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Theorem prdsmgp 17131
Description: The multiplicative monoid of a product is the product of the multiplicative monoids of the factors. (Contributed by Mario Carneiro, 11-Mar-2015.)
Hypotheses
Ref Expression
prdsmgp.y  |-  Y  =  ( S X_s R )
prdsmgp.m  |-  M  =  (mulGrp `  Y )
prdsmgp.z  |-  Z  =  ( S X_s (mulGrp  o.  R )
)
prdsmgp.i  |-  ( ph  ->  I  e.  V )
prdsmgp.s  |-  ( ph  ->  S  e.  W )
prdsmgp.r  |-  ( ph  ->  R  Fn  I )
Assertion
Ref Expression
prdsmgp  |-  ( ph  ->  ( ( Base `  M
)  =  ( Base `  Z )  /\  ( +g  `  M )  =  ( +g  `  Z
) ) )

Proof of Theorem prdsmgp
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2467 . . . . . 6  |-  (mulGrp `  ( R `  x ) )  =  (mulGrp `  ( R `  x ) )
2 eqid 2467 . . . . . 6  |-  ( Base `  ( R `  x
) )  =  (
Base `  ( R `  x ) )
31, 2mgpbas 17019 . . . . 5  |-  ( Base `  ( R `  x
) )  =  (
Base `  (mulGrp `  ( R `  x )
) )
4 prdsmgp.r . . . . . . . 8  |-  ( ph  ->  R  Fn  I )
5 fvco2 5949 . . . . . . . 8  |-  ( ( R  Fn  I  /\  x  e.  I )  ->  ( (mulGrp  o.  R
) `  x )  =  (mulGrp `  ( R `  x ) ) )
64, 5sylan 471 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  (
(mulGrp  o.  R ) `  x )  =  (mulGrp `  ( R `  x
) ) )
76eqcomd 2475 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  (mulGrp `  ( R `  x
) )  =  ( (mulGrp  o.  R ) `  x ) )
87fveq2d 5876 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  ( Base `  (mulGrp `  ( R `  x )
) )  =  (
Base `  ( (mulGrp  o.  R ) `  x
) ) )
93, 8syl5eq 2520 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  ( Base `  ( R `  x ) )  =  ( Base `  (
(mulGrp  o.  R ) `  x ) ) )
109ixpeq2dva 7496 . . 3  |-  ( ph  -> 
X_ x  e.  I 
( Base `  ( R `  x ) )  = 
X_ x  e.  I 
( Base `  ( (mulGrp  o.  R ) `  x
) ) )
11 prdsmgp.y . . . 4  |-  Y  =  ( S X_s R )
12 prdsmgp.m . . . . . 6  |-  M  =  (mulGrp `  Y )
13 eqid 2467 . . . . . 6  |-  ( Base `  Y )  =  (
Base `  Y )
1412, 13mgpbas 17019 . . . . 5  |-  ( Base `  Y )  =  (
Base `  M )
1514eqcomi 2480 . . . 4  |-  ( Base `  M )  =  (
Base `  Y )
16 prdsmgp.s . . . 4  |-  ( ph  ->  S  e.  W )
17 prdsmgp.i . . . 4  |-  ( ph  ->  I  e.  V )
1811, 15, 16, 17, 4prdsbas2 14741 . . 3  |-  ( ph  ->  ( Base `  M
)  =  X_ x  e.  I  ( Base `  ( R `  x
) ) )
19 prdsmgp.z . . . 4  |-  Z  =  ( S X_s (mulGrp  o.  R )
)
20 eqid 2467 . . . 4  |-  ( Base `  Z )  =  (
Base `  Z )
21 fnmgp 17015 . . . . . 6  |- mulGrp  Fn  _V
2221a1i 11 . . . . 5  |-  ( ph  -> mulGrp 
Fn  _V )
23 ssv 3529 . . . . . 6  |-  ran  R  C_ 
_V
2423a1i 11 . . . . 5  |-  ( ph  ->  ran  R  C_  _V )
25 fnco 5695 . . . . 5  |-  ( (mulGrp 
Fn  _V  /\  R  Fn  I  /\  ran  R  C_  _V )  ->  (mulGrp  o.  R )  Fn  I
)
2622, 4, 24, 25syl3anc 1228 . . . 4  |-  ( ph  ->  (mulGrp  o.  R )  Fn  I )
2719, 20, 16, 17, 26prdsbas2 14741 . . 3  |-  ( ph  ->  ( Base `  Z
)  =  X_ x  e.  I  ( Base `  ( (mulGrp  o.  R
) `  x )
) )
2810, 18, 273eqtr4d 2518 . 2  |-  ( ph  ->  ( Base `  M
)  =  ( Base `  Z ) )
29 eqid 2467 . . . 4  |-  ( .r
`  Y )  =  ( .r `  Y
)
3012, 29mgpplusg 17017 . . 3  |-  ( .r
`  Y )  =  ( +g  `  M
)
31 eqid 2467 . . . . . . . . 9  |-  (mulGrp `  ( R `  z ) )  =  (mulGrp `  ( R `  z ) )
32 eqid 2467 . . . . . . . . 9  |-  ( .r
`  ( R `  z ) )  =  ( .r `  ( R `  z )
)
3331, 32mgpplusg 17017 . . . . . . . 8  |-  ( .r
`  ( R `  z ) )  =  ( +g  `  (mulGrp `  ( R `  z
) ) )
34 fvco2 5949 . . . . . . . . . . 11  |-  ( ( R  Fn  I  /\  z  e.  I )  ->  ( (mulGrp  o.  R
) `  z )  =  (mulGrp `  ( R `  z ) ) )
354, 34sylan 471 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  I )  ->  (
(mulGrp  o.  R ) `  z )  =  (mulGrp `  ( R `  z
) ) )
3635eqcomd 2475 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  I )  ->  (mulGrp `  ( R `  z
) )  =  ( (mulGrp  o.  R ) `  z ) )
3736fveq2d 5876 . . . . . . . 8  |-  ( (
ph  /\  z  e.  I )  ->  ( +g  `  (mulGrp `  ( R `  z )
) )  =  ( +g  `  ( (mulGrp 
o.  R ) `  z ) ) )
3833, 37syl5eq 2520 . . . . . . 7  |-  ( (
ph  /\  z  e.  I )  ->  ( .r `  ( R `  z ) )  =  ( +g  `  (
(mulGrp  o.  R ) `  z ) ) )
3938oveqd 6312 . . . . . 6  |-  ( (
ph  /\  z  e.  I )  ->  (
( x `  z
) ( .r `  ( R `  z ) ) ( y `  z ) )  =  ( ( x `  z ) ( +g  `  ( (mulGrp  o.  R
) `  z )
) ( y `  z ) ) )
4039mpteq2dva 4539 . . . . 5  |-  ( ph  ->  ( z  e.  I  |->  ( ( x `  z ) ( .r
`  ( R `  z ) ) ( y `  z ) ) )  =  ( z  e.  I  |->  ( ( x `  z
) ( +g  `  (
(mulGrp  o.  R ) `  z ) ) ( y `  z ) ) ) )
4128, 28, 40mpt2eq123dv 6354 . . . 4  |-  ( ph  ->  ( x  e.  (
Base `  M ) ,  y  e.  ( Base `  M )  |->  ( z  e.  I  |->  ( ( x `  z
) ( .r `  ( R `  z ) ) ( y `  z ) ) ) )  =  ( x  e.  ( Base `  Z
) ,  y  e.  ( Base `  Z
)  |->  ( z  e.  I  |->  ( ( x `
 z ) ( +g  `  ( (mulGrp 
o.  R ) `  z ) ) ( y `  z ) ) ) ) )
42 fnex 6138 . . . . . 6  |-  ( ( R  Fn  I  /\  I  e.  V )  ->  R  e.  _V )
434, 17, 42syl2anc 661 . . . . 5  |-  ( ph  ->  R  e.  _V )
44 fndm 5686 . . . . . 6  |-  ( R  Fn  I  ->  dom  R  =  I )
454, 44syl 16 . . . . 5  |-  ( ph  ->  dom  R  =  I )
4611, 16, 43, 15, 45, 29prdsmulr 14731 . . . 4  |-  ( ph  ->  ( .r `  Y
)  =  ( x  e.  ( Base `  M
) ,  y  e.  ( Base `  M
)  |->  ( z  e.  I  |->  ( ( x `
 z ) ( .r `  ( R `
 z ) ) ( y `  z
) ) ) ) )
47 fnex 6138 . . . . . 6  |-  ( ( (mulGrp  o.  R )  Fn  I  /\  I  e.  V )  ->  (mulGrp  o.  R )  e.  _V )
4826, 17, 47syl2anc 661 . . . . 5  |-  ( ph  ->  (mulGrp  o.  R )  e.  _V )
49 fndm 5686 . . . . . 6  |-  ( (mulGrp 
o.  R )  Fn  I  ->  dom  (mulGrp  o.  R )  =  I )
5026, 49syl 16 . . . . 5  |-  ( ph  ->  dom  (mulGrp  o.  R
)  =  I )
51 eqid 2467 . . . . 5  |-  ( +g  `  Z )  =  ( +g  `  Z )
5219, 16, 48, 20, 50, 51prdsplusg 14730 . . . 4  |-  ( ph  ->  ( +g  `  Z
)  =  ( x  e.  ( Base `  Z
) ,  y  e.  ( Base `  Z
)  |->  ( z  e.  I  |->  ( ( x `
 z ) ( +g  `  ( (mulGrp 
o.  R ) `  z ) ) ( y `  z ) ) ) ) )
5341, 46, 523eqtr4d 2518 . . 3  |-  ( ph  ->  ( .r `  Y
)  =  ( +g  `  Z ) )
5430, 53syl5eqr 2522 . 2  |-  ( ph  ->  ( +g  `  M
)  =  ( +g  `  Z ) )
5528, 54jca 532 1  |-  ( ph  ->  ( ( Base `  M
)  =  ( Base `  Z )  /\  ( +g  `  M )  =  ( +g  `  Z
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3118    C_ wss 3481    |-> cmpt 4511   dom cdm 5005   ran crn 5006    o. ccom 5009    Fn wfn 5589   ` cfv 5594  (class class class)co 6295    |-> cmpt2 6297   X_cixp 7481   Basecbs 14507   +g cplusg 14572   .rcmulr 14573   X_scprds 14718  mulGrpcmgp 17013
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-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-prds 14720  df-mgp 17014
This theorem is referenced by:  prdsringd  17133  prdscrngd  17134  prds1  17135  pwsmgp  17139
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