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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 2733 | . . 3 β’ (Baseβπ ) = (Baseβπ ) | |
4 | 3 | a1i 11 | . 2 β’ ((π β Ring β§ π β πΌ) β (Baseβπ ) = (Baseβπ )) |
5 | eqid 2733 | . 2 β’ (+gβπ ) = (+gβπ ) | |
6 | eqid 2733 | . 2 β’ (.rβπ ) = (.rβπ ) | |
7 | qus1.o | . 2 β’ 1 = (1rβπ ) | |
8 | eqid 2733 | . . . . . . 7 β’ (LIdealβπ ) = (LIdealβπ ) | |
9 | eqid 2733 | . . . . . . 7 β’ (opprβπ ) = (opprβπ ) | |
10 | eqid 2733 | . . . . . . 7 β’ (LIdealβ(opprβπ )) = (LIdealβ(opprβπ )) | |
11 | qusring.i | . . . . . . 7 β’ πΌ = (2Idealβπ ) | |
12 | 8, 9, 10, 11 | 2idlval 20719 | . . . . . 6 β’ πΌ = ((LIdealβπ ) β© (LIdealβ(opprβπ ))) |
13 | 12 | elin2 4158 | . . . . 5 β’ (π β πΌ β (π β (LIdealβπ ) β§ π β (LIdealβ(opprβπ )))) |
14 | 13 | simplbi 499 | . . . 4 β’ (π β πΌ β π β (LIdealβπ )) |
15 | 8 | lidlsubg 20701 | . . . 4 β’ ((π β Ring β§ π β (LIdealβπ )) β π β (SubGrpβπ )) |
16 | 14, 15 | sylan2 594 | . . 3 β’ ((π β Ring β§ π β πΌ) β π β (SubGrpβπ )) |
17 | eqid 2733 | . . . 4 β’ (π ~QG π) = (π ~QG π) | |
18 | 3, 17 | eqger 18985 | . . 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 20007 | . . . . . 6 β’ (π β Ring β π β Abel) | |
21 | 20 | adantr 482 | . . . . 5 β’ ((π β Ring β§ π β πΌ) β π β Abel) |
22 | ablnsg 19630 | . . . . 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 2837 | . . 3 β’ ((π β Ring β§ π β πΌ) β π β (NrmSGrpβπ )) |
25 | 3, 17, 5 | eqgcpbl 18989 | . . 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 20720 | . 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 484 | . 2 β’ ((π β Ring β§ π β πΌ) β π β Ring) | |
29 | 2, 4, 5, 6, 7, 19, 26, 27, 28 | qusring2 20051 | 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 397 = wceq 1542 β wcel 2107 class class class wbr 5106 βcfv 6497 (class class class)co 7358 Er wer 8648 [cec 8649 Basecbs 17088 +gcplusg 17138 .rcmulr 17139 /s cqus 17392 SubGrpcsubg 18927 NrmSGrpcnsg 18928 ~QG cqg 18929 Abelcabl 19568 1rcur 19918 Ringcrg 19969 opprcoppr 20053 LIdealclidl 20647 2Idealc2idl 20717 |
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 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
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 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-tpos 8158 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8651 df-ec 8653 df-qs 8657 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-sup 9383 df-inf 9384 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-3 12222 df-4 12223 df-5 12224 df-6 12225 df-7 12226 df-8 12227 df-9 12228 df-n0 12419 df-z 12505 df-dec 12624 df-uz 12769 df-fz 13431 df-struct 17024 df-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-ress 17118 df-plusg 17151 df-mulr 17152 df-sca 17154 df-vsca 17155 df-ip 17156 df-tset 17157 df-ple 17158 df-ds 17160 df-0g 17328 df-imas 17395 df-qus 17396 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-grp 18756 df-minusg 18757 df-sbg 18758 df-subg 18930 df-nsg 18931 df-eqg 18932 df-cmn 19569 df-abl 19570 df-mgp 19902 df-ur 19919 df-ring 19971 df-oppr 20054 df-subrg 20234 df-lmod 20338 df-lss 20408 df-sra 20649 df-rgmod 20650 df-lidl 20651 df-2idl 20718 |
This theorem is referenced by: qusring 20722 qusrhm 20723 |
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