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Mirrors > Home > MPE Home > Th. List > prdscrngd | Structured version Visualization version GIF version |
Description: A product of commutative rings is a commutative ring. Since the resulting ring will have zero divisors in all nontrivial cases, this cannot be strengthened much further. (Contributed by Mario Carneiro, 11-Mar-2015.) |
Ref | Expression |
---|---|
prdscrngd.y | β’ π = (πXsπ ) |
prdscrngd.i | β’ (π β πΌ β π) |
prdscrngd.s | β’ (π β π β π) |
prdscrngd.r | β’ (π β π :πΌβΆCRing) |
Ref | Expression |
---|---|
prdscrngd | β’ (π β π β CRing) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prdscrngd.y | . . 3 β’ π = (πXsπ ) | |
2 | prdscrngd.i | . . 3 β’ (π β πΌ β π) | |
3 | prdscrngd.s | . . 3 β’ (π β π β π) | |
4 | prdscrngd.r | . . . 4 β’ (π β π :πΌβΆCRing) | |
5 | crngring 19984 | . . . . 5 β’ (π₯ β CRing β π₯ β Ring) | |
6 | 5 | ssriv 3952 | . . . 4 β’ CRing β Ring |
7 | fss 6689 | . . . 4 β’ ((π :πΌβΆCRing β§ CRing β Ring) β π :πΌβΆRing) | |
8 | 4, 6, 7 | sylancl 587 | . . 3 β’ (π β π :πΌβΆRing) |
9 | 1, 2, 3, 8 | prdsringd 20044 | . 2 β’ (π β π β Ring) |
10 | eqid 2733 | . . . 4 β’ (πXs(mulGrp β π )) = (πXs(mulGrp β π )) | |
11 | fnmgp 19906 | . . . . . . 7 β’ mulGrp Fn V | |
12 | ssv 3972 | . . . . . . 7 β’ CRing β V | |
13 | fnssres 6628 | . . . . . . 7 β’ ((mulGrp Fn V β§ CRing β V) β (mulGrp βΎ CRing) Fn CRing) | |
14 | 11, 12, 13 | mp2an 691 | . . . . . 6 β’ (mulGrp βΎ CRing) Fn CRing |
15 | fvres 6865 | . . . . . . . 8 β’ (π₯ β CRing β ((mulGrp βΎ CRing)βπ₯) = (mulGrpβπ₯)) | |
16 | eqid 2733 | . . . . . . . . 9 β’ (mulGrpβπ₯) = (mulGrpβπ₯) | |
17 | 16 | crngmgp 19980 | . . . . . . . 8 β’ (π₯ β CRing β (mulGrpβπ₯) β CMnd) |
18 | 15, 17 | eqeltrd 2834 | . . . . . . 7 β’ (π₯ β CRing β ((mulGrp βΎ CRing)βπ₯) β CMnd) |
19 | 18 | rgen 3063 | . . . . . 6 β’ βπ₯ β CRing ((mulGrp βΎ CRing)βπ₯) β CMnd |
20 | ffnfv 7070 | . . . . . 6 β’ ((mulGrp βΎ CRing):CRingβΆCMnd β ((mulGrp βΎ CRing) Fn CRing β§ βπ₯ β CRing ((mulGrp βΎ CRing)βπ₯) β CMnd)) | |
21 | 14, 19, 20 | mpbir2an 710 | . . . . 5 β’ (mulGrp βΎ CRing):CRingβΆCMnd |
22 | fco2 6699 | . . . . 5 β’ (((mulGrp βΎ CRing):CRingβΆCMnd β§ π :πΌβΆCRing) β (mulGrp β π ):πΌβΆCMnd) | |
23 | 21, 4, 22 | sylancr 588 | . . . 4 β’ (π β (mulGrp β π ):πΌβΆCMnd) |
24 | 10, 2, 3, 23 | prdscmnd 19647 | . . 3 β’ (π β (πXs(mulGrp β π )) β CMnd) |
25 | eqidd 2734 | . . . 4 β’ (π β (Baseβ(mulGrpβπ)) = (Baseβ(mulGrpβπ))) | |
26 | eqid 2733 | . . . . . 6 β’ (mulGrpβπ) = (mulGrpβπ) | |
27 | 4 | ffnd 6673 | . . . . . 6 β’ (π β π Fn πΌ) |
28 | 1, 26, 10, 2, 3, 27 | prdsmgp 20042 | . . . . 5 β’ (π β ((Baseβ(mulGrpβπ)) = (Baseβ(πXs(mulGrp β π ))) β§ (+gβ(mulGrpβπ)) = (+gβ(πXs(mulGrp β π ))))) |
29 | 28 | simpld 496 | . . . 4 β’ (π β (Baseβ(mulGrpβπ)) = (Baseβ(πXs(mulGrp β π )))) |
30 | 28 | simprd 497 | . . . . 5 β’ (π β (+gβ(mulGrpβπ)) = (+gβ(πXs(mulGrp β π )))) |
31 | 30 | oveqdr 7389 | . . . 4 β’ ((π β§ (π₯ β (Baseβ(mulGrpβπ)) β§ π¦ β (Baseβ(mulGrpβπ)))) β (π₯(+gβ(mulGrpβπ))π¦) = (π₯(+gβ(πXs(mulGrp β π )))π¦)) |
32 | 25, 29, 31 | cmnpropd 19581 | . . 3 β’ (π β ((mulGrpβπ) β CMnd β (πXs(mulGrp β π )) β CMnd)) |
33 | 24, 32 | mpbird 257 | . 2 β’ (π β (mulGrpβπ) β CMnd) |
34 | 26 | iscrng 19979 | . 2 β’ (π β CRing β (π β Ring β§ (mulGrpβπ) β CMnd)) |
35 | 9, 33, 34 | sylanbrc 584 | 1 β’ (π β π β CRing) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 βwral 3061 Vcvv 3447 β wss 3914 βΎ cres 5639 β ccom 5641 Fn wfn 6495 βΆwf 6496 βcfv 6500 (class class class)co 7361 Basecbs 17091 +gcplusg 17141 Xscprds 17335 CMndccmn 19570 mulGrpcmgp 19904 Ringcrg 19972 CRingccrg 19973 |
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 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 |
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 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-er 8654 df-map 8773 df-ixp 8842 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-sup 9386 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-nn 12162 df-2 12224 df-3 12225 df-4 12226 df-5 12227 df-6 12228 df-7 12229 df-8 12230 df-9 12231 df-n0 12422 df-z 12508 df-dec 12627 df-uz 12772 df-fz 13434 df-struct 17027 df-sets 17044 df-slot 17062 df-ndx 17074 df-base 17092 df-plusg 17154 df-mulr 17155 df-sca 17157 df-vsca 17158 df-ip 17159 df-tset 17160 df-ple 17161 df-ds 17163 df-hom 17165 df-cco 17166 df-0g 17331 df-prds 17337 df-mgm 18505 df-sgrp 18554 df-mnd 18565 df-grp 18759 df-minusg 18760 df-cmn 19572 df-mgp 19905 df-ring 19974 df-cring 19975 |
This theorem is referenced by: pwscrng 20049 |
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