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Theorem prdscrngd 17134
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 17081 . . . . 5  |-  ( x  e.  CRing  ->  x  e.  Ring )
65ssriv 3513 . . . 4  |-  CRing  C_  Ring
7 fss 5745 . . . 4  |-  ( ( R : I --> CRing  /\  CRing  C_ 
Ring )  ->  R : I --> Ring )
84, 6, 7sylancl 662 . . 3  |-  ( ph  ->  R : I --> Ring )
91, 2, 3, 8prdsringd 17133 . 2  |-  ( ph  ->  Y  e.  Ring )
10 eqid 2467 . . . 4  |-  ( S
X_s (mulGrp  o.  R )
)  =  ( S
X_s (mulGrp  o.  R )
)
11 fnmgp 17015 . . . . . . 7  |- mulGrp  Fn  _V
12 ssv 3529 . . . . . . 7  |-  CRing  C_  _V
13 fnssres 5700 . . . . . . 7  |-  ( (mulGrp 
Fn  _V  /\  CRing  C_  _V )  ->  (mulGrp  |`  CRing )  Fn 
CRing )
1411, 12, 13mp2an 672 . . . . . 6  |-  (mulGrp  |`  CRing )  Fn 
CRing
15 fvres 5886 . . . . . . . 8  |-  ( x  e.  CRing  ->  ( (mulGrp  |` 
CRing ) `  x )  =  (mulGrp `  x
) )
16 eqid 2467 . . . . . . . . 9  |-  (mulGrp `  x )  =  (mulGrp `  x )
1716crngmgp 17078 . . . . . . . 8  |-  ( x  e.  CRing  ->  (mulGrp `  x
)  e. CMnd )
1815, 17eqeltrd 2555 . . . . . . 7  |-  ( x  e.  CRing  ->  ( (mulGrp  |` 
CRing ) `  x )  e. CMnd )
1918rgen 2827 . . . . . 6  |-  A. x  e.  CRing  ( (mulGrp  |`  CRing ) `  x )  e. CMnd
20 ffnfv 6058 . . . . . 6  |-  ( (mulGrp  |` 
CRing ) : CRing -->CMnd  <->  ( (mulGrp  |`  CRing )  Fn 
CRing  /\  A. x  e. 
CRing  ( (mulGrp  |`  CRing ) `  x )  e. CMnd )
)
2114, 19, 20mpbir2an 918 . . . . 5  |-  (mulGrp  |`  CRing ) :
CRing
-->CMnd
22 fco2 5748 . . . . 5  |-  ( ( (mulGrp  |`  CRing ) : CRing -->CMnd  /\  R : I --> CRing )  -> 
(mulGrp  o.  R ) : I -->CMnd )
2321, 4, 22sylancr 663 . . . 4  |-  ( ph  ->  (mulGrp  o.  R ) : I -->CMnd )
2410, 2, 3, 23prdscmnd 16740 . . 3  |-  ( ph  ->  ( S X_s (mulGrp  o.  R )
)  e. CMnd )
25 eqidd 2468 . . . 4  |-  ( ph  ->  ( Base `  (mulGrp `  Y ) )  =  ( Base `  (mulGrp `  Y ) ) )
26 eqid 2467 . . . . . 6  |-  (mulGrp `  Y )  =  (mulGrp `  Y )
27 ffn 5737 . . . . . . 7  |-  ( R : I --> CRing  ->  R  Fn  I )
284, 27syl 16 . . . . . 6  |-  ( ph  ->  R  Fn  I )
291, 26, 10, 2, 3, 28prdsmgp 17131 . . . . 5  |-  ( ph  ->  ( ( Base `  (mulGrp `  Y ) )  =  ( Base `  ( S X_s (mulGrp  o.  R )
) )  /\  ( +g  `  (mulGrp `  Y
) )  =  ( +g  `  ( S
X_s (mulGrp  o.  R )
) ) ) )
3029simpld 459 . . . 4  |-  ( ph  ->  ( Base `  (mulGrp `  Y ) )  =  ( Base `  ( S X_s (mulGrp  o.  R )
) ) )
3129simprd 463 . . . . 5  |-  ( ph  ->  ( +g  `  (mulGrp `  Y ) )  =  ( +g  `  ( S X_s (mulGrp  o.  R )
) ) )
3231proplem3 14963 . . . 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 16680 . . 3  |-  ( ph  ->  ( (mulGrp `  Y
)  e. CMnd  <->  ( S X_s (mulGrp  o.  R ) )  e. CMnd
) )
3424, 33mpbird 232 . 2  |-  ( ph  ->  (mulGrp `  Y )  e. CMnd )
3526iscrng 17077 . 2  |-  ( Y  e.  CRing 
<->  ( Y  e.  Ring  /\  (mulGrp `  Y )  e. CMnd ) )
369, 34, 35sylanbrc 664 1  |-  ( ph  ->  Y  e.  CRing )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2817   _Vcvv 3118    C_ wss 3481    |` cres 5007    o. ccom 5009    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6295   Basecbs 14507   +g cplusg 14572   X_scprds 14718  CMndccmn 16671  mulGrpcmgp 17013   Ringcrg 17070   CRingccrg 17071
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-ixp 7482  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-sup 7913  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-fz 11685  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-plusg 14585  df-mulr 14586  df-sca 14588  df-vsca 14589  df-ip 14590  df-tset 14591  df-ple 14592  df-ds 14594  df-hom 14596  df-cco 14597  df-0g 14714  df-prds 14720  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-grp 15929  df-minusg 15930  df-cmn 16673  df-mgp 17014  df-ring 17072  df-cring 17073
This theorem is referenced by:  pwscrng  17138
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