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Mirrors > Home > MPE Home > Th. List > rrgeq0 | Structured version Visualization version GIF version |
Description: Left-multiplication by a left regular element does not change zeroness. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
Ref | Expression |
---|---|
rrgval.e | β’ πΈ = (RLRegβπ ) |
rrgval.b | β’ π΅ = (Baseβπ ) |
rrgval.t | β’ Β· = (.rβπ ) |
rrgval.z | β’ 0 = (0gβπ ) |
Ref | Expression |
---|---|
rrgeq0 | β’ ((π β Ring β§ π β πΈ β§ π β π΅) β ((π Β· π) = 0 β π = 0 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rrgval.e | . . . 4 β’ πΈ = (RLRegβπ ) | |
2 | rrgval.b | . . . 4 β’ π΅ = (Baseβπ ) | |
3 | rrgval.t | . . . 4 β’ Β· = (.rβπ ) | |
4 | rrgval.z | . . . 4 β’ 0 = (0gβπ ) | |
5 | 1, 2, 3, 4 | rrgeq0i 20775 | . . 3 β’ ((π β πΈ β§ π β π΅) β ((π Β· π) = 0 β π = 0 )) |
6 | 5 | 3adant1 1131 | . 2 β’ ((π β Ring β§ π β πΈ β§ π β π΅) β ((π Β· π) = 0 β π = 0 )) |
7 | simp1 1137 | . . . 4 β’ ((π β Ring β§ π β πΈ β§ π β π΅) β π β Ring) | |
8 | 1, 2, 3, 4 | rrgval 20773 | . . . . . 6 β’ πΈ = {π₯ β π΅ β£ βπ¦ β π΅ ((π₯ Β· π¦) = 0 β π¦ = 0 )} |
9 | 8 | ssrab3 4041 | . . . . 5 β’ πΈ β π΅ |
10 | simp2 1138 | . . . . 5 β’ ((π β Ring β§ π β πΈ β§ π β π΅) β π β πΈ) | |
11 | 9, 10 | sselid 3943 | . . . 4 β’ ((π β Ring β§ π β πΈ β§ π β π΅) β π β π΅) |
12 | 2, 3, 4 | ringrz 20017 | . . . 4 β’ ((π β Ring β§ π β π΅) β (π Β· 0 ) = 0 ) |
13 | 7, 11, 12 | syl2anc 585 | . . 3 β’ ((π β Ring β§ π β πΈ β§ π β π΅) β (π Β· 0 ) = 0 ) |
14 | oveq2 7366 | . . . 4 β’ (π = 0 β (π Β· π) = (π Β· 0 )) | |
15 | 14 | eqeq1d 2735 | . . 3 β’ (π = 0 β ((π Β· π) = 0 β (π Β· 0 ) = 0 )) |
16 | 13, 15 | syl5ibrcom 247 | . 2 β’ ((π β Ring β§ π β πΈ β§ π β π΅) β (π = 0 β (π Β· π) = 0 )) |
17 | 6, 16 | impbid 211 | 1 β’ ((π β Ring β§ π β πΈ β§ π β π΅) β ((π Β· π) = 0 β π = 0 )) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ w3a 1088 = wceq 1542 β wcel 2107 βwral 3061 βcfv 6497 (class class class)co 7358 Basecbs 17088 .rcmulr 17139 0gc0g 17326 Ringcrg 19969 RLRegcrlreg 20765 |
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-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-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-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 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-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-plusg 17151 df-0g 17328 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-grp 18756 df-mgp 19902 df-ring 19971 df-rlreg 20769 |
This theorem is referenced by: rrgsupp 20777 |
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