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| Mirrors > Home > MPE Home > Th. List > mircinv | Structured version Visualization version GIF version | ||
| Description: The center point is invariant of a point inversion. (Contributed by Thierry Arnoux, 25-Aug-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) |
| Ref | Expression |
|---|---|
| mircinv | ⊢ (𝜑 → (𝑀‘𝐴) = 𝐴) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqid 2737 | . 2 ⊢ 𝐴 = 𝐴 | |
| 2 | mirval.p | . . 3 ⊢ 𝑃 = (Base‘𝐺) | |
| 3 | mirval.d | . . 3 ⊢ − = (dist‘𝐺) | |
| 4 | mirval.i | . . 3 ⊢ 𝐼 = (Itv‘𝐺) | |
| 5 | mirval.l | . . 3 ⊢ 𝐿 = (LineG‘𝐺) | |
| 6 | mirval.s | . . 3 ⊢ 𝑆 = (pInvG‘𝐺) | |
| 7 | mirval.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
| 8 | mirval.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
| 9 | mirfv.m | . . 3 ⊢ 𝑀 = (𝑆‘𝐴) | |
| 10 | 2, 3, 4, 5, 6, 7, 8, 9, 8 | mirinv 28674 | . 2 ⊢ (𝜑 → ((𝑀‘𝐴) = 𝐴 ↔ 𝐴 = 𝐴)) |
| 11 | 1, 10 | mpbiri 258 | 1 ⊢ (𝜑 → (𝑀‘𝐴) = 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 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: mirln 28684 mirconn 28686 mirbtwnhl 28688 midexlem 28700 ragtrivb 28710 colperpexlem1 28738 colperpexlem3 28740 midex 28745 lmieu 28792 |
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