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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 2821 | . . . 4 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊) | |
3 | tgrpgrp.g | . . . 4 ⊢ 𝐺 = ((TGrp‘𝐾)‘𝑊) | |
4 | eqid 2821 | . . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
5 | 1, 2, 3, 4 | tgrpbase 37876 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (Base‘𝐺) = ((LTrn‘𝐾)‘𝑊)) |
6 | 5 | eqcomd 2827 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ((LTrn‘𝐾)‘𝑊) = (Base‘𝐺)) |
7 | eqidd 2822 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (+g‘𝐺) = (+g‘𝐺)) | |
8 | 1, 3 | tgrpgrp 37880 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐺 ∈ Grp) |
9 | 1, 2 | ltrncom 37868 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑓 ∘ 𝑔) = (𝑔 ∘ 𝑓)) |
10 | eqid 2821 | . . . . . 6 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
11 | 1, 2, 3, 10 | tgrpov 37878 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊))) → (𝑓(+g‘𝐺)𝑔) = (𝑓 ∘ 𝑔)) |
12 | 11 | 3expa 1114 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊))) → (𝑓(+g‘𝐺)𝑔) = (𝑓 ∘ 𝑔)) |
13 | 12 | 3impb 1111 | . . 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 37878 | . . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ (𝑔 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊))) → (𝑔(+g‘𝐺)𝑓) = (𝑔 ∘ 𝑓)) |
15 | 14 | 3expa 1114 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑔 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊))) → (𝑔(+g‘𝐺)𝑓) = (𝑔 ∘ 𝑓)) |
16 | 15 | 3impb 1111 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑔(+g‘𝐺)𝑓) = (𝑔 ∘ 𝑓)) |
17 | 16 | 3com23 1122 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑔(+g‘𝐺)𝑓) = (𝑔 ∘ 𝑓)) |
18 | 9, 13, 17 | 3eqtr4d 2866 | . 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 18914 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐺 ∈ Abel) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ∘ ccom 5554 ‘cfv 6350 (class class class)co 7150 Basecbs 16477 +gcplusg 16559 Abelcabl 18901 HLchlt 36480 LHypclh 37114 LTrncltrn 37231 TGrpctgrp 37872 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-riotaBAD 36083 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-iin 4915 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-undef 7933 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-plusg 16572 df-0g 16709 df-proset 17532 df-poset 17550 df-plt 17562 df-lub 17578 df-glb 17579 df-join 17580 df-meet 17581 df-p0 17643 df-p1 17644 df-lat 17650 df-clat 17712 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-grp 18100 df-cmn 18902 df-abl 18903 df-oposet 36306 df-ol 36308 df-oml 36309 df-covers 36396 df-ats 36397 df-atl 36428 df-cvlat 36452 df-hlat 36481 df-llines 36628 df-lplanes 36629 df-lvols 36630 df-lines 36631 df-psubsp 36633 df-pmap 36634 df-padd 36926 df-lhyp 37118 df-laut 37119 df-ldil 37234 df-ltrn 37235 df-trl 37289 df-tgrp 37873 |
This theorem is referenced by: dvaabl 38154 |
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