![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > erngdvlem2-rN | Structured version Visualization version GIF version |
Description: Lemma for eringring 40389. (Contributed by NM, 6-Aug-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ernggrp.h-r | β’ π» = (LHypβπΎ) |
ernggrp.d-r | β’ π· = ((EDRingRβπΎ)βπ) |
ernggrplem.b-r | β’ π΅ = (BaseβπΎ) |
ernggrplem.t-r | β’ π = ((LTrnβπΎ)βπ) |
ernggrplem.e-r | β’ πΈ = ((TEndoβπΎ)βπ) |
ernggrplem.p-r | β’ π = (π β πΈ, π β πΈ β¦ (π β π β¦ ((πβπ) β (πβπ)))) |
ernggrplem.o-r | β’ π = (π β π β¦ ( I βΎ π΅)) |
ernggrplem.i-r | β’ πΌ = (π β πΈ β¦ (π β π β¦ β‘(πβπ))) |
Ref | Expression |
---|---|
erngdvlem2-rN | β’ ((πΎ β HL β§ π β π») β π· β Abel) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ernggrp.h-r | . . . 4 β’ π» = (LHypβπΎ) | |
2 | ernggrplem.t-r | . . . 4 β’ π = ((LTrnβπΎ)βπ) | |
3 | ernggrplem.e-r | . . . 4 β’ πΈ = ((TEndoβπΎ)βπ) | |
4 | ernggrp.d-r | . . . 4 β’ π· = ((EDRingRβπΎ)βπ) | |
5 | eqid 2727 | . . . 4 β’ (Baseβπ·) = (Baseβπ·) | |
6 | 1, 2, 3, 4, 5 | erngbase-rN 40206 | . . 3 β’ ((πΎ β HL β§ π β π») β (Baseβπ·) = πΈ) |
7 | 6 | eqcomd 2733 | . 2 β’ ((πΎ β HL β§ π β π») β πΈ = (Baseβπ·)) |
8 | ernggrplem.p-r | . . 3 β’ π = (π β πΈ, π β πΈ β¦ (π β π β¦ ((πβπ) β (πβπ)))) | |
9 | eqid 2727 | . . . 4 β’ (+gβπ·) = (+gβπ·) | |
10 | 1, 2, 3, 4, 9 | erngfplus-rN 40207 | . . 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 2786 | . 2 β’ ((πΎ β HL β§ π β π») β π = (+gβπ·)) |
12 | ernggrplem.b-r | . . 3 β’ π΅ = (BaseβπΎ) | |
13 | ernggrplem.o-r | . . 3 β’ π = (π β π β¦ ( I βΎ π΅)) | |
14 | ernggrplem.i-r | . . 3 β’ πΌ = (π β πΈ β¦ (π β π β¦ β‘(πβπ))) | |
15 | 1, 4, 12, 2, 3, 8, 13, 14 | erngdvlem1-rN 40393 | . 2 β’ ((πΎ β HL β§ π β π») β π· β Grp) |
16 | 1, 2, 3, 8 | tendoplcom 40179 | . 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 19734 | 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 1534 β wcel 2099 β¦ cmpt 5225 I cid 5569 β‘ccnv 5671 βΎ cres 5674 β ccom 5676 βcfv 6542 β cmpo 7416 Basecbs 17165 +gcplusg 17218 Abelcabl 19720 HLchlt 38746 LHypclh 39381 LTrncltrn 39498 TEndoctendo 40149 EDRingRcedring-rN 40151 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 ax-riotaBAD 38349 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 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 5143 df-opab 5205 df-mpt 5226 df-tr 5260 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 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7863 df-1st 7985 df-2nd 7986 df-undef 8270 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-er 8716 df-map 8836 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-nn 12229 df-2 12291 df-3 12292 df-n0 12489 df-z 12575 df-uz 12839 df-fz 13503 df-struct 17101 df-slot 17136 df-ndx 17148 df-base 17166 df-plusg 17231 df-mulr 17232 df-0g 17408 df-proset 18272 df-poset 18290 df-plt 18307 df-lub 18323 df-glb 18324 df-join 18325 df-meet 18326 df-p0 18402 df-p1 18403 df-lat 18409 df-clat 18476 df-mgm 18585 df-sgrp 18664 df-mnd 18680 df-grp 18878 df-cmn 19721 df-abl 19722 df-oposet 38572 df-ol 38574 df-oml 38575 df-covers 38662 df-ats 38663 df-atl 38694 df-cvlat 38718 df-hlat 38747 df-llines 38895 df-lplanes 38896 df-lvols 38897 df-lines 38898 df-psubsp 38900 df-pmap 38901 df-padd 39193 df-lhyp 39385 df-laut 39386 df-ldil 39501 df-ltrn 39502 df-trl 39556 df-tendo 40152 df-edring-rN 40153 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |