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Mirrors > Home > MPE Home > Th. List > midcom | Structured version Visualization version GIF version |
Description: Commutativity rule for the midpoint. (Contributed by Thierry Arnoux, 2-Dec-2019.) |
Ref | Expression |
---|---|
ismid.p | ⊢ 𝑃 = (Base‘𝐺) |
ismid.d | ⊢ − = (dist‘𝐺) |
ismid.i | ⊢ 𝐼 = (Itv‘𝐺) |
ismid.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
ismid.1 | ⊢ (𝜑 → 𝐺DimTarskiG≥2) |
midcl.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
midcl.2 | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
Ref | Expression |
---|---|
midcom | ⊢ (𝜑 → (𝐴(midG‘𝐺)𝐵) = (𝐵(midG‘𝐺)𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ismid.p | . . 3 ⊢ 𝑃 = (Base‘𝐺) | |
2 | ismid.d | . . 3 ⊢ − = (dist‘𝐺) | |
3 | ismid.i | . . 3 ⊢ 𝐼 = (Itv‘𝐺) | |
4 | eqid 2736 | . . 3 ⊢ (LineG‘𝐺) = (LineG‘𝐺) | |
5 | eqid 2736 | . . 3 ⊢ (pInvG‘𝐺) = (pInvG‘𝐺) | |
6 | ismid.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
7 | ismid.1 | . . . 4 ⊢ (𝜑 → 𝐺DimTarskiG≥2) | |
8 | midcl.2 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
9 | midcl.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
10 | 1, 2, 3, 6, 7, 8, 9 | midcl 27183 | . . 3 ⊢ (𝜑 → (𝐵(midG‘𝐺)𝐴) ∈ 𝑃) |
11 | eqid 2736 | . . 3 ⊢ ((pInvG‘𝐺)‘(𝐵(midG‘𝐺)𝐴)) = ((pInvG‘𝐺)‘(𝐵(midG‘𝐺)𝐴)) | |
12 | eqidd 2737 | . . . 4 ⊢ (𝜑 → (𝐵(midG‘𝐺)𝐴) = (𝐵(midG‘𝐺)𝐴)) | |
13 | 1, 2, 3, 6, 7, 8, 9, 12 | midcgr 27186 | . . 3 ⊢ (𝜑 → ((𝐵(midG‘𝐺)𝐴) − 𝐵) = ((𝐵(midG‘𝐺)𝐴) − 𝐴)) |
14 | 1, 2, 3, 6, 7, 8, 9 | midbtwn 27185 | . . 3 ⊢ (𝜑 → (𝐵(midG‘𝐺)𝐴) ∈ (𝐵𝐼𝐴)) |
15 | 1, 2, 3, 4, 5, 6, 10, 11, 9, 8, 13, 14 | ismir 27065 | . 2 ⊢ (𝜑 → 𝐵 = (((pInvG‘𝐺)‘(𝐵(midG‘𝐺)𝐴))‘𝐴)) |
16 | 1, 2, 3, 6, 7, 9, 8, 5, 10 | ismidb 27184 | . 2 ⊢ (𝜑 → (𝐵 = (((pInvG‘𝐺)‘(𝐵(midG‘𝐺)𝐴))‘𝐴) ↔ (𝐴(midG‘𝐺)𝐵) = (𝐵(midG‘𝐺)𝐴))) |
17 | 15, 16 | mpbid 231 | 1 ⊢ (𝜑 → (𝐴(midG‘𝐺)𝐵) = (𝐵(midG‘𝐺)𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2104 class class class wbr 5081 ‘cfv 6458 (class class class)co 7307 2c2 12074 Basecbs 16957 distcds 17016 TarskiGcstrkg 26833 DimTarskiG≥cstrkgld 26837 Itvcitv 26839 LineGclng 26840 pInvGcmir 27058 midGcmid 27178 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-rep 5218 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7620 ax-cnex 10973 ax-resscn 10974 ax-1cn 10975 ax-icn 10976 ax-addcl 10977 ax-addrcl 10978 ax-mulcl 10979 ax-mulrcl 10980 ax-mulcom 10981 ax-addass 10982 ax-mulass 10983 ax-distr 10984 ax-i2m1 10985 ax-1ne0 10986 ax-1rid 10987 ax-rnegex 10988 ax-rrecex 10989 ax-cnre 10990 ax-pre-lttri 10991 ax-pre-lttrn 10992 ax-pre-ltadd 10993 ax-pre-mulgt0 10994 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3285 df-reu 3286 df-rab 3287 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-tp 4570 df-op 4572 df-uni 4845 df-int 4887 df-iun 4933 df-br 5082 df-opab 5144 df-mpt 5165 df-tr 5199 df-id 5500 df-eprel 5506 df-po 5514 df-so 5515 df-fr 5555 df-we 5557 df-xp 5606 df-rel 5607 df-cnv 5608 df-co 5609 df-dm 5610 df-rn 5611 df-res 5612 df-ima 5613 df-pred 6217 df-ord 6284 df-on 6285 df-lim 6286 df-suc 6287 df-iota 6410 df-fun 6460 df-fn 6461 df-f 6462 df-f1 6463 df-fo 6464 df-f1o 6465 df-fv 6466 df-riota 7264 df-ov 7310 df-oprab 7311 df-mpo 7312 df-om 7745 df-1st 7863 df-2nd 7864 df-frecs 8128 df-wrecs 8159 df-recs 8233 df-rdg 8272 df-1o 8328 df-oadd 8332 df-er 8529 df-map 8648 df-pm 8649 df-en 8765 df-dom 8766 df-sdom 8767 df-fin 8768 df-dju 9703 df-card 9741 df-pnf 11057 df-mnf 11058 df-xr 11059 df-ltxr 11060 df-le 11061 df-sub 11253 df-neg 11254 df-nn 12020 df-2 12082 df-3 12083 df-n0 12280 df-xnn0 12352 df-z 12366 df-uz 12629 df-fz 13286 df-fzo 13429 df-hash 14091 df-word 14263 df-concat 14319 df-s1 14346 df-s2 14606 df-s3 14607 df-trkgc 26854 df-trkgb 26855 df-trkgcb 26856 df-trkgld 26858 df-trkg 26859 df-cgrg 26917 df-leg 26989 df-mir 27059 df-rag 27100 df-perpg 27102 df-mid 27180 |
This theorem is referenced by: lmicom 27194 hypcgrlem1 27205 hypcgrlem2 27206 hypcgr 27207 |
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