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Mirrors > Home > MPE Home > Th. List > qus1 | Structured version Visualization version GIF version |
Description: The multiplicative identity of the quotient ring. (Contributed by Mario Carneiro, 14-Jun-2015.) |
Ref | Expression |
---|---|
qusring.u | β’ π = (π /s (π ~QG π)) |
qusring.i | β’ πΌ = (2Idealβπ ) |
qus1.o | β’ 1 = (1rβπ ) |
Ref | Expression |
---|---|
qus1 | β’ ((π β Ring β§ π β πΌ) β (π β Ring β§ [ 1 ](π ~QG π) = (1rβπ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | qusring.u | . . 3 β’ π = (π /s (π ~QG π)) | |
2 | 1 | a1i 11 | . 2 β’ ((π β Ring β§ π β πΌ) β π = (π /s (π ~QG π))) |
3 | eqid 2728 | . . 3 β’ (Baseβπ ) = (Baseβπ ) | |
4 | 3 | a1i 11 | . 2 β’ ((π β Ring β§ π β πΌ) β (Baseβπ ) = (Baseβπ )) |
5 | eqid 2728 | . 2 β’ (+gβπ ) = (+gβπ ) | |
6 | eqid 2728 | . 2 β’ (.rβπ ) = (.rβπ ) | |
7 | qus1.o | . 2 β’ 1 = (1rβπ ) | |
8 | eqid 2728 | . . . . . . 7 β’ (LIdealβπ ) = (LIdealβπ ) | |
9 | eqid 2728 | . . . . . . 7 β’ (opprβπ ) = (opprβπ ) | |
10 | eqid 2728 | . . . . . . 7 β’ (LIdealβ(opprβπ )) = (LIdealβ(opprβπ )) | |
11 | qusring.i | . . . . . . 7 β’ πΌ = (2Idealβπ ) | |
12 | 8, 9, 10, 11 | 2idlval 21138 | . . . . . 6 β’ πΌ = ((LIdealβπ ) β© (LIdealβ(opprβπ ))) |
13 | 12 | elin2 4193 | . . . . 5 β’ (π β πΌ β (π β (LIdealβπ ) β§ π β (LIdealβ(opprβπ )))) |
14 | 13 | simplbi 497 | . . . 4 β’ (π β πΌ β π β (LIdealβπ )) |
15 | 8 | lidlsubg 21112 | . . . 4 β’ ((π β Ring β§ π β (LIdealβπ )) β π β (SubGrpβπ )) |
16 | 14, 15 | sylan2 592 | . . 3 β’ ((π β Ring β§ π β πΌ) β π β (SubGrpβπ )) |
17 | eqid 2728 | . . . 4 β’ (π ~QG π) = (π ~QG π) | |
18 | 3, 17 | eqger 19126 | . . 3 β’ (π β (SubGrpβπ ) β (π ~QG π) Er (Baseβπ )) |
19 | 16, 18 | syl 17 | . 2 β’ ((π β Ring β§ π β πΌ) β (π ~QG π) Er (Baseβπ )) |
20 | ringabl 20210 | . . . . . 6 β’ (π β Ring β π β Abel) | |
21 | 20 | adantr 480 | . . . . 5 β’ ((π β Ring β§ π β πΌ) β π β Abel) |
22 | ablnsg 19795 | . . . . 5 β’ (π β Abel β (NrmSGrpβπ ) = (SubGrpβπ )) | |
23 | 21, 22 | syl 17 | . . . 4 β’ ((π β Ring β§ π β πΌ) β (NrmSGrpβπ ) = (SubGrpβπ )) |
24 | 16, 23 | eleqtrrd 2832 | . . 3 β’ ((π β Ring β§ π β πΌ) β π β (NrmSGrpβπ )) |
25 | 3, 17, 5 | eqgcpbl 19130 | . . 3 β’ (π β (NrmSGrpβπ ) β ((π(π ~QG π)π β§ π(π ~QG π)π) β (π(+gβπ )π)(π ~QG π)(π(+gβπ )π))) |
26 | 24, 25 | syl 17 | . 2 β’ ((π β Ring β§ π β πΌ) β ((π(π ~QG π)π β§ π(π ~QG π)π) β (π(+gβπ )π)(π ~QG π)(π(+gβπ )π))) |
27 | 3, 17, 11, 6 | 2idlcpbl 21159 | . 2 β’ ((π β Ring β§ π β πΌ) β ((π(π ~QG π)π β§ π(π ~QG π)π) β (π(.rβπ )π)(π ~QG π)(π(.rβπ )π))) |
28 | simpl 482 | . 2 β’ ((π β Ring β§ π β πΌ) β π β Ring) | |
29 | 2, 4, 5, 6, 7, 19, 26, 27, 28 | qusring2 20263 | 1 β’ ((π β Ring β§ π β πΌ) β (π β Ring β§ [ 1 ](π ~QG π) = (1rβπ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1534 β wcel 2099 class class class wbr 5142 βcfv 6542 (class class class)co 7414 Er wer 8715 [cec 8716 Basecbs 17173 +gcplusg 17226 .rcmulr 17227 /s cqus 17480 SubGrpcsubg 19068 NrmSGrpcnsg 19069 ~QG cqg 19070 Abelcabl 19729 1rcur 20114 Ringcrg 20166 opprcoppr 20265 LIdealclidl 21095 2Idealc2idl 21136 |
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 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 |
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 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 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-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 7865 df-1st 7987 df-2nd 7988 df-tpos 8225 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-ec 8720 df-qs 8724 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-sup 9459 df-inf 9460 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-nn 12237 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-7 12304 df-8 12305 df-9 12306 df-n0 12497 df-z 12583 df-dec 12702 df-uz 12847 df-fz 13511 df-struct 17109 df-sets 17126 df-slot 17144 df-ndx 17156 df-base 17174 df-ress 17203 df-plusg 17239 df-mulr 17240 df-sca 17242 df-vsca 17243 df-ip 17244 df-tset 17245 df-ple 17246 df-ds 17248 df-0g 17416 df-imas 17483 df-qus 17484 df-mgm 18593 df-sgrp 18672 df-mnd 18688 df-grp 18886 df-minusg 18887 df-sbg 18888 df-subg 19071 df-nsg 19072 df-eqg 19073 df-cmn 19730 df-abl 19731 df-mgp 20068 df-rng 20086 df-ur 20115 df-ring 20168 df-oppr 20266 df-subrg 20501 df-lmod 20738 df-lss 20809 df-sra 21051 df-rgmod 21052 df-lidl 21097 df-2idl 21137 |
This theorem is referenced by: qusring 21162 qusrhm 21163 rhmquskerlem 33134 rhmqusnsg 33137 qsnzr 33165 qsdrngilem 33199 qsdrnglem2 33201 |
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