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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 2622 | . 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 25555 | . . 3 ⊢ (𝜑 → 𝑀:𝑃⟶𝑃) |
14 | 13, 6 | ffvelrnd 6360 | . 2 ⊢ (𝜑 → (𝑀‘𝑋) ∈ 𝑃) |
15 | 13, 7 | ffvelrnd 6360 | . 2 ⊢ (𝜑 → (𝑀‘𝑌) ∈ 𝑃) |
16 | 13, 8 | ffvelrnd 6360 | . 2 ⊢ (𝜑 → (𝑀‘𝑍) ∈ 𝑃) |
17 | 1, 2, 3, 9, 10, 5, 11, 12, 6, 7 | miriso 25565 | . . . 4 ⊢ (𝜑 → ((𝑀‘𝑋) − (𝑀‘𝑌)) = (𝑋 − 𝑌)) |
18 | 17 | eqcomd 2628 | . . 3 ⊢ (𝜑 → (𝑋 − 𝑌) = ((𝑀‘𝑋) − (𝑀‘𝑌))) |
19 | 1, 2, 3, 9, 10, 5, 11, 12, 7, 8 | miriso 25565 | . . . 4 ⊢ (𝜑 → ((𝑀‘𝑌) − (𝑀‘𝑍)) = (𝑌 − 𝑍)) |
20 | 19 | eqcomd 2628 | . . 3 ⊢ (𝜑 → (𝑌 − 𝑍) = ((𝑀‘𝑌) − (𝑀‘𝑍))) |
21 | 1, 2, 3, 9, 10, 5, 11, 12, 8, 6 | miriso 25565 | . . . 4 ⊢ (𝜑 → ((𝑀‘𝑍) − (𝑀‘𝑋)) = (𝑍 − 𝑋)) |
22 | 21 | eqcomd 2628 | . . 3 ⊢ (𝜑 → (𝑍 − 𝑋) = ((𝑀‘𝑍) − (𝑀‘𝑋))) |
23 | 1, 2, 4, 5, 6, 7, 8, 14, 15, 16, 18, 20, 22 | trgcgr 25411 | . 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 25425 | 1 ⊢ (𝜑 → (𝑀‘𝑌) ∈ ((𝑀‘𝑋)𝐼(𝑀‘𝑍))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 distcds 15950 TarskiGcstrkg 25329 Itvcitv 25335 LineGclng 25336 cgrGccgrg 25405 pInvGcmir 25547 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-xnn0 11364 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-hash 13118 df-word 13299 df-concat 13301 df-s1 13302 df-s2 13593 df-s3 13594 df-trkgc 25347 df-trkgb 25348 df-trkgcb 25349 df-trkg 25352 df-cgrg 25406 df-mir 25548 |
This theorem is referenced by: mirbtwnb 25567 mirmir2 25569 mirhl 25574 mirauto 25579 krippenlem 25585 colperpexlem1 25622 opphllem2 25640 |
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