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Theorem prdsmgp 16692
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 2441 . . . . . 6  |-  (mulGrp `  ( R `  x ) )  =  (mulGrp `  ( R `  x ) )
2 eqid 2441 . . . . . 6  |-  ( Base `  ( R `  x
) )  =  (
Base `  ( R `  x ) )
31, 2mgpbas 16587 . . . . 5  |-  ( Base `  ( R `  x
) )  =  (
Base `  (mulGrp `  ( R `  x )
) )
4 prdsmgp.r . . . . . . . 8  |-  ( ph  ->  R  Fn  I )
5 fvco2 5763 . . . . . . . 8  |-  ( ( R  Fn  I  /\  x  e.  I )  ->  ( (mulGrp  o.  R
) `  x )  =  (mulGrp `  ( R `  x ) ) )
64, 5sylan 468 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  (
(mulGrp  o.  R ) `  x )  =  (mulGrp `  ( R `  x
) ) )
76eqcomd 2446 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  (mulGrp `  ( R `  x
) )  =  ( (mulGrp  o.  R ) `  x ) )
87fveq2d 5692 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  ( Base `  (mulGrp `  ( R `  x )
) )  =  (
Base `  ( (mulGrp  o.  R ) `  x
) ) )
93, 8syl5eq 2485 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  ( Base `  ( R `  x ) )  =  ( Base `  (
(mulGrp  o.  R ) `  x ) ) )
109ixpeq2dva 7274 . . 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 2441 . . . . . 6  |-  ( Base `  Y )  =  (
Base `  Y )
1412, 13mgpbas 16587 . . . . 5  |-  ( Base `  Y )  =  (
Base `  M )
1514eqcomi 2445 . . . 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 14403 . . 3  |-  ( ph  ->  ( Base `  M
)  =  X_ x  e.  I  ( Base `  ( R `  x
) ) )
19 prdsmgp.z . . . 4  |-  Z  =  ( S X_s (mulGrp  o.  R )
)
20 eqid 2441 . . . 4  |-  ( Base `  Z )  =  (
Base `  Z )
21 fnmgp 16583 . . . . . 6  |- mulGrp  Fn  _V
2221a1i 11 . . . . 5  |-  ( ph  -> mulGrp 
Fn  _V )
23 ssv 3373 . . . . . 6  |-  ran  R  C_ 
_V
2423a1i 11 . . . . 5  |-  ( ph  ->  ran  R  C_  _V )
25 fnco 5516 . . . . 5  |-  ( (mulGrp 
Fn  _V  /\  R  Fn  I  /\  ran  R  C_  _V )  ->  (mulGrp  o.  R )  Fn  I
)
2622, 4, 24, 25syl3anc 1213 . . . 4  |-  ( ph  ->  (mulGrp  o.  R )  Fn  I )
2719, 20, 16, 17, 26prdsbas2 14403 . . 3  |-  ( ph  ->  ( Base `  Z
)  =  X_ x  e.  I  ( Base `  ( (mulGrp  o.  R
) `  x )
) )
2810, 18, 273eqtr4d 2483 . 2  |-  ( ph  ->  ( Base `  M
)  =  ( Base `  Z ) )
29 eqid 2441 . . . 4  |-  ( .r
`  Y )  =  ( .r `  Y
)
3012, 29mgpplusg 16585 . . 3  |-  ( .r
`  Y )  =  ( +g  `  M
)
31 eqid 2441 . . . . . . . . 9  |-  (mulGrp `  ( R `  z ) )  =  (mulGrp `  ( R `  z ) )
32 eqid 2441 . . . . . . . . 9  |-  ( .r
`  ( R `  z ) )  =  ( .r `  ( R `  z )
)
3331, 32mgpplusg 16585 . . . . . . . 8  |-  ( .r
`  ( R `  z ) )  =  ( +g  `  (mulGrp `  ( R `  z
) ) )
34 fvco2 5763 . . . . . . . . . . 11  |-  ( ( R  Fn  I  /\  z  e.  I )  ->  ( (mulGrp  o.  R
) `  z )  =  (mulGrp `  ( R `  z ) ) )
354, 34sylan 468 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  I )  ->  (
(mulGrp  o.  R ) `  z )  =  (mulGrp `  ( R `  z
) ) )
3635eqcomd 2446 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  I )  ->  (mulGrp `  ( R `  z
) )  =  ( (mulGrp  o.  R ) `  z ) )
3736fveq2d 5692 . . . . . . . 8  |-  ( (
ph  /\  z  e.  I )  ->  ( +g  `  (mulGrp `  ( R `  z )
) )  =  ( +g  `  ( (mulGrp 
o.  R ) `  z ) ) )
3833, 37syl5eq 2485 . . . . . . 7  |-  ( (
ph  /\  z  e.  I )  ->  ( .r `  ( R `  z ) )  =  ( +g  `  (
(mulGrp  o.  R ) `  z ) ) )
3938oveqd 6107 . . . . . 6  |-  ( (
ph  /\  z  e.  I )  ->  (
( x `  z
) ( .r `  ( R `  z ) ) ( y `  z ) )  =  ( ( x `  z ) ( +g  `  ( (mulGrp  o.  R
) `  z )
) ( y `  z ) ) )
4039mpteq2dva 4375 . . . . 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 6147 . . . 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 5941 . . . . . 6  |-  ( ( R  Fn  I  /\  I  e.  V )  ->  R  e.  _V )
434, 17, 42syl2anc 656 . . . . 5  |-  ( ph  ->  R  e.  _V )
44 fndm 5507 . . . . . 6  |-  ( R  Fn  I  ->  dom  R  =  I )
454, 44syl 16 . . . . 5  |-  ( ph  ->  dom  R  =  I )
4611, 16, 43, 15, 45, 29prdsmulr 14393 . . . 4  |-  ( ph  ->  ( .r `  Y
)  =  ( x  e.  ( Base `  M
) ,  y  e.  ( Base `  M
)  |->  ( z  e.  I  |->  ( ( x `
 z ) ( .r `  ( R `
 z ) ) ( y `  z
) ) ) ) )
47 fnex 5941 . . . . . 6  |-  ( ( (mulGrp  o.  R )  Fn  I  /\  I  e.  V )  ->  (mulGrp  o.  R )  e.  _V )
4826, 17, 47syl2anc 656 . . . . 5  |-  ( ph  ->  (mulGrp  o.  R )  e.  _V )
49 fndm 5507 . . . . . 6  |-  ( (mulGrp 
o.  R )  Fn  I  ->  dom  (mulGrp  o.  R )  =  I )
5026, 49syl 16 . . . . 5  |-  ( ph  ->  dom  (mulGrp  o.  R
)  =  I )
51 eqid 2441 . . . . 5  |-  ( +g  `  Z )  =  ( +g  `  Z )
5219, 16, 48, 20, 50, 51prdsplusg 14392 . . . 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 2483 . . 3  |-  ( ph  ->  ( .r `  Y
)  =  ( +g  `  Z ) )
5430, 53syl5eqr 2487 . 2  |-  ( ph  ->  ( +g  `  M
)  =  ( +g  `  Z ) )
5528, 54jca 529 1  |-  ( ph  ->  ( ( Base `  M
)  =  ( Base `  Z )  /\  ( +g  `  M )  =  ( +g  `  Z
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1364    e. wcel 1761   _Vcvv 2970    C_ wss 3325    e. cmpt 4347   dom cdm 4836   ran crn 4837    o. ccom 4840    Fn wfn 5410   ` cfv 5415  (class class class)co 6090    e. cmpt2 6092   X_cixp 7259   Basecbs 14170   +g cplusg 14234   .rcmulr 14235   X_scprds 14380  mulGrpcmgp 16581
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-map 7212  df-ixp 7260  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-sup 7687  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-fz 11434  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-plusg 14247  df-mulr 14248  df-sca 14250  df-vsca 14251  df-ip 14252  df-tset 14253  df-ple 14254  df-ds 14256  df-hom 14258  df-cco 14259  df-prds 14382  df-mgp 16582
This theorem is referenced by:  prdsrngd  16694  prdscrngd  16695  prds1  16696  pwsmgp  16700
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