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Mirrors > Home > MPE Home > Th. List > mirbtwni | Structured version Visualization version GIF version |
Description: Point inversion preserves betweenness, first half of Theorem 7.15 of [Schwabhauser] p. 51. (Contributed by Thierry Arnoux, 9-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) |
miriso.1 | ⊢ (𝜑 → 𝑋 ∈ 𝑃) |
miriso.2 | ⊢ (𝜑 → 𝑌 ∈ 𝑃) |
mirbtwni.z | ⊢ (𝜑 → 𝑍 ∈ 𝑃) |
mirbtwni.b | ⊢ (𝜑 → 𝑌 ∈ (𝑋𝐼𝑍)) |
Ref | Expression |
---|---|
mirbtwni | ⊢ (𝜑 → (𝑀‘𝑌) ∈ ((𝑀‘𝑋)𝐼(𝑀‘𝑍))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mirval.p | . 2 ⊢ 𝑃 = (Base‘𝐺) | |
2 | mirval.d | . 2 ⊢ − = (dist‘𝐺) | |
3 | mirval.i | . 2 ⊢ 𝐼 = (Itv‘𝐺) | |
4 | eqid 2736 | . 2 ⊢ (cgrG‘𝐺) = (cgrG‘𝐺) | |
5 | mirval.g | . 2 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
6 | miriso.1 | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝑃) | |
7 | miriso.2 | . 2 ⊢ (𝜑 → 𝑌 ∈ 𝑃) | |
8 | mirbtwni.z | . 2 ⊢ (𝜑 → 𝑍 ∈ 𝑃) | |
9 | mirval.l | . . . 4 ⊢ 𝐿 = (LineG‘𝐺) | |
10 | mirval.s | . . . 4 ⊢ 𝑆 = (pInvG‘𝐺) | |
11 | mirval.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
12 | mirfv.m | . . . 4 ⊢ 𝑀 = (𝑆‘𝐴) | |
13 | 1, 2, 3, 9, 10, 5, 11, 12 | mirf 26705 | . . 3 ⊢ (𝜑 → 𝑀:𝑃⟶𝑃) |
14 | 13, 6 | ffvelrnd 6883 | . 2 ⊢ (𝜑 → (𝑀‘𝑋) ∈ 𝑃) |
15 | 13, 7 | ffvelrnd 6883 | . 2 ⊢ (𝜑 → (𝑀‘𝑌) ∈ 𝑃) |
16 | 13, 8 | ffvelrnd 6883 | . 2 ⊢ (𝜑 → (𝑀‘𝑍) ∈ 𝑃) |
17 | 1, 2, 3, 9, 10, 5, 11, 12, 6, 7 | miriso 26715 | . . . 4 ⊢ (𝜑 → ((𝑀‘𝑋) − (𝑀‘𝑌)) = (𝑋 − 𝑌)) |
18 | 17 | eqcomd 2742 | . . 3 ⊢ (𝜑 → (𝑋 − 𝑌) = ((𝑀‘𝑋) − (𝑀‘𝑌))) |
19 | 1, 2, 3, 9, 10, 5, 11, 12, 7, 8 | miriso 26715 | . . . 4 ⊢ (𝜑 → ((𝑀‘𝑌) − (𝑀‘𝑍)) = (𝑌 − 𝑍)) |
20 | 19 | eqcomd 2742 | . . 3 ⊢ (𝜑 → (𝑌 − 𝑍) = ((𝑀‘𝑌) − (𝑀‘𝑍))) |
21 | 1, 2, 3, 9, 10, 5, 11, 12, 8, 6 | miriso 26715 | . . . 4 ⊢ (𝜑 → ((𝑀‘𝑍) − (𝑀‘𝑋)) = (𝑍 − 𝑋)) |
22 | 21 | eqcomd 2742 | . . 3 ⊢ (𝜑 → (𝑍 − 𝑋) = ((𝑀‘𝑍) − (𝑀‘𝑋))) |
23 | 1, 2, 4, 5, 6, 7, 8, 14, 15, 16, 18, 20, 22 | trgcgr 26561 | . 2 ⊢ (𝜑 → 〈“𝑋𝑌𝑍”〉(cgrG‘𝐺)〈“(𝑀‘𝑋)(𝑀‘𝑌)(𝑀‘𝑍)”〉) |
24 | mirbtwni.b | . 2 ⊢ (𝜑 → 𝑌 ∈ (𝑋𝐼𝑍)) | |
25 | 1, 2, 3, 4, 5, 6, 7, 8, 14, 15, 16, 23, 24 | tgbtwnxfr 26575 | 1 ⊢ (𝜑 → (𝑀‘𝑌) ∈ ((𝑀‘𝑋)𝐼(𝑀‘𝑍))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2112 ‘cfv 6358 (class class class)co 7191 Basecbs 16666 distcds 16758 TarskiGcstrkg 26475 Itvcitv 26481 LineGclng 26482 cgrGccgrg 26555 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-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 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-nel 3037 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-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 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-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 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-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-oadd 8184 df-er 8369 df-pm 8489 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-dju 9482 df-card 9520 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-2 11858 df-3 11859 df-n0 12056 df-xnn0 12128 df-z 12142 df-uz 12404 df-fz 13061 df-fzo 13204 df-hash 13862 df-word 14035 df-concat 14091 df-s1 14118 df-s2 14378 df-s3 14379 df-trkgc 26493 df-trkgb 26494 df-trkgcb 26495 df-trkg 26498 df-cgrg 26556 df-mir 26698 |
This theorem is referenced by: mirbtwnb 26717 mirmir2 26719 mirhl 26724 mirauto 26729 krippenlem 26735 colperpexlem1 26775 opphllem2 26793 |
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