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Theorem prdsmgp 15718
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 2438 . . . . . 6  |-  (mulGrp `  ( R `  x ) )  =  (mulGrp `  ( R `  x ) )
2 eqid 2438 . . . . . 6  |-  ( Base `  ( R `  x
) )  =  (
Base `  ( R `  x ) )
31, 2mgpbas 15656 . . . . 5  |-  ( Base `  ( R `  x
) )  =  (
Base `  (mulGrp `  ( R `  x )
) )
4 prdsmgp.r . . . . . . . 8  |-  ( ph  ->  R  Fn  I )
5 fvco2 5800 . . . . . . . 8  |-  ( ( R  Fn  I  /\  x  e.  I )  ->  ( (mulGrp  o.  R
) `  x )  =  (mulGrp `  ( R `  x ) ) )
64, 5sylan 459 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  (
(mulGrp  o.  R ) `  x )  =  (mulGrp `  ( R `  x
) ) )
76eqcomd 2443 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  (mulGrp `  ( R `  x
) )  =  ( (mulGrp  o.  R ) `  x ) )
87fveq2d 5734 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  ( Base `  (mulGrp `  ( R `  x )
) )  =  (
Base `  ( (mulGrp  o.  R ) `  x
) ) )
93, 8syl5eq 2482 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  ( Base `  ( R `  x ) )  =  ( Base `  (
(mulGrp  o.  R ) `  x ) ) )
109ixpeq2dva 7079 . . 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 2438 . . . . . 6  |-  ( Base `  Y )  =  (
Base `  Y )
1412, 13mgpbas 15656 . . . . 5  |-  ( Base `  Y )  =  (
Base `  M )
1514eqcomi 2442 . . . 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 13693 . . 3  |-  ( ph  ->  ( Base `  M
)  =  X_ x  e.  I  ( Base `  ( R `  x
) ) )
19 prdsmgp.z . . . 4  |-  Z  =  ( S X_s (mulGrp  o.  R )
)
20 eqid 2438 . . . 4  |-  ( Base `  Z )  =  (
Base `  Z )
21 fnmgp 15652 . . . . . 6  |- mulGrp  Fn  _V
2221a1i 11 . . . . 5  |-  ( ph  -> mulGrp 
Fn  _V )
23 ssv 3370 . . . . . 6  |-  ran  R  C_ 
_V
2423a1i 11 . . . . 5  |-  ( ph  ->  ran  R  C_  _V )
25 fnco 5555 . . . . 5  |-  ( (mulGrp 
Fn  _V  /\  R  Fn  I  /\  ran  R  C_  _V )  ->  (mulGrp  o.  R )  Fn  I
)
2622, 4, 24, 25syl3anc 1185 . . . 4  |-  ( ph  ->  (mulGrp  o.  R )  Fn  I )
2719, 20, 16, 17, 26prdsbas2 13693 . . 3  |-  ( ph  ->  ( Base `  Z
)  =  X_ x  e.  I  ( Base `  ( (mulGrp  o.  R
) `  x )
) )
2810, 18, 273eqtr4d 2480 . 2  |-  ( ph  ->  ( Base `  M
)  =  ( Base `  Z ) )
29 eqid 2438 . . . 4  |-  ( .r
`  Y )  =  ( .r `  Y
)
3012, 29mgpplusg 15654 . . 3  |-  ( .r
`  Y )  =  ( +g  `  M
)
31 eqid 2438 . . . . . . . . 9  |-  (mulGrp `  ( R `  z ) )  =  (mulGrp `  ( R `  z ) )
32 eqid 2438 . . . . . . . . 9  |-  ( .r
`  ( R `  z ) )  =  ( .r `  ( R `  z )
)
3331, 32mgpplusg 15654 . . . . . . . 8  |-  ( .r
`  ( R `  z ) )  =  ( +g  `  (mulGrp `  ( R `  z
) ) )
34 fvco2 5800 . . . . . . . . . . 11  |-  ( ( R  Fn  I  /\  z  e.  I )  ->  ( (mulGrp  o.  R
) `  z )  =  (mulGrp `  ( R `  z ) ) )
354, 34sylan 459 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  I )  ->  (
(mulGrp  o.  R ) `  z )  =  (mulGrp `  ( R `  z
) ) )
3635eqcomd 2443 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  I )  ->  (mulGrp `  ( R `  z
) )  =  ( (mulGrp  o.  R ) `  z ) )
3736fveq2d 5734 . . . . . . . 8  |-  ( (
ph  /\  z  e.  I )  ->  ( +g  `  (mulGrp `  ( R `  z )
) )  =  ( +g  `  ( (mulGrp 
o.  R ) `  z ) ) )
3833, 37syl5eq 2482 . . . . . . 7  |-  ( (
ph  /\  z  e.  I )  ->  ( .r `  ( R `  z ) )  =  ( +g  `  (
(mulGrp  o.  R ) `  z ) ) )
3938oveqd 6100 . . . . . 6  |-  ( (
ph  /\  z  e.  I )  ->  (
( x `  z
) ( .r `  ( R `  z ) ) ( y `  z ) )  =  ( ( x `  z ) ( +g  `  ( (mulGrp  o.  R
) `  z )
) ( y `  z ) ) )
4039mpteq2dva 4297 . . . . 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 6138 . . . 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 5963 . . . . . 6  |-  ( ( R  Fn  I  /\  I  e.  V )  ->  R  e.  _V )
434, 17, 42syl2anc 644 . . . . 5  |-  ( ph  ->  R  e.  _V )
44 fndm 5546 . . . . . 6  |-  ( R  Fn  I  ->  dom  R  =  I )
454, 44syl 16 . . . . 5  |-  ( ph  ->  dom  R  =  I )
4611, 16, 43, 15, 45, 29prdsmulr 13684 . . . 4  |-  ( ph  ->  ( .r `  Y
)  =  ( x  e.  ( Base `  M
) ,  y  e.  ( Base `  M
)  |->  ( z  e.  I  |->  ( ( x `
 z ) ( .r `  ( R `
 z ) ) ( y `  z
) ) ) ) )
47 fnex 5963 . . . . . 6  |-  ( ( (mulGrp  o.  R )  Fn  I  /\  I  e.  V )  ->  (mulGrp  o.  R )  e.  _V )
4826, 17, 47syl2anc 644 . . . . 5  |-  ( ph  ->  (mulGrp  o.  R )  e.  _V )
49 fndm 5546 . . . . . 6  |-  ( (mulGrp 
o.  R )  Fn  I  ->  dom  (mulGrp  o.  R )  =  I )
5026, 49syl 16 . . . . 5  |-  ( ph  ->  dom  (mulGrp  o.  R
)  =  I )
51 eqid 2438 . . . . 5  |-  ( +g  `  Z )  =  ( +g  `  Z )
5219, 16, 48, 20, 50, 51prdsplusg 13683 . . . 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 2480 . . 3  |-  ( ph  ->  ( .r `  Y
)  =  ( +g  `  Z ) )
5430, 53syl5eqr 2484 . 2  |-  ( ph  ->  ( +g  `  M
)  =  ( +g  `  Z ) )
5528, 54jca 520 1  |-  ( ph  ->  ( ( Base `  M
)  =  ( Base `  Z )  /\  ( +g  `  M )  =  ( +g  `  Z
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   _Vcvv 2958    C_ wss 3322    e. cmpt 4268   dom cdm 4880   ran crn 4881    o. ccom 4884    Fn wfn 5451   ` cfv 5456  (class class class)co 6083    e. cmpt2 6085   X_cixp 7065   Basecbs 13471   +g cplusg 13531   .rcmulr 13532   X_scprds 13671  mulGrpcmgp 15650
This theorem is referenced by:  prdsrngd  15720  prdscrngd  15721  prds1  15722  pwsmgp  15726
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-oadd 6730  df-er 6907  df-map 7022  df-ixp 7066  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-sup 7448  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-2 10060  df-3 10061  df-4 10062  df-5 10063  df-6 10064  df-7 10065  df-8 10066  df-9 10067  df-10 10068  df-n0 10224  df-z 10285  df-dec 10385  df-uz 10491  df-fz 11046  df-struct 13473  df-ndx 13474  df-slot 13475  df-base 13476  df-sets 13477  df-plusg 13544  df-mulr 13545  df-sca 13547  df-vsca 13548  df-tset 13550  df-ple 13551  df-ds 13553  df-hom 13555  df-cco 13556  df-prds 13673  df-mgp 15651
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