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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 39863. (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 2733 | . . . 4 β’ (Baseβπ·) = (Baseβπ·) | |
6 | 1, 2, 3, 4, 5 | erngbase 39672 | . . 3 β’ ((πΎ β HL β§ π β π») β (Baseβπ·) = πΈ) |
7 | 6 | eqcomd 2739 | . 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 2733 | . . . 4 β’ (+gβπ·) = (+gβπ·) | |
10 | 1, 2, 3, 4, 9 | erngfplus 39673 | . . 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 2792 | . 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 39859 | . 2 β’ ((πΎ β HL β§ π β π») β π· β Grp) |
16 | 1, 2, 3, 8 | tendoplcom 39653 | . 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 19663 | 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 397 = wceq 1542 β wcel 2107 β¦ cmpt 5232 I cid 5574 β‘ccnv 5676 βΎ cres 5679 β ccom 5681 βcfv 6544 β cmpo 7411 Basecbs 17144 +gcplusg 17197 Abelcabl 19649 HLchlt 38220 LHypclh 38855 LTrncltrn 38972 TEndoctendo 39623 EDRingcedring 39624 |
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 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-riotaBAD 37823 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-undef 8258 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-2 12275 df-3 12276 df-n0 12473 df-z 12559 df-uz 12823 df-fz 13485 df-struct 17080 df-slot 17115 df-ndx 17127 df-base 17145 df-plusg 17210 df-mulr 17211 df-0g 17387 df-proset 18248 df-poset 18266 df-plt 18283 df-lub 18299 df-glb 18300 df-join 18301 df-meet 18302 df-p0 18378 df-p1 18379 df-lat 18385 df-clat 18452 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-grp 18822 df-cmn 19650 df-abl 19651 df-oposet 38046 df-ol 38048 df-oml 38049 df-covers 38136 df-ats 38137 df-atl 38168 df-cvlat 38192 df-hlat 38221 df-llines 38369 df-lplanes 38370 df-lvols 38371 df-lines 38372 df-psubsp 38374 df-pmap 38375 df-padd 38667 df-lhyp 38859 df-laut 38860 df-ldil 38975 df-ltrn 38976 df-trl 39030 df-tendo 39626 df-edring 39628 |
This theorem is referenced by: (None) |
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