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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 2737 | . . . 4 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊) | |
3 | tgrpgrp.g | . . . 4 ⊢ 𝐺 = ((TGrp‘𝐾)‘𝑊) | |
4 | eqid 2737 | . . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
5 | 1, 2, 3, 4 | tgrpbase 39238 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (Base‘𝐺) = ((LTrn‘𝐾)‘𝑊)) |
6 | 5 | eqcomd 2743 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ((LTrn‘𝐾)‘𝑊) = (Base‘𝐺)) |
7 | eqidd 2738 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (+g‘𝐺) = (+g‘𝐺)) | |
8 | 1, 3 | tgrpgrp 39242 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐺 ∈ Grp) |
9 | 1, 2 | ltrncom 39230 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑓 ∘ 𝑔) = (𝑔 ∘ 𝑓)) |
10 | eqid 2737 | . . . . . 6 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
11 | 1, 2, 3, 10 | tgrpov 39240 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊))) → (𝑓(+g‘𝐺)𝑔) = (𝑓 ∘ 𝑔)) |
12 | 11 | 3expa 1119 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊))) → (𝑓(+g‘𝐺)𝑔) = (𝑓 ∘ 𝑔)) |
13 | 12 | 3impb 1116 | . . 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 39240 | . . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ (𝑔 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊))) → (𝑔(+g‘𝐺)𝑓) = (𝑔 ∘ 𝑓)) |
15 | 14 | 3expa 1119 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑔 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊))) → (𝑔(+g‘𝐺)𝑓) = (𝑔 ∘ 𝑓)) |
16 | 15 | 3impb 1116 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑔(+g‘𝐺)𝑓) = (𝑔 ∘ 𝑓)) |
17 | 16 | 3com23 1127 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑔(+g‘𝐺)𝑓) = (𝑔 ∘ 𝑓)) |
18 | 9, 13, 17 | 3eqtr4d 2787 | . 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 19584 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐺 ∈ Abel) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∘ ccom 5642 ‘cfv 6501 (class class class)co 7362 Basecbs 17090 +gcplusg 17140 Abelcabl 19570 HLchlt 37841 LHypclh 38476 LTrncltrn 38593 TGrpctgrp 39234 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 ax-riotaBAD 37444 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-iin 4962 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-undef 8209 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-er 8655 df-map 8774 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-nn 12161 df-2 12223 df-n0 12421 df-z 12507 df-uz 12771 df-fz 13432 df-struct 17026 df-slot 17061 df-ndx 17073 df-base 17091 df-plusg 17153 df-0g 17330 df-proset 18191 df-poset 18209 df-plt 18226 df-lub 18242 df-glb 18243 df-join 18244 df-meet 18245 df-p0 18321 df-p1 18322 df-lat 18328 df-clat 18395 df-mgm 18504 df-sgrp 18553 df-mnd 18564 df-grp 18758 df-cmn 19571 df-abl 19572 df-oposet 37667 df-ol 37669 df-oml 37670 df-covers 37757 df-ats 37758 df-atl 37789 df-cvlat 37813 df-hlat 37842 df-llines 37990 df-lplanes 37991 df-lvols 37992 df-lines 37993 df-psubsp 37995 df-pmap 37996 df-padd 38288 df-lhyp 38480 df-laut 38481 df-ldil 38596 df-ltrn 38597 df-trl 38651 df-tgrp 39235 |
This theorem is referenced by: dvaabl 39516 |
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