Itvcitv 26481 LineGclng 26482 pInvGcmir 26697 |
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 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
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 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-trkgc 26493 df-trkgb 26494 df-trkgcb 26495 df-trkg 26498 df-mir 26698 |
This theorem is referenced by: mirmir 26707 mireq 26710 mirinv 26711 miriso 26715 mirmir2 26719 mirauto 26729 colmid 26733 krippenlem 26735 midexlem 26737 mideulem2 26779 opphllem 26780 midcom 26827 trgcopyeulem 26850 |
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