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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 40376. (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 2726 | . . . 4 β’ (Baseβπ·) = (Baseβπ·) | |
6 | 1, 2, 3, 4, 5 | erngbase 40185 | . . 3 β’ ((πΎ β HL β§ π β π») β (Baseβπ·) = πΈ) |
7 | 6 | eqcomd 2732 | . 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 2726 | . . . 4 β’ (+gβπ·) = (+gβπ·) | |
10 | 1, 2, 3, 4, 9 | erngfplus 40186 | . . 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 2785 | . 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 40372 | . 2 β’ ((πΎ β HL β§ π β π») β π· β Grp) |
16 | 1, 2, 3, 8 | tendoplcom 40166 | . 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 19715 | 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 395 = wceq 1533 β wcel 2098 β¦ cmpt 5224 I cid 5566 β‘ccnv 5668 βΎ cres 5671 β ccom 5673 βcfv 6537 β cmpo 7407 Basecbs 17153 +gcplusg 17206 Abelcabl 19701 HLchlt 38733 LHypclh 39368 LTrncltrn 39485 TEndoctendo 40136 EDRingcedring 40137 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-riotaBAD 38336 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-undef 8259 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-n0 12477 df-z 12563 df-uz 12827 df-fz 13491 df-struct 17089 df-slot 17124 df-ndx 17136 df-base 17154 df-plusg 17219 df-mulr 17220 df-0g 17396 df-proset 18260 df-poset 18278 df-plt 18295 df-lub 18311 df-glb 18312 df-join 18313 df-meet 18314 df-p0 18390 df-p1 18391 df-lat 18397 df-clat 18464 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-grp 18866 df-cmn 19702 df-abl 19703 df-oposet 38559 df-ol 38561 df-oml 38562 df-covers 38649 df-ats 38650 df-atl 38681 df-cvlat 38705 df-hlat 38734 df-llines 38882 df-lplanes 38883 df-lvols 38884 df-lines 38885 df-psubsp 38887 df-pmap 38888 df-padd 39180 df-lhyp 39372 df-laut 39373 df-ldil 39488 df-ltrn 39489 df-trl 39543 df-tendo 40139 df-edring 40141 |
This theorem is referenced by: (None) |
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