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Mirrors > Home > MPE Home > Th. List > mircl | Structured version Visualization version GIF version |
Description: Closure of the point inversion function. (Contributed by Thierry Arnoux, 20-Oct-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) |
mircl.x | ⊢ (𝜑 → 𝑋 ∈ 𝑃) |
Ref | Expression |
---|---|
mircl | ⊢ (𝜑 → (𝑀‘𝑋) ∈ 𝑃) |
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 | 1, 2, 3, 4, 5, 6, 7, 8 | mirf 28579 | . 2 ⊢ (𝜑 → 𝑀:𝑃⟶𝑃) |
10 | mircl.x | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝑃) | |
11 | 9, 10 | ffvelcdmd 7098 | 1 ⊢ (𝜑 → (𝑀‘𝑋) ∈ 𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ‘cfv 6553 Basecbs 17208 distcds 17270 TarskiGcstrkg 28346 Itvcitv 28352 LineGclng 28353 pInvGcmir 28571 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pr 5432 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4325 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5579 df-xp 5687 df-rel 5688 df-cnv 5689 df-co 5690 df-dm 5691 df-rn 5692 df-res 5693 df-ima 5694 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7379 df-ov 7426 df-trkgc 28367 df-trkgb 28368 df-trkgcb 28369 df-trkg 28372 df-mir 28572 |
This theorem is referenced by: mirmir 28581 mirreu 28583 mireq 28584 miriso 28589 mirmir2 28593 mirln 28595 mirconn 28597 mirhl 28598 mirbtwnhl 28599 mirhl2 28600 mircgrextend 28601 mirtrcgr 28602 miduniq 28604 miduniq1 28605 miduniq2 28606 ragcom 28617 ragcol 28618 ragmir 28619 mirrag 28620 ragflat2 28622 ragflat 28623 ragcgr 28626 footexALT 28637 footexlem1 28638 footexlem2 28639 footex 28640 colperpexlem1 28649 colperpexlem3 28651 mideulem2 28653 opphllem 28654 opphllem2 28667 opphllem3 28668 opphllem4 28669 opphllem6 28671 opphl 28673 colhp 28689 mirmid 28702 lmieu 28703 lmimid 28713 lmiisolem 28715 hypcgrlem1 28718 hypcgrlem2 28719 hypcgr 28720 sacgr 28750 |
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