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Theorem prdscrngd 17782
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.)
Hypotheses
Ref Expression
prdscrngd.y  |-  Y  =  ( S X_s R )
prdscrngd.i  |-  ( ph  ->  I  e.  W )
prdscrngd.s  |-  ( ph  ->  S  e.  V )
prdscrngd.r  |-  ( ph  ->  R : I --> CRing )
Assertion
Ref Expression
prdscrngd  |-  ( ph  ->  Y  e.  CRing )

Proof of Theorem prdscrngd
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prdscrngd.y . . 3  |-  Y  =  ( S X_s R )
2 prdscrngd.i . . 3  |-  ( ph  ->  I  e.  W )
3 prdscrngd.s . . 3  |-  ( ph  ->  S  e.  V )
4 prdscrngd.r . . . 4  |-  ( ph  ->  R : I --> CRing )
5 crngring 17732 . . . . 5  |-  ( x  e.  CRing  ->  x  e.  Ring )
65ssriv 3465 . . . 4  |-  CRing  C_  Ring
7 fss 5745 . . . 4  |-  ( ( R : I --> CRing  /\  CRing  C_ 
Ring )  ->  R : I --> Ring )
84, 6, 7sylancl 666 . . 3  |-  ( ph  ->  R : I --> Ring )
91, 2, 3, 8prdsringd 17781 . 2  |-  ( ph  ->  Y  e.  Ring )
10 eqid 2420 . . . 4  |-  ( S
X_s (mulGrp  o.  R )
)  =  ( S
X_s (mulGrp  o.  R )
)
11 fnmgp 17666 . . . . . . 7  |- mulGrp  Fn  _V
12 ssv 3481 . . . . . . 7  |-  CRing  C_  _V
13 fnssres 5698 . . . . . . 7  |-  ( (mulGrp 
Fn  _V  /\  CRing  C_  _V )  ->  (mulGrp  |`  CRing )  Fn 
CRing )
1411, 12, 13mp2an 676 . . . . . 6  |-  (mulGrp  |`  CRing )  Fn 
CRing
15 fvres 5886 . . . . . . . 8  |-  ( x  e.  CRing  ->  ( (mulGrp  |` 
CRing ) `  x )  =  (mulGrp `  x
) )
16 eqid 2420 . . . . . . . . 9  |-  (mulGrp `  x )  =  (mulGrp `  x )
1716crngmgp 17729 . . . . . . . 8  |-  ( x  e.  CRing  ->  (mulGrp `  x
)  e. CMnd )
1815, 17eqeltrd 2508 . . . . . . 7  |-  ( x  e.  CRing  ->  ( (mulGrp  |` 
CRing ) `  x )  e. CMnd )
1918rgen 2783 . . . . . 6  |-  A. x  e.  CRing  ( (mulGrp  |`  CRing ) `  x )  e. CMnd
20 ffnfv 6055 . . . . . 6  |-  ( (mulGrp  |` 
CRing ) : CRing -->CMnd  <->  ( (mulGrp  |`  CRing )  Fn 
CRing  /\  A. x  e. 
CRing  ( (mulGrp  |`  CRing ) `  x )  e. CMnd )
)
2114, 19, 20mpbir2an 928 . . . . 5  |-  (mulGrp  |`  CRing ) :
CRing
-->CMnd
22 fco2 5748 . . . . 5  |-  ( ( (mulGrp  |`  CRing ) : CRing -->CMnd  /\  R : I --> CRing )  -> 
(mulGrp  o.  R ) : I -->CMnd )
2321, 4, 22sylancr 667 . . . 4  |-  ( ph  ->  (mulGrp  o.  R ) : I -->CMnd )
2410, 2, 3, 23prdscmnd 17440 . . 3  |-  ( ph  ->  ( S X_s (mulGrp  o.  R )
)  e. CMnd )
25 eqidd 2421 . . . 4  |-  ( ph  ->  ( Base `  (mulGrp `  Y ) )  =  ( Base `  (mulGrp `  Y ) ) )
26 eqid 2420 . . . . . 6  |-  (mulGrp `  Y )  =  (mulGrp `  Y )
27 ffn 5737 . . . . . . 7  |-  ( R : I --> CRing  ->  R  Fn  I )
284, 27syl 17 . . . . . 6  |-  ( ph  ->  R  Fn  I )
291, 26, 10, 2, 3, 28prdsmgp 17779 . . . . 5  |-  ( ph  ->  ( ( Base `  (mulGrp `  Y ) )  =  ( Base `  ( S X_s (mulGrp  o.  R )
) )  /\  ( +g  `  (mulGrp `  Y
) )  =  ( +g  `  ( S
X_s (mulGrp  o.  R )
) ) ) )
3029simpld 460 . . . 4  |-  ( ph  ->  ( Base `  (mulGrp `  Y ) )  =  ( Base `  ( S X_s (mulGrp  o.  R )
) ) )
3129simprd 464 . . . . 5  |-  ( ph  ->  ( +g  `  (mulGrp `  Y ) )  =  ( +g  `  ( S X_s (mulGrp  o.  R )
) ) )
3231oveqdr 6320 . . . 4  |-  ( (
ph  /\  ( x  e.  ( Base `  (mulGrp `  Y ) )  /\  y  e.  ( Base `  (mulGrp `  Y )
) ) )  -> 
( x ( +g  `  (mulGrp `  Y )
) y )  =  ( x ( +g  `  ( S X_s (mulGrp  o.  R )
) ) y ) )
3325, 30, 32cmnpropd 17380 . . 3  |-  ( ph  ->  ( (mulGrp `  Y
)  e. CMnd  <->  ( S X_s (mulGrp  o.  R ) )  e. CMnd
) )
3424, 33mpbird 235 . 2  |-  ( ph  ->  (mulGrp `  Y )  e. CMnd )
3526iscrng 17728 . 2  |-  ( Y  e.  CRing 
<->  ( Y  e.  Ring  /\  (mulGrp `  Y )  e. CMnd ) )
369, 34, 35sylanbrc 668 1  |-  ( ph  ->  Y  e.  CRing )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1867   A.wral 2773   _Vcvv 3078    C_ wss 3433    |` cres 4847    o. ccom 4849    Fn wfn 5587   -->wf 5588   ` cfv 5592  (class class class)co 6296   Basecbs 15081   +g cplusg 15150   X_scprds 15304  CMndccmn 17371  mulGrpcmgp 17664   Ringcrg 17721   CRingccrg 17722
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-cnex 9584  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-pre-mulgt0 9605
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-int 4250  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-1o 7181  df-oadd 7185  df-er 7362  df-map 7473  df-ixp 7522  df-en 7569  df-dom 7570  df-sdom 7571  df-fin 7572  df-sup 7953  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-sub 9851  df-neg 9852  df-nn 10599  df-2 10657  df-3 10658  df-4 10659  df-5 10660  df-6 10661  df-7 10662  df-8 10663  df-9 10664  df-10 10665  df-n0 10859  df-z 10927  df-dec 11041  df-uz 11149  df-fz 11772  df-struct 15083  df-ndx 15084  df-slot 15085  df-base 15086  df-sets 15087  df-plusg 15163  df-mulr 15164  df-sca 15166  df-vsca 15167  df-ip 15168  df-tset 15169  df-ple 15170  df-ds 15172  df-hom 15174  df-cco 15175  df-0g 15300  df-prds 15306  df-mgm 16440  df-sgrp 16479  df-mnd 16489  df-grp 16625  df-minusg 16626  df-cmn 17373  df-mgp 17665  df-ring 17723  df-cring 17724
This theorem is referenced by:  pwscrng  17786
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