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Theorem rhmsscmap2 40529
Description: The unital ring homomorphisms between unital rings (in a universe) are a subcategory subset of the mappings between base sets of unital rings (in the same universe). (Contributed by AV, 6-Mar-2020.)
Hypotheses
Ref Expression
rhmsscmap.u  |-  ( ph  ->  U  e.  V )
rhmsscmap.r  |-  ( ph  ->  R  =  ( Ring 
i^i  U ) )
Assertion
Ref Expression
rhmsscmap2  |-  ( ph  ->  ( RingHom  |`  ( R  X.  R ) )  C_cat  (
x  e.  R , 
y  e.  R  |->  ( ( Base `  y
)  ^m  ( Base `  x ) ) ) )
Distinct variable group:    x, R, y
Allowed substitution hints:    ph( x, y)    U( x, y)    V( x, y)

Proof of Theorem rhmsscmap2
Dummy variables  a 
b  h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssid 3437 . . 3  |-  R  C_  R
21a1i 11 . 2  |-  ( ph  ->  R  C_  R )
3 eqid 2471 . . . . . . 7  |-  ( Base `  a )  =  (
Base `  a )
4 eqid 2471 . . . . . . 7  |-  ( Base `  b )  =  (
Base `  b )
53, 4rhmf 18032 . . . . . 6  |-  ( h  e.  ( a RingHom  b
)  ->  h :
( Base `  a ) --> ( Base `  b )
)
6 simpr 468 . . . . . . . 8  |-  ( ( ( ph  /\  (
a  e.  R  /\  b  e.  R )
)  /\  h :
( Base `  a ) --> ( Base `  b )
)  ->  h :
( Base `  a ) --> ( Base `  b )
)
7 fvex 5889 . . . . . . . . . 10  |-  ( Base `  b )  e.  _V
8 fvex 5889 . . . . . . . . . 10  |-  ( Base `  a )  e.  _V
97, 8pm3.2i 462 . . . . . . . . 9  |-  ( (
Base `  b )  e.  _V  /\  ( Base `  a )  e.  _V )
10 elmapg 7503 . . . . . . . . 9  |-  ( ( ( Base `  b
)  e.  _V  /\  ( Base `  a )  e.  _V )  ->  (
h  e.  ( (
Base `  b )  ^m  ( Base `  a
) )  <->  h :
( Base `  a ) --> ( Base `  b )
) )
119, 10mp1i 13 . . . . . . . 8  |-  ( ( ( ph  /\  (
a  e.  R  /\  b  e.  R )
)  /\  h :
( Base `  a ) --> ( Base `  b )
)  ->  ( h  e.  ( ( Base `  b
)  ^m  ( Base `  a ) )  <->  h :
( Base `  a ) --> ( Base `  b )
) )
126, 11mpbird 240 . . . . . . 7  |-  ( ( ( ph  /\  (
a  e.  R  /\  b  e.  R )
)  /\  h :
( Base `  a ) --> ( Base `  b )
)  ->  h  e.  ( ( Base `  b
)  ^m  ( Base `  a ) ) )
1312ex 441 . . . . . 6  |-  ( (
ph  /\  ( a  e.  R  /\  b  e.  R ) )  -> 
( h : (
Base `  a ) --> ( Base `  b )  ->  h  e.  ( (
Base `  b )  ^m  ( Base `  a
) ) ) )
145, 13syl5 32 . . . . 5  |-  ( (
ph  /\  ( a  e.  R  /\  b  e.  R ) )  -> 
( h  e.  ( a RingHom  b )  ->  h  e.  ( ( Base `  b )  ^m  ( Base `  a )
) ) )
1514ssrdv 3424 . . . 4  |-  ( (
ph  /\  ( a  e.  R  /\  b  e.  R ) )  -> 
( a RingHom  b )  C_  ( ( Base `  b
)  ^m  ( Base `  a ) ) )
16 ovres 6455 . . . . 5  |-  ( ( a  e.  R  /\  b  e.  R )  ->  ( a ( RingHom  |`  ( R  X.  R ) ) b )  =  ( a RingHom  b ) )
1716adantl 473 . . . 4  |-  ( (
ph  /\  ( a  e.  R  /\  b  e.  R ) )  -> 
( a ( RingHom  |`  ( R  X.  R ) ) b )  =  ( a RingHom  b ) )
18 eqidd 2472 . . . . . 6  |-  ( ( a  e.  R  /\  b  e.  R )  ->  ( x  e.  R ,  y  e.  R  |->  ( ( Base `  y
)  ^m  ( Base `  x ) ) )  =  ( x  e.  R ,  y  e.  R  |->  ( ( Base `  y )  ^m  ( Base `  x ) ) ) )
19 fveq2 5879 . . . . . . . 8  |-  ( y  =  b  ->  ( Base `  y )  =  ( Base `  b
) )
20 fveq2 5879 . . . . . . . 8  |-  ( x  =  a  ->  ( Base `  x )  =  ( Base `  a
) )
2119, 20oveqan12rd 6328 . . . . . . 7  |-  ( ( x  =  a  /\  y  =  b )  ->  ( ( Base `  y
)  ^m  ( Base `  x ) )  =  ( ( Base `  b
)  ^m  ( Base `  a ) ) )
2221adantl 473 . . . . . 6  |-  ( ( ( a  e.  R  /\  b  e.  R
)  /\  ( x  =  a  /\  y  =  b ) )  ->  ( ( Base `  y )  ^m  ( Base `  x ) )  =  ( ( Base `  b )  ^m  ( Base `  a ) ) )
23 simpl 464 . . . . . 6  |-  ( ( a  e.  R  /\  b  e.  R )  ->  a  e.  R )
24 simpr 468 . . . . . 6  |-  ( ( a  e.  R  /\  b  e.  R )  ->  b  e.  R )
25 ovex 6336 . . . . . . 7  |-  ( (
Base `  b )  ^m  ( Base `  a
) )  e.  _V
2625a1i 11 . . . . . 6  |-  ( ( a  e.  R  /\  b  e.  R )  ->  ( ( Base `  b
)  ^m  ( Base `  a ) )  e. 
_V )
2718, 22, 23, 24, 26ovmpt2d 6443 . . . . 5  |-  ( ( a  e.  R  /\  b  e.  R )  ->  ( a ( x  e.  R ,  y  e.  R  |->  ( (
Base `  y )  ^m  ( Base `  x
) ) ) b )  =  ( (
Base `  b )  ^m  ( Base `  a
) ) )
2827adantl 473 . . . 4  |-  ( (
ph  /\  ( a  e.  R  /\  b  e.  R ) )  -> 
( a ( x  e.  R ,  y  e.  R  |->  ( (
Base `  y )  ^m  ( Base `  x
) ) ) b )  =  ( (
Base `  b )  ^m  ( Base `  a
) ) )
2915, 17, 283sstr4d 3461 . . 3  |-  ( (
ph  /\  ( a  e.  R  /\  b  e.  R ) )  -> 
( a ( RingHom  |`  ( R  X.  R ) ) b )  C_  (
a ( x  e.  R ,  y  e.  R  |->  ( ( Base `  y )  ^m  ( Base `  x ) ) ) b ) )
3029ralrimivva 2814 . 2  |-  ( ph  ->  A. a  e.  R  A. b  e.  R  ( a ( RingHom  |`  ( R  X.  R ) ) b )  C_  (
a ( x  e.  R ,  y  e.  R  |->  ( ( Base `  y )  ^m  ( Base `  x ) ) ) b ) )
31 rhmfn 40426 . . . . 5  |- RingHom  Fn  ( Ring  X.  Ring )
3231a1i 11 . . . 4  |-  ( ph  -> RingHom 
Fn  ( Ring  X.  Ring ) )
33 rhmsscmap.r . . . . . 6  |-  ( ph  ->  R  =  ( Ring 
i^i  U ) )
34 inss1 3643 . . . . . 6  |-  ( Ring 
i^i  U )  C_  Ring
3533, 34syl6eqss 3468 . . . . 5  |-  ( ph  ->  R  C_  Ring )
36 xpss12 4945 . . . . 5  |-  ( ( R  C_  Ring  /\  R  C_ 
Ring )  ->  ( R  X.  R )  C_  ( Ring  X.  Ring )
)
3735, 35, 36syl2anc 673 . . . 4  |-  ( ph  ->  ( R  X.  R
)  C_  ( Ring  X. 
Ring ) )
38 fnssres 5699 . . . 4  |-  ( ( RingHom  Fn  ( Ring  X.  Ring )  /\  ( R  X.  R
)  C_  ( Ring  X. 
Ring ) )  -> 
( RingHom  |`  ( R  X.  R ) )  Fn  ( R  X.  R
) )
3932, 37, 38syl2anc 673 . . 3  |-  ( ph  ->  ( RingHom  |`  ( R  X.  R ) )  Fn  ( R  X.  R
) )
40 eqid 2471 . . . . 5  |-  ( x  e.  R ,  y  e.  R  |->  ( (
Base `  y )  ^m  ( Base `  x
) ) )  =  ( x  e.  R ,  y  e.  R  |->  ( ( Base `  y
)  ^m  ( Base `  x ) ) )
41 ovex 6336 . . . . 5  |-  ( (
Base `  y )  ^m  ( Base `  x
) )  e.  _V
4240, 41fnmpt2i 6881 . . . 4  |-  ( x  e.  R ,  y  e.  R  |->  ( (
Base `  y )  ^m  ( Base `  x
) ) )  Fn  ( R  X.  R
)
4342a1i 11 . . 3  |-  ( ph  ->  ( x  e.  R ,  y  e.  R  |->  ( ( Base `  y
)  ^m  ( Base `  x ) ) )  Fn  ( R  X.  R ) )
44 incom 3616 . . . . 5  |-  ( Ring 
i^i  U )  =  ( U  i^i  Ring )
45 rhmsscmap.u . . . . . 6  |-  ( ph  ->  U  e.  V )
46 inex1g 4539 . . . . . 6  |-  ( U  e.  V  ->  ( U  i^i  Ring )  e.  _V )
4745, 46syl 17 . . . . 5  |-  ( ph  ->  ( U  i^i  Ring )  e.  _V )
4844, 47syl5eqel 2553 . . . 4  |-  ( ph  ->  ( Ring  i^i  U )  e.  _V )
4933, 48eqeltrd 2549 . . 3  |-  ( ph  ->  R  e.  _V )
5039, 43, 49isssc 15803 . 2  |-  ( ph  ->  ( ( RingHom  |`  ( R  X.  R ) ) 
C_cat  ( x  e.  R ,  y  e.  R  |->  ( ( Base `  y
)  ^m  ( Base `  x ) ) )  <-> 
( R  C_  R  /\  A. a  e.  R  A. b  e.  R  ( a ( RingHom  |`  ( R  X.  R ) ) b )  C_  (
a ( x  e.  R ,  y  e.  R  |->  ( ( Base `  y )  ^m  ( Base `  x ) ) ) b ) ) ) )
512, 30, 50mpbir2and 936 1  |-  ( ph  ->  ( RingHom  |`  ( R  X.  R ) )  C_cat  (
x  e.  R , 
y  e.  R  |->  ( ( Base `  y
)  ^m  ( Base `  x ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904   A.wral 2756   _Vcvv 3031    i^i cin 3389    C_ wss 3390   class class class wbr 4395    X. cxp 4837    |` cres 4841    Fn wfn 5584   -->wf 5585   ` cfv 5589  (class class class)co 6308    |-> cmpt2 6310    ^m cmap 7490   Basecbs 15199    C_cat cssc 15790   Ringcrg 17858   RingHom crh 18018
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-map 7492  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-plusg 15281  df-0g 15418  df-ssc 15793  df-mhm 16660  df-ghm 16959  df-mgp 17802  df-ur 17814  df-ring 17860  df-rnghom 18021
This theorem is referenced by: (None)
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