![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > erngdvlem1 | Structured version Visualization version GIF version |
Description: Lemma for eringring 40521. (Contributed by NM, 4-Aug-2013.) |
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 |
---|---|
erngdvlem1 | β’ ((πΎ β HL β§ π β π») β π· β Grp) |
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 2725 | . . . 4 β’ (Baseβπ·) = (Baseβπ·) | |
6 | 1, 2, 3, 4, 5 | erngbase 40330 | . . 3 β’ ((πΎ β HL β§ π β π») β (Baseβπ·) = πΈ) |
7 | 6 | eqcomd 2731 | . 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 2725 | . . . 4 β’ (+gβπ·) = (+gβπ·) | |
10 | 1, 2, 3, 4, 9 | erngfplus 40331 | . . 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 2784 | . 2 β’ ((πΎ β HL β§ π β π») β π = (+gβπ·)) |
12 | 1, 2, 3, 8 | tendoplcl 40310 | . 2 β’ (((πΎ β HL β§ π β π») β§ π β πΈ β§ π‘ β πΈ) β (π ππ‘) β πΈ) |
13 | 1, 2, 3, 8 | tendoplass 40312 | . 2 β’ (((πΎ β HL β§ π β π») β§ (π β πΈ β§ π‘ β πΈ β§ π’ β πΈ)) β ((π ππ‘)ππ’) = (π π(π‘ππ’))) |
14 | erngdv.b | . . 3 β’ π΅ = (BaseβπΎ) | |
15 | erngdv.o | . . 3 β’ 0 = (π β π β¦ ( I βΎ π΅)) | |
16 | 14, 1, 2, 3, 15 | tendo0cl 40319 | . 2 β’ ((πΎ β HL β§ π β π») β 0 β πΈ) |
17 | 14, 1, 2, 3, 15, 8 | tendo0pl 40320 | . 2 β’ (((πΎ β HL β§ π β π») β§ π β πΈ) β ( 0 ππ ) = π ) |
18 | erngdv.i | . . 3 β’ πΌ = (π β πΈ β¦ (π β π β¦ β‘(πβπ))) | |
19 | 1, 2, 3, 18 | tendoicl 40325 | . 2 β’ (((πΎ β HL β§ π β π») β§ π β πΈ) β (πΌβπ ) β πΈ) |
20 | 1, 2, 3, 18, 14, 8, 15 | tendoipl 40326 | . 2 β’ (((πΎ β HL β§ π β π») β§ π β πΈ) β ((πΌβπ )ππ ) = 0 ) |
21 | 7, 11, 12, 13, 16, 17, 19, 20 | isgrpd 18919 | 1 β’ ((πΎ β HL β§ π β π») β π· β Grp) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 β¦ cmpt 5226 I cid 5569 β‘ccnv 5671 βΎ cres 5674 β ccom 5676 βcfv 6543 β cmpo 7418 Basecbs 17179 +gcplusg 17232 Grpcgrp 18894 HLchlt 38878 LHypclh 39513 LTrncltrn 39630 TEndoctendo 40281 EDRingcedring 40282 |
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 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-riotaBAD 38481 |
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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-undef 8277 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-map 8845 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-3 12306 df-n0 12503 df-z 12589 df-uz 12853 df-fz 13517 df-struct 17115 df-slot 17150 df-ndx 17162 df-base 17180 df-plusg 17245 df-mulr 17246 df-0g 17422 df-proset 18286 df-poset 18304 df-plt 18321 df-lub 18337 df-glb 18338 df-join 18339 df-meet 18340 df-p0 18416 df-p1 18417 df-lat 18423 df-clat 18490 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-grp 18897 df-oposet 38704 df-ol 38706 df-oml 38707 df-covers 38794 df-ats 38795 df-atl 38826 df-cvlat 38850 df-hlat 38879 df-llines 39027 df-lplanes 39028 df-lvols 39029 df-lines 39030 df-psubsp 39032 df-pmap 39033 df-padd 39325 df-lhyp 39517 df-laut 39518 df-ldil 39633 df-ltrn 39634 df-trl 39688 df-tendo 40284 df-edring 40286 |
This theorem is referenced by: erngdvlem2N 40518 erngdvlem3 40519 erngdvlem4 40520 |
Copyright terms: Public domain | W3C validator |