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Theorem prdscrngd 15396
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 crngrng 15351 . . . . 5  |-  ( x  e.  CRing  ->  x  e.  Ring )
65ssriv 3184 . . . 4  |-  CRing  C_  Ring
7 fss 5397 . . . 4  |-  ( ( R : I --> CRing  /\  CRing  C_ 
Ring )  ->  R : I --> Ring )
84, 6, 7sylancl 643 . . 3  |-  ( ph  ->  R : I --> Ring )
91, 2, 3, 8prdsrngd 15395 . 2  |-  ( ph  ->  Y  e.  Ring )
10 eqid 2283 . . . 4  |-  ( S
X_s (mulGrp  o.  R )
)  =  ( S
X_s (mulGrp  o.  R )
)
11 fnmgp 15327 . . . . . . 7  |- mulGrp  Fn  _V
12 ssv 3198 . . . . . . 7  |-  CRing  C_  _V
13 fnssres 5357 . . . . . . 7  |-  ( (mulGrp 
Fn  _V  /\  CRing  C_  _V )  ->  (mulGrp  |`  CRing )  Fn 
CRing )
1411, 12, 13mp2an 653 . . . . . 6  |-  (mulGrp  |`  CRing )  Fn 
CRing
15 fvres 5542 . . . . . . . 8  |-  ( x  e.  CRing  ->  ( (mulGrp  |` 
CRing ) `  x )  =  (mulGrp `  x
) )
16 eqid 2283 . . . . . . . . 9  |-  (mulGrp `  x )  =  (mulGrp `  x )
1716crngmgp 15349 . . . . . . . 8  |-  ( x  e.  CRing  ->  (mulGrp `  x
)  e. CMnd )
1815, 17eqeltrd 2357 . . . . . . 7  |-  ( x  e.  CRing  ->  ( (mulGrp  |` 
CRing ) `  x )  e. CMnd )
1918rgen 2608 . . . . . 6  |-  A. x  e.  CRing  ( (mulGrp  |`  CRing ) `  x )  e. CMnd
20 ffnfv 5685 . . . . . 6  |-  ( (mulGrp  |` 
CRing ) : CRing -->CMnd  <->  ( (mulGrp  |`  CRing )  Fn 
CRing  /\  A. x  e. 
CRing  ( (mulGrp  |`  CRing ) `  x )  e. CMnd )
)
2114, 19, 20mpbir2an 886 . . . . 5  |-  (mulGrp  |`  CRing ) :
CRing
-->CMnd
22 fco2 5399 . . . . 5  |-  ( ( (mulGrp  |`  CRing ) : CRing -->CMnd  /\  R : I --> CRing )  -> 
(mulGrp  o.  R ) : I -->CMnd )
2321, 4, 22sylancr 644 . . . 4  |-  ( ph  ->  (mulGrp  o.  R ) : I -->CMnd )
2410, 2, 3, 23prdscmnd 15153 . . 3  |-  ( ph  ->  ( S X_s (mulGrp  o.  R )
)  e. CMnd )
25 eqidd 2284 . . . 4  |-  ( ph  ->  ( Base `  (mulGrp `  Y ) )  =  ( Base `  (mulGrp `  Y ) ) )
26 eqid 2283 . . . . . 6  |-  (mulGrp `  Y )  =  (mulGrp `  Y )
27 ffn 5389 . . . . . . 7  |-  ( R : I --> CRing  ->  R  Fn  I )
284, 27syl 15 . . . . . 6  |-  ( ph  ->  R  Fn  I )
291, 26, 10, 2, 3, 28prdsmgp 15393 . . . . 5  |-  ( ph  ->  ( ( Base `  (mulGrp `  Y ) )  =  ( Base `  ( S X_s (mulGrp  o.  R )
) )  /\  ( +g  `  (mulGrp `  Y
) )  =  ( +g  `  ( S
X_s (mulGrp  o.  R )
) ) ) )
3029simpld 445 . . . 4  |-  ( ph  ->  ( Base `  (mulGrp `  Y ) )  =  ( Base `  ( S X_s (mulGrp  o.  R )
) ) )
3129simprd 449 . . . . 5  |-  ( ph  ->  ( +g  `  (mulGrp `  Y ) )  =  ( +g  `  ( S X_s (mulGrp  o.  R )
) ) )
3231proplem3 13593 . . . 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 15098 . . 3  |-  ( ph  ->  ( (mulGrp `  Y
)  e. CMnd  <->  ( S X_s (mulGrp  o.  R ) )  e. CMnd
) )
3424, 33mpbird 223 . 2  |-  ( ph  ->  (mulGrp `  Y )  e. CMnd )
3526iscrng 15348 . 2  |-  ( Y  e.  CRing 
<->  ( Y  e.  Ring  /\  (mulGrp `  Y )  e. CMnd ) )
369, 34, 35sylanbrc 645 1  |-  ( ph  ->  Y  e.  CRing )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788    C_ wss 3152    |` cres 4691    o. ccom 4693    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   X_scprds 13346  CMndccmn 15089  mulGrpcmgp 15325   Ringcrg 15337   CRingccrg 15338
This theorem is referenced by:  pwscrng  15400
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-prds 13348  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-cmn 15091  df-mgp 15326  df-rng 15340  df-cring 15341
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