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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > erngdvlem2N | Structured version Visualization version GIF version |
Description: Lemma for eringring 40497. (Contributed by NM, 6-Aug-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ernggrp.h | β’ π» = (LHypβπΎ) |
ernggrp.d | β’ π· = ((EDRingβπΎ)βπ) |
erngdv.b | β’ π΅ = (BaseβπΎ) |
erngdv.t | β’ π = ((LTrnβπΎ)βπ) |
erngdv.e | β’ πΈ = ((TEndoβπΎ)βπ) |
erngdv.p | β’ π = (π β πΈ, π β πΈ β¦ (π β π β¦ ((πβπ) β (πβπ)))) |
erngdv.o | β’ 0 = (π β π β¦ ( I βΎ π΅)) |
erngdv.i | β’ πΌ = (π β πΈ β¦ (π β π β¦ β‘(πβπ))) |
Ref | Expression |
---|---|
erngdvlem2N | β’ ((πΎ β HL β§ π β π») β π· β Abel) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ernggrp.h | . . . 4 β’ π» = (LHypβπΎ) | |
2 | erngdv.t | . . . 4 β’ π = ((LTrnβπΎ)βπ) | |
3 | erngdv.e | . . . 4 β’ πΈ = ((TEndoβπΎ)βπ) | |
4 | ernggrp.d | . . . 4 β’ π· = ((EDRingβπΎ)βπ) | |
5 | eqid 2728 | . . . 4 β’ (Baseβπ·) = (Baseβπ·) | |
6 | 1, 2, 3, 4, 5 | erngbase 40306 | . . 3 β’ ((πΎ β HL β§ π β π») β (Baseβπ·) = πΈ) |
7 | 6 | eqcomd 2734 | . 2 β’ ((πΎ β HL β§ π β π») β πΈ = (Baseβπ·)) |
8 | erngdv.p | . . 3 β’ π = (π β πΈ, π β πΈ β¦ (π β π β¦ ((πβπ) β (πβπ)))) | |
9 | eqid 2728 | . . . 4 β’ (+gβπ·) = (+gβπ·) | |
10 | 1, 2, 3, 4, 9 | erngfplus 40307 | . . 3 β’ ((πΎ β HL β§ π β π») β (+gβπ·) = (π β πΈ, π β πΈ β¦ (π β π β¦ ((πβπ) β (πβπ))))) |
11 | 8, 10 | eqtr4id 2787 | . 2 β’ ((πΎ β HL β§ π β π») β π = (+gβπ·)) |
12 | erngdv.b | . . 3 β’ π΅ = (BaseβπΎ) | |
13 | erngdv.o | . . 3 β’ 0 = (π β π β¦ ( I βΎ π΅)) | |
14 | erngdv.i | . . 3 β’ πΌ = (π β πΈ β¦ (π β π β¦ β‘(πβπ))) | |
15 | 1, 4, 12, 2, 3, 8, 13, 14 | erngdvlem1 40493 | . 2 β’ ((πΎ β HL β§ π β π») β π· β Grp) |
16 | 1, 2, 3, 8 | tendoplcom 40287 | . 2 β’ (((πΎ β HL β§ π β π») β§ π β πΈ β§ π‘ β πΈ) β (π ππ‘) = (π‘ππ )) |
17 | 7, 11, 15, 16 | isabld 19757 | 1 β’ ((πΎ β HL β§ π β π») β π· β Abel) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 β¦ cmpt 5235 I cid 5579 β‘ccnv 5681 βΎ cres 5684 β ccom 5686 βcfv 6553 β cmpo 7428 Basecbs 17187 +gcplusg 17240 Abelcabl 19743 HLchlt 38854 LHypclh 39489 LTrncltrn 39606 TEndoctendo 40257 EDRingcedring 40258 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-riotaBAD 38457 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-undef 8285 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-map 8853 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-3 12314 df-n0 12511 df-z 12597 df-uz 12861 df-fz 13525 df-struct 17123 df-slot 17158 df-ndx 17170 df-base 17188 df-plusg 17253 df-mulr 17254 df-0g 17430 df-proset 18294 df-poset 18312 df-plt 18329 df-lub 18345 df-glb 18346 df-join 18347 df-meet 18348 df-p0 18424 df-p1 18425 df-lat 18431 df-clat 18498 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-grp 18900 df-cmn 19744 df-abl 19745 df-oposet 38680 df-ol 38682 df-oml 38683 df-covers 38770 df-ats 38771 df-atl 38802 df-cvlat 38826 df-hlat 38855 df-llines 39003 df-lplanes 39004 df-lvols 39005 df-lines 39006 df-psubsp 39008 df-pmap 39009 df-padd 39301 df-lhyp 39493 df-laut 39494 df-ldil 39609 df-ltrn 39610 df-trl 39664 df-tendo 40260 df-edring 40262 |
This theorem is referenced by: (None) |
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