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| Mirrors > Home > MPE Home > Th. List > qusmulrng | Structured version Visualization version GIF version | ||
| Description: Value of the multiplication operation in a quotient ring of a non-unital ring. Formerly part of proof for quscrng 21177. Similar to qusmul2 21173. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 28-Feb-2025.) |
| Ref | Expression |
|---|---|
| qusmulrng.e | β’ βΌ = (π ~QG π) |
| qusmulrng.h | β’ π» = (π /s βΌ ) |
| qusmulrng.b | β’ π΅ = (Baseβπ ) |
| qusmulrng.p | β’ Β· = (.rβπ ) |
| qusmulrng.a | β’ β = (.rβπ») |
| Ref | Expression |
|---|---|
| qusmulrng | β’ (((π β Rng β§ π β (2Idealβπ ) β§ π β (SubGrpβπ )) β§ (π β π΅ β§ π β π΅)) β ([π] βΌ β [π] βΌ ) = [(π Β· π)] βΌ ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | qusmulrng.h | . . . 4 β’ π» = (π /s βΌ ) | |
| 2 | 1 | a1i 11 | . . 3 β’ ((π β Rng β§ π β (2Idealβπ ) β§ π β (SubGrpβπ )) β π» = (π /s βΌ )) |
| 3 | qusmulrng.b | . . . 4 β’ π΅ = (Baseβπ ) | |
| 4 | 3 | a1i 11 | . . 3 β’ ((π β Rng β§ π β (2Idealβπ ) β§ π β (SubGrpβπ )) β π΅ = (Baseβπ )) |
| 5 | qusmulrng.e | . . . . 5 β’ βΌ = (π ~QG π) | |
| 6 | 3, 5 | eqger 19135 | . . . 4 β’ (π β (SubGrpβπ ) β βΌ Er π΅) |
| 7 | 6 | 3ad2ant3 1132 | . . 3 β’ ((π β Rng β§ π β (2Idealβπ ) β§ π β (SubGrpβπ )) β βΌ Er π΅) |
| 8 | simp1 1133 | . . 3 β’ ((π β Rng β§ π β (2Idealβπ ) β§ π β (SubGrpβπ )) β π β Rng) | |
| 9 | eqid 2725 | . . . 4 β’ (2Idealβπ ) = (2Idealβπ ) | |
| 10 | qusmulrng.p | . . . 4 β’ Β· = (.rβπ ) | |
| 11 | 3, 5, 9, 10 | 2idlcpblrng 21167 | . . 3 β’ ((π β Rng β§ π β (2Idealβπ ) β§ π β (SubGrpβπ )) β ((π βΌ π β§ π βΌ π) β (π Β· π) βΌ (π Β· π))) |
| 12 | 8 | anim1i 613 | . . . . 5 β’ (((π β Rng β§ π β (2Idealβπ ) β§ π β (SubGrpβπ )) β§ (π β π΅ β§ π β π΅)) β (π β Rng β§ (π β π΅ β§ π β π΅))) |
| 13 | 3anass 1092 | . . . . 5 β’ ((π β Rng β§ π β π΅ β§ π β π΅) β (π β Rng β§ (π β π΅ β§ π β π΅))) | |
| 14 | 12, 13 | sylibr 233 | . . . 4 β’ (((π β Rng β§ π β (2Idealβπ ) β§ π β (SubGrpβπ )) β§ (π β π΅ β§ π β π΅)) β (π β Rng β§ π β π΅ β§ π β π΅)) |
| 15 | 3, 10 | rngcl 20106 | . . . 4 β’ ((π β Rng β§ π β π΅ β§ π β π΅) β (π Β· π) β π΅) |
| 16 | 14, 15 | syl 17 | . . 3 β’ (((π β Rng β§ π β (2Idealβπ ) β§ π β (SubGrpβπ )) β§ (π β π΅ β§ π β π΅)) β (π Β· π) β π΅) |
| 17 | qusmulrng.a | . . 3 β’ β = (.rβπ») | |
| 18 | 2, 4, 7, 8, 11, 16, 10, 17 | qusmulval 17534 | . 2 β’ (((π β Rng β§ π β (2Idealβπ ) β§ π β (SubGrpβπ )) β§ π β π΅ β§ π β π΅) β ([π] βΌ β [π] βΌ ) = [(π Β· π)] βΌ ) |
| 19 | 18 | 3expb 1117 | 1 β’ (((π β Rng β§ π β (2Idealβπ ) β§ π β (SubGrpβπ )) β§ (π β π΅ β§ π β π΅)) β ([π] βΌ β [π] βΌ ) = [(π Β· π)] βΌ ) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 βcfv 6541 (class class class)co 7414 Er wer 8718 [cec 8719 Basecbs 17177 .rcmulr 17231 /s cqus 17484 SubGrpcsubg 19077 ~QG cqg 19079 Rngcrng 20094 2Idealc2idl 21145 |
| 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 5278 ax-sep 5292 ax-nul 5299 ax-pow 5357 ax-pr 5421 ax-un 7736 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 |
| 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 3958 df-nul 4317 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-tp 4627 df-op 4629 df-uni 4902 df-iun 4991 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5568 df-eprel 5574 df-po 5582 df-so 5583 df-fr 5625 df-we 5627 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7867 df-1st 7989 df-2nd 7990 df-tpos 8228 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-er 8721 df-ec 8723 df-qs 8727 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-sup 9463 df-inf 9464 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12501 df-z 12587 df-dec 12706 df-uz 12851 df-fz 13515 df-struct 17113 df-sets 17130 df-slot 17148 df-ndx 17160 df-base 17178 df-ress 17207 df-plusg 17243 df-mulr 17244 df-sca 17246 df-vsca 17247 df-ip 17248 df-tset 17249 df-ple 17250 df-ds 17252 df-0g 17420 df-imas 17487 df-qus 17488 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-grp 18895 df-minusg 18896 df-sbg 18897 df-subg 19080 df-eqg 19082 df-cmn 19739 df-abl 19740 df-mgp 20077 df-rng 20095 df-oppr 20275 df-lss 20818 df-sra 21060 df-rgmod 21061 df-lidl 21106 df-2idl 21146 |
| This theorem is referenced by: quscrng 21177 rngqiprnglinlem2 21184 pzriprnglem12 21420 |
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