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Theorem rngcresringcat 40540
Description: The restriction of the category of non-unital rings to the set of unital ring homomorphisms is the category of unital rings. (Contributed by AV, 16-Mar-2020.)
Hypotheses
Ref Expression
rhmsubcrngc.c  |-  C  =  (RngCat `  U )
rhmsubcrngc.u  |-  ( ph  ->  U  e.  V )
rhmsubcrngc.b  |-  ( ph  ->  B  =  ( Ring 
i^i  U ) )
rhmsubcrngc.h  |-  ( ph  ->  H  =  ( RingHom  |`  ( B  X.  B ) ) )
Assertion
Ref Expression
rngcresringcat  |-  ( ph  ->  ( C  |`cat  H )  =  (RingCat `  U )
)

Proof of Theorem rngcresringcat
Dummy variable  r is distinct from all other variables.
StepHypRef Expression
1 rhmsubcrngc.c . . . 4  |-  C  =  (RngCat `  U )
2 rhmsubcrngc.u . . . 4  |-  ( ph  ->  U  e.  V )
3 eqidd 2472 . . . 4  |-  ( ph  ->  ( U  i^i Rng )  =  ( U  i^i Rng ) )
4 eqidd 2472 . . . 4  |-  ( ph  ->  ( RngHomo  |`  ( ( U  i^i Rng )  X.  ( U  i^i Rng ) ) )  =  ( RngHomo  |`  (
( U  i^i Rng )  X.  ( U  i^i Rng )
) ) )
5 eqidd 2472 . . . 4  |-  ( ph  ->  (comp `  (ExtStrCat `  U
) )  =  (comp `  (ExtStrCat `  U )
) )
61, 2, 3, 4, 5dfrngc2 40482 . . 3  |-  ( ph  ->  C  =  { <. (
Base `  ndx ) ,  ( U  i^i Rng ) >. ,  <. ( Hom  `  ndx ) ,  ( RngHomo  |`  (
( U  i^i Rng )  X.  ( U  i^i Rng )
) ) >. ,  <. (comp `  ndx ) ,  (comp `  (ExtStrCat `  U )
) >. } )
7 inex1g 4539 . . . 4  |-  ( U  e.  V  ->  ( U  i^i Rng )  e.  _V )
82, 7syl 17 . . 3  |-  ( ph  ->  ( U  i^i Rng )  e.  _V )
9 rnghmfn 40398 . . . . 5  |- RngHomo  Fn  (Rng  X. Rng )
10 fnfun 5683 . . . . 5  |-  ( RngHomo  Fn  (Rng  X. Rng )  ->  Fun RngHomo  )
119, 10mp1i 13 . . . 4  |-  ( ph  ->  Fun RngHomo  )
12 sqxpexg 6615 . . . . 5  |-  ( ( U  i^i Rng )  e.  _V  ->  ( ( U  i^i Rng )  X.  ( U  i^i Rng ) )  e. 
_V )
138, 12syl 17 . . . 4  |-  ( ph  ->  ( ( U  i^i Rng )  X.  ( U  i^i Rng ) )  e.  _V )
14 resfunexg 6146 . . . 4  |-  ( ( Fun RngHomo  /\  ( ( U  i^i Rng )  X.  ( U  i^i Rng ) )  e. 
_V )  ->  ( RngHomo  |`  ( ( U  i^i Rng )  X.  ( U  i^i Rng ) ) )  e.  _V )
1511, 13, 14syl2anc 673 . . 3  |-  ( ph  ->  ( RngHomo  |`  ( ( U  i^i Rng )  X.  ( U  i^i Rng ) ) )  e.  _V )
16 fvex 5889 . . . 4  |-  (comp `  (ExtStrCat `  U ) )  e.  _V
1716a1i 11 . . 3  |-  ( ph  ->  (comp `  (ExtStrCat `  U
) )  e.  _V )
18 rhmsubcrngc.b . . . . 5  |-  ( ph  ->  B  =  ( Ring 
i^i  U ) )
19 incom 3616 . . . . 5  |-  ( Ring 
i^i  U )  =  ( U  i^i  Ring )
2018, 19syl6eq 2521 . . . 4  |-  ( ph  ->  B  =  ( U  i^i  Ring ) )
21 inex1g 4539 . . . . 5  |-  ( U  e.  V  ->  ( U  i^i  Ring )  e.  _V )
222, 21syl 17 . . . 4  |-  ( ph  ->  ( U  i^i  Ring )  e.  _V )
2320, 22eqeltrd 2549 . . 3  |-  ( ph  ->  B  e.  _V )
24 rhmsubcrngc.h . . . 4  |-  ( ph  ->  H  =  ( RingHom  |`  ( B  X.  B ) ) )
25 rhmfn 40426 . . . . . 6  |- RingHom  Fn  ( Ring  X.  Ring )
26 fnfun 5683 . . . . . 6  |-  ( RingHom  Fn  ( Ring  X.  Ring )  ->  Fun RingHom  )
2725, 26mp1i 13 . . . . 5  |-  ( ph  ->  Fun RingHom  )
28 sqxpexg 6615 . . . . . 6  |-  ( B  e.  _V  ->  ( B  X.  B )  e. 
_V )
2923, 28syl 17 . . . . 5  |-  ( ph  ->  ( B  X.  B
)  e.  _V )
30 resfunexg 6146 . . . . 5  |-  ( ( Fun RingHom  /\  ( B  X.  B )  e.  _V )  ->  ( RingHom  |`  ( B  X.  B ) )  e.  _V )
3127, 29, 30syl2anc 673 . . . 4  |-  ( ph  ->  ( RingHom  |`  ( B  X.  B ) )  e. 
_V )
3224, 31eqeltrd 2549 . . 3  |-  ( ph  ->  H  e.  _V )
33 ringrng 40387 . . . . . . 7  |-  ( r  e.  Ring  ->  r  e. Rng )
3433a1i 11 . . . . . 6  |-  ( ph  ->  ( r  e.  Ring  -> 
r  e. Rng ) )
3534ssrdv 3424 . . . . 5  |-  ( ph  ->  Ring  C_ Rng )
36 ssrin 3648 . . . . 5  |-  ( Ring  C_ Rng  ->  ( Ring  i^i  U )  C_  (Rng  i^i  U ) )
3735, 36syl 17 . . . 4  |-  ( ph  ->  ( Ring  i^i  U ) 
C_  (Rng  i^i  U
) )
38 incom 3616 . . . . 5  |-  ( U  i^i Rng )  =  (Rng 
i^i  U )
3938a1i 11 . . . 4  |-  ( ph  ->  ( U  i^i Rng )  =  (Rng  i^i  U ) )
4037, 18, 393sstr4d 3461 . . 3  |-  ( ph  ->  B  C_  ( U  i^i Rng ) )
416, 8, 15, 17, 23, 32, 40estrres 16102 . 2  |-  ( ph  ->  ( ( Cs  B ) sSet  <. ( Hom  `  ndx ) ,  H >. )  =  { <. ( Base `  ndx ) ,  B >. ,  <. ( Hom  `  ndx ) ,  H >. ,  <. (comp ` 
ndx ) ,  (comp `  (ExtStrCat `  U )
) >. } )
42 eqid 2471 . . 3  |-  ( C  |`cat 
H )  =  ( C  |`cat  H )
431a1i 11 . . . 4  |-  ( ph  ->  C  =  (RngCat `  U ) )
44 fvex 5889 . . . . 5  |-  (RngCat `  U )  e.  _V
4544a1i 11 . . . 4  |-  ( ph  ->  (RngCat `  U )  e.  _V )
4643, 45eqeltrd 2549 . . 3  |-  ( ph  ->  C  e.  _V )
4720, 24rhmresfn 40519 . . 3  |-  ( ph  ->  H  Fn  ( B  X.  B ) )
4842, 46, 23, 47rescval2 15811 . 2  |-  ( ph  ->  ( C  |`cat  H )  =  ( ( Cs  B ) sSet  <. ( Hom  `  ndx ) ,  H >. ) )
49 eqid 2471 . . 3  |-  (RingCat `  U
)  =  (RingCat `  U
)
5049, 2, 20, 24, 5dfringc2 40528 . 2  |-  ( ph  ->  (RingCat `  U )  =  { <. ( Base `  ndx ) ,  B >. , 
<. ( Hom  `  ndx ) ,  H >. , 
<. (comp `  ndx ) ,  (comp `  (ExtStrCat `  U
) ) >. } )
5141, 48, 503eqtr4d 2515 1  |-  ( ph  ->  ( C  |`cat  H )  =  (RingCat `  U )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1452    e. wcel 1904   _Vcvv 3031    i^i cin 3389    C_ wss 3390   {ctp 3963   <.cop 3965    X. cxp 4837    |` cres 4841   Fun wfun 5583    Fn wfn 5584   ` cfv 5589  (class class class)co 6308   ndxcnx 15196   sSet csts 15197   Basecbs 15199   ↾s cress 15200   Hom chom 15279  compcco 15280    |`cat cresc 15791  ExtStrCatcestrc 16085   Ringcrg 17858   RingHom crh 18018  Rngcrng 40382   RngHomo crngh 40393  RngCatcrngc 40467  RingCatcringc 40513
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-fal 1458  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-rmo 2764  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-int 4227  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-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  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-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-fz 11811  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-hom 15292  df-cco 15293  df-0g 15418  df-resc 15794  df-estrc 16086  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-mhm 16660  df-grp 16751  df-minusg 16752  df-ghm 16959  df-cmn 17510  df-abl 17511  df-mgp 17802  df-ur 17814  df-ring 17860  df-rnghom 18021  df-rng0 40383  df-rnghomo 40395  df-rngc 40469  df-ringc 40515
This theorem is referenced by: (None)
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