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| Mirrors > Home > MPE Home > Th. List > Mathboxes > 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 20870. Similar to qusmul2 20867. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 28-Feb-2025.) TODO: Use in proof of quscrng 20870 if moved to main. |
| 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 19052 | . . . 4 β’ (π β (SubGrpβπ ) β βΌ Er π΅) |
| 7 | 6 | 3ad2ant3 1135 | . . 3 β’ ((π β Rng β§ π β (2Idealβπ ) β§ π β (SubGrpβπ )) β βΌ Er π΅) |
| 8 | simp1 1136 | . . 3 β’ ((π β Rng β§ π β (2Idealβπ ) β§ π β (SubGrpβπ )) β π β Rng) | |
| 9 | eqid 2732 | . . . 4 β’ (2Idealβπ ) = (2Idealβπ ) | |
| 10 | qusmulrng.p | . . . 4 β’ Β· = (.rβπ ) | |
| 11 | 3, 5, 9, 10 | 2idlcpblrng 46747 | . . 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 615 | . . . . 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 1095 | . . . . 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 46649 | . . . 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 17497 | . 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 1120 | 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 396 β§ w3a 1087 = wceq 1541 β wcel 2106 βcfv 6540 (class class class)co 7405 Er wer 8696 [cec 8697 Basecbs 17140 .rcmulr 17194 /s cqus 17447 SubGrpcsubg 18994 ~QG cqg 18996 2Idealc2idl 20848 Rngcrng 46634 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
| This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-tpos 8207 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-ec 8701 df-qs 8705 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-inf 9434 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-fz 13481 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-0g 17383 df-imas 17450 df-qus 17451 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-grp 18818 df-minusg 18819 df-sbg 18820 df-subg 18997 df-eqg 18999 df-cmn 19644 df-abl 19645 df-mgp 19982 df-oppr 20142 df-lss 20535 df-sra 20777 df-rgmod 20778 df-lidl 20779 df-2idl 20849 df-rng 46635 |
| This theorem is referenced by: rngqiprnglinlem2 46757 |
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