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Theorem prdsmgp 17238
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 17126 . . . . 5  |-  ( Base `  ( R `  x
) )  =  (
Base `  (mulGrp `  ( R `  x )
) )
4 prdsmgp.r . . . . . . . 8  |-  ( ph  ->  R  Fn  I )
5 fvco2 5933 . . . . . . . 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 2451 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  (mulGrp `  ( R `  x
) )  =  ( (mulGrp  o.  R ) `  x ) )
87fveq2d 5860 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  ( Base `  (mulGrp `  ( R `  x )
) )  =  (
Base `  ( (mulGrp  o.  R ) `  x
) ) )
93, 8syl5eq 2496 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  ( Base `  ( R `  x ) )  =  ( Base `  (
(mulGrp  o.  R ) `  x ) ) )
109ixpeq2dva 7486 . . 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 17126 . . . . 5  |-  ( Base `  Y )  =  (
Base `  M )
1514eqcomi 2456 . . . 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 14848 . . 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 17122 . . . . . 6  |- mulGrp  Fn  _V
2221a1i 11 . . . . 5  |-  ( ph  -> mulGrp 
Fn  _V )
23 ssv 3509 . . . . . 6  |-  ran  R  C_ 
_V
2423a1i 11 . . . . 5  |-  ( ph  ->  ran  R  C_  _V )
25 fnco 5679 . . . . 5  |-  ( (mulGrp 
Fn  _V  /\  R  Fn  I  /\  ran  R  C_  _V )  ->  (mulGrp  o.  R )  Fn  I
)
2622, 4, 24, 25syl3anc 1229 . . . 4  |-  ( ph  ->  (mulGrp  o.  R )  Fn  I )
2719, 20, 16, 17, 26prdsbas2 14848 . . 3  |-  ( ph  ->  ( Base `  Z
)  =  X_ x  e.  I  ( Base `  ( (mulGrp  o.  R
) `  x )
) )
2810, 18, 273eqtr4d 2494 . 2  |-  ( ph  ->  ( Base `  M
)  =  ( Base `  Z ) )
29 eqid 2443 . . . 4  |-  ( .r
`  Y )  =  ( .r `  Y
)
3012, 29mgpplusg 17124 . . 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 17124 . . . . . . . 8  |-  ( .r
`  ( R `  z ) )  =  ( +g  `  (mulGrp `  ( R `  z
) ) )
34 fvco2 5933 . . . . . . . . . . 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 2451 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  I )  ->  (mulGrp `  ( R `  z
) )  =  ( (mulGrp  o.  R ) `  z ) )
3736fveq2d 5860 . . . . . . . 8  |-  ( (
ph  /\  z  e.  I )  ->  ( +g  `  (mulGrp `  ( R `  z )
) )  =  ( +g  `  ( (mulGrp 
o.  R ) `  z ) ) )
3833, 37syl5eq 2496 . . . . . . 7  |-  ( (
ph  /\  z  e.  I )  ->  ( .r `  ( R `  z ) )  =  ( +g  `  (
(mulGrp  o.  R ) `  z ) ) )
3938oveqd 6298 . . . . . 6  |-  ( (
ph  /\  z  e.  I )  ->  (
( x `  z
) ( .r `  ( R `  z ) ) ( y `  z ) )  =  ( ( x `  z ) ( +g  `  ( (mulGrp  o.  R
) `  z )
) ( y `  z ) ) )
4039mpteq2dva 4523 . . . . 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 6344 . . . 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 6124 . . . . . 6  |-  ( ( R  Fn  I  /\  I  e.  V )  ->  R  e.  _V )
434, 17, 42syl2anc 661 . . . . 5  |-  ( ph  ->  R  e.  _V )
44 fndm 5670 . . . . . 6  |-  ( R  Fn  I  ->  dom  R  =  I )
454, 44syl 16 . . . . 5  |-  ( ph  ->  dom  R  =  I )
4611, 16, 43, 15, 45, 29prdsmulr 14838 . . . 4  |-  ( ph  ->  ( .r `  Y
)  =  ( x  e.  ( Base `  M
) ,  y  e.  ( Base `  M
)  |->  ( z  e.  I  |->  ( ( x `
 z ) ( .r `  ( R `
 z ) ) ( y `  z
) ) ) ) )
47 fnex 6124 . . . . . 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 5670 . . . . . 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 14837 . . . 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 2494 . . 3  |-  ( ph  ->  ( .r `  Y
)  =  ( +g  `  Z ) )
5430, 53syl5eqr 2498 . 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 1383    e. wcel 1804   _Vcvv 3095    C_ wss 3461    |-> cmpt 4495   dom cdm 4989   ran crn 4990    o. ccom 4993    Fn wfn 5573   ` cfv 5578  (class class class)co 6281    |-> cmpt2 6283   X_cixp 7471   Basecbs 14614   +g cplusg 14679   .rcmulr 14680   X_scprds 14825  mulGrpcmgp 17120
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-map 7424  df-ixp 7472  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-sup 7903  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10544  df-2 10601  df-3 10602  df-4 10603  df-5 10604  df-6 10605  df-7 10606  df-8 10607  df-9 10608  df-10 10609  df-n0 10803  df-z 10872  df-dec 10987  df-uz 11093  df-fz 11684  df-struct 14616  df-ndx 14617  df-slot 14618  df-base 14619  df-sets 14620  df-plusg 14692  df-mulr 14693  df-sca 14695  df-vsca 14696  df-ip 14697  df-tset 14698  df-ple 14699  df-ds 14701  df-hom 14703  df-cco 14704  df-prds 14827  df-mgp 17121
This theorem is referenced by:  prdsringd  17240  prdscrngd  17241  prds1  17242  pwsmgp  17246
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