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Mirrors > Home > MPE Home > Th. List > Mathboxes > tgrpabl | Structured version Visualization version GIF version |
Description: The translation group is an Abelian group. Lemma G of [Crawley] p. 116. (Contributed by NM, 6-Jun-2013.) |
Ref | Expression |
---|---|
tgrpgrp.h | ⊢ 𝐻 = (LHyp‘𝐾) |
tgrpgrp.g | ⊢ 𝐺 = ((TGrp‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
tgrpabl | ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐺 ∈ Abel) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tgrpgrp.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | eqid 2825 | . . . 4 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊) | |
3 | tgrpgrp.g | . . . 4 ⊢ 𝐺 = ((TGrp‘𝐾)‘𝑊) | |
4 | eqid 2825 | . . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
5 | 1, 2, 3, 4 | tgrpbase 36820 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (Base‘𝐺) = ((LTrn‘𝐾)‘𝑊)) |
6 | 5 | eqcomd 2831 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ((LTrn‘𝐾)‘𝑊) = (Base‘𝐺)) |
7 | eqidd 2826 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (+g‘𝐺) = (+g‘𝐺)) | |
8 | 1, 3 | tgrpgrp 36824 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐺 ∈ Grp) |
9 | 1, 2 | ltrncom 36812 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑓 ∘ 𝑔) = (𝑔 ∘ 𝑓)) |
10 | eqid 2825 | . . . . . 6 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
11 | 1, 2, 3, 10 | tgrpov 36822 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊))) → (𝑓(+g‘𝐺)𝑔) = (𝑓 ∘ 𝑔)) |
12 | 11 | 3expa 1151 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊))) → (𝑓(+g‘𝐺)𝑔) = (𝑓 ∘ 𝑔)) |
13 | 12 | 3impb 1147 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑓(+g‘𝐺)𝑔) = (𝑓 ∘ 𝑔)) |
14 | 1, 2, 3, 10 | tgrpov 36822 | . . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ (𝑔 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊))) → (𝑔(+g‘𝐺)𝑓) = (𝑔 ∘ 𝑓)) |
15 | 14 | 3expa 1151 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑔 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊))) → (𝑔(+g‘𝐺)𝑓) = (𝑔 ∘ 𝑓)) |
16 | 15 | 3impb 1147 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑔(+g‘𝐺)𝑓) = (𝑔 ∘ 𝑓)) |
17 | 16 | 3com23 1160 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑔(+g‘𝐺)𝑓) = (𝑔 ∘ 𝑓)) |
18 | 9, 13, 17 | 3eqtr4d 2871 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑓(+g‘𝐺)𝑔) = (𝑔(+g‘𝐺)𝑓)) |
19 | 6, 7, 8, 18 | isabld 18566 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐺 ∈ Abel) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∧ w3a 1111 = wceq 1656 ∈ wcel 2164 ∘ ccom 5350 ‘cfv 6127 (class class class)co 6910 Basecbs 16229 +gcplusg 16312 Abelcabl 18554 HLchlt 35424 LHypclh 36058 LTrncltrn 36175 TGrpctgrp 36816 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 ax-riotaBAD 35027 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-iin 4745 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-1st 7433 df-2nd 7434 df-undef 7669 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-1o 7831 df-oadd 7835 df-er 8014 df-map 8129 df-en 8229 df-dom 8230 df-sdom 8231 df-fin 8232 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-nn 11358 df-2 11421 df-n0 11626 df-z 11712 df-uz 11976 df-fz 12627 df-struct 16231 df-ndx 16232 df-slot 16233 df-base 16235 df-plusg 16325 df-0g 16462 df-proset 17288 df-poset 17306 df-plt 17318 df-lub 17334 df-glb 17335 df-join 17336 df-meet 17337 df-p0 17399 df-p1 17400 df-lat 17406 df-clat 17468 df-mgm 17602 df-sgrp 17644 df-mnd 17655 df-grp 17786 df-cmn 18555 df-abl 18556 df-oposet 35250 df-ol 35252 df-oml 35253 df-covers 35340 df-ats 35341 df-atl 35372 df-cvlat 35396 df-hlat 35425 df-llines 35572 df-lplanes 35573 df-lvols 35574 df-lines 35575 df-psubsp 35577 df-pmap 35578 df-padd 35870 df-lhyp 36062 df-laut 36063 df-ldil 36178 df-ltrn 36179 df-trl 36233 df-tgrp 36817 |
This theorem is referenced by: dvaabl 37098 |
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