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Theorem prdsmgp 16707
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 2443 . . . . . 6  |-  (mulGrp `  ( R `  x ) )  =  (mulGrp `  ( R `  x ) )
2 eqid 2443 . . . . . 6  |-  ( Base `  ( R `  x
) )  =  (
Base `  ( R `  x ) )
31, 2mgpbas 16602 . . . . 5  |-  ( Base `  ( R `  x
) )  =  (
Base `  (mulGrp `  ( R `  x )
) )
4 prdsmgp.r . . . . . . . 8  |-  ( ph  ->  R  Fn  I )
5 fvco2 5771 . . . . . . . 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 2448 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  (mulGrp `  ( R `  x
) )  =  ( (mulGrp  o.  R ) `  x ) )
87fveq2d 5700 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  ( Base `  (mulGrp `  ( R `  x )
) )  =  (
Base `  ( (mulGrp  o.  R ) `  x
) ) )
93, 8syl5eq 2487 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  ( Base `  ( R `  x ) )  =  ( Base `  (
(mulGrp  o.  R ) `  x ) ) )
109ixpeq2dva 7283 . . 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 2443 . . . . . 6  |-  ( Base `  Y )  =  (
Base `  Y )
1412, 13mgpbas 16602 . . . . 5  |-  ( Base `  Y )  =  (
Base `  M )
1514eqcomi 2447 . . . 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 14412 . . 3  |-  ( ph  ->  ( Base `  M
)  =  X_ x  e.  I  ( Base `  ( R `  x
) ) )
19 prdsmgp.z . . . 4  |-  Z  =  ( S X_s (mulGrp  o.  R )
)
20 eqid 2443 . . . 4  |-  ( Base `  Z )  =  (
Base `  Z )
21 fnmgp 16598 . . . . . 6  |- mulGrp  Fn  _V
2221a1i 11 . . . . 5  |-  ( ph  -> mulGrp 
Fn  _V )
23 ssv 3381 . . . . . 6  |-  ran  R  C_ 
_V
2423a1i 11 . . . . 5  |-  ( ph  ->  ran  R  C_  _V )
25 fnco 5524 . . . . 5  |-  ( (mulGrp 
Fn  _V  /\  R  Fn  I  /\  ran  R  C_  _V )  ->  (mulGrp  o.  R )  Fn  I
)
2622, 4, 24, 25syl3anc 1218 . . . 4  |-  ( ph  ->  (mulGrp  o.  R )  Fn  I )
2719, 20, 16, 17, 26prdsbas2 14412 . . 3  |-  ( ph  ->  ( Base `  Z
)  =  X_ x  e.  I  ( Base `  ( (mulGrp  o.  R
) `  x )
) )
2810, 18, 273eqtr4d 2485 . 2  |-  ( ph  ->  ( Base `  M
)  =  ( Base `  Z ) )
29 eqid 2443 . . . 4  |-  ( .r
`  Y )  =  ( .r `  Y
)
3012, 29mgpplusg 16600 . . 3  |-  ( .r
`  Y )  =  ( +g  `  M
)
31 eqid 2443 . . . . . . . . 9  |-  (mulGrp `  ( R `  z ) )  =  (mulGrp `  ( R `  z ) )
32 eqid 2443 . . . . . . . . 9  |-  ( .r
`  ( R `  z ) )  =  ( .r `  ( R `  z )
)
3331, 32mgpplusg 16600 . . . . . . . 8  |-  ( .r
`  ( R `  z ) )  =  ( +g  `  (mulGrp `  ( R `  z
) ) )
34 fvco2 5771 . . . . . . . . . . 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 2448 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  I )  ->  (mulGrp `  ( R `  z
) )  =  ( (mulGrp  o.  R ) `  z ) )
3736fveq2d 5700 . . . . . . . 8  |-  ( (
ph  /\  z  e.  I )  ->  ( +g  `  (mulGrp `  ( R `  z )
) )  =  ( +g  `  ( (mulGrp 
o.  R ) `  z ) ) )
3833, 37syl5eq 2487 . . . . . . 7  |-  ( (
ph  /\  z  e.  I )  ->  ( .r `  ( R `  z ) )  =  ( +g  `  (
(mulGrp  o.  R ) `  z ) ) )
3938oveqd 6113 . . . . . 6  |-  ( (
ph  /\  z  e.  I )  ->  (
( x `  z
) ( .r `  ( R `  z ) ) ( y `  z ) )  =  ( ( x `  z ) ( +g  `  ( (mulGrp  o.  R
) `  z )
) ( y `  z ) ) )
4039mpteq2dva 4383 . . . . 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 6153 . . . 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 5949 . . . . . 6  |-  ( ( R  Fn  I  /\  I  e.  V )  ->  R  e.  _V )
434, 17, 42syl2anc 661 . . . . 5  |-  ( ph  ->  R  e.  _V )
44 fndm 5515 . . . . . 6  |-  ( R  Fn  I  ->  dom  R  =  I )
454, 44syl 16 . . . . 5  |-  ( ph  ->  dom  R  =  I )
4611, 16, 43, 15, 45, 29prdsmulr 14402 . . . 4  |-  ( ph  ->  ( .r `  Y
)  =  ( x  e.  ( Base `  M
) ,  y  e.  ( Base `  M
)  |->  ( z  e.  I  |->  ( ( x `
 z ) ( .r `  ( R `
 z ) ) ( y `  z
) ) ) ) )
47 fnex 5949 . . . . . 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 5515 . . . . . 6  |-  ( (mulGrp 
o.  R )  Fn  I  ->  dom  (mulGrp  o.  R )  =  I )
5026, 49syl 16 . . . . 5  |-  ( ph  ->  dom  (mulGrp  o.  R
)  =  I )
51 eqid 2443 . . . . 5  |-  ( +g  `  Z )  =  ( +g  `  Z )
5219, 16, 48, 20, 50, 51prdsplusg 14401 . . . 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 2485 . . 3  |-  ( ph  ->  ( .r `  Y
)  =  ( +g  `  Z ) )
5430, 53syl5eqr 2489 . 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 1369    e. wcel 1756   _Vcvv 2977    C_ wss 3333    e. cmpt 4355   dom cdm 4845   ran crn 4846    o. ccom 4849    Fn wfn 5418   ` cfv 5423  (class class class)co 6096    e. cmpt2 6098   X_cixp 7268   Basecbs 14179   +g cplusg 14243   .rcmulr 14244   X_scprds 14389  mulGrpcmgp 16596
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-map 7221  df-ixp 7269  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-sup 7696  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-2 10385  df-3 10386  df-4 10387  df-5 10388  df-6 10389  df-7 10390  df-8 10391  df-9 10392  df-10 10393  df-n0 10585  df-z 10652  df-dec 10761  df-uz 10867  df-fz 11443  df-struct 14181  df-ndx 14182  df-slot 14183  df-base 14184  df-sets 14185  df-plusg 14256  df-mulr 14257  df-sca 14259  df-vsca 14260  df-ip 14261  df-tset 14262  df-ple 14263  df-ds 14265  df-hom 14267  df-cco 14268  df-prds 14391  df-mgp 16597
This theorem is referenced by:  prdsrngd  16709  prdscrngd  16710  prds1  16711  pwsmgp  16715
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