Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rngcrescrhmALTV Structured version   Visualization version   Unicode version

Theorem rngcrescrhmALTV 40614
Description: The category of non-unital rings (in a universe) restricted to the ring homomorphisms between unital rings (in the same universe). (Contributed by AV, 1-Mar-2020.) (New usage is discouraged.)
Hypotheses
Ref Expression
rngcrescrhmALTV.u  |-  ( ph  ->  U  e.  V )
rngcrescrhmALTV.c  |-  C  =  (RngCatALTV `  U )
rngcrescrhmALTV.r  |-  ( ph  ->  R  =  ( Ring 
i^i  U ) )
rngcrescrhmALTV.h  |-  H  =  ( RingHom  |`  ( R  X.  R ) )
Assertion
Ref Expression
rngcrescrhmALTV  |-  ( ph  ->  ( C  |`cat  H )  =  ( ( Cs  R ) sSet  <. ( Hom  `  ndx ) ,  H >. ) )

Proof of Theorem rngcrescrhmALTV
StepHypRef Expression
1 eqid 2471 . 2  |-  ( C  |`cat 
H )  =  ( C  |`cat  H )
2 rngcrescrhmALTV.c . . . 4  |-  C  =  (RngCatALTV `  U )
3 fvex 5889 . . . 4  |-  (RngCatALTV `  U
)  e.  _V
42, 3eqeltri 2545 . . 3  |-  C  e. 
_V
54a1i 11 . 2  |-  ( ph  ->  C  e.  _V )
6 rngcrescrhmALTV.r . . . 4  |-  ( ph  ->  R  =  ( Ring 
i^i  U ) )
7 incom 3616 . . . 4  |-  ( Ring 
i^i  U )  =  ( U  i^i  Ring )
86, 7syl6eq 2521 . . 3  |-  ( ph  ->  R  =  ( U  i^i  Ring ) )
9 rngcrescrhmALTV.u . . . 4  |-  ( ph  ->  U  e.  V )
10 inex1g 4539 . . . 4  |-  ( U  e.  V  ->  ( U  i^i  Ring )  e.  _V )
119, 10syl 17 . . 3  |-  ( ph  ->  ( U  i^i  Ring )  e.  _V )
128, 11eqeltrd 2549 . 2  |-  ( ph  ->  R  e.  _V )
13 inss1 3643 . . . . . 6  |-  ( Ring 
i^i  U )  C_  Ring
146, 13syl6eqss 3468 . . . . 5  |-  ( ph  ->  R  C_  Ring )
15 xpss12 4945 . . . . 5  |-  ( ( R  C_  Ring  /\  R  C_ 
Ring )  ->  ( R  X.  R )  C_  ( Ring  X.  Ring )
)
1614, 14, 15syl2anc 673 . . . 4  |-  ( ph  ->  ( R  X.  R
)  C_  ( Ring  X. 
Ring ) )
17 rhmfn 40426 . . . . 5  |- RingHom  Fn  ( Ring  X.  Ring )
18 fnssresb 5698 . . . . 5  |-  ( RingHom  Fn  ( Ring  X.  Ring )  ->  ( ( RingHom  |`  ( R  X.  R ) )  Fn  ( R  X.  R )  <->  ( R  X.  R )  C_  ( Ring  X.  Ring ) ) )
1917, 18mp1i 13 . . . 4  |-  ( ph  ->  ( ( RingHom  |`  ( R  X.  R ) )  Fn  ( R  X.  R )  <->  ( R  X.  R )  C_  ( Ring  X.  Ring ) ) )
2016, 19mpbird 240 . . 3  |-  ( ph  ->  ( RingHom  |`  ( R  X.  R ) )  Fn  ( R  X.  R
) )
21 rngcrescrhmALTV.h . . . 4  |-  H  =  ( RingHom  |`  ( R  X.  R ) )
2221fneq1i 5680 . . 3  |-  ( H  Fn  ( R  X.  R )  <->  ( RingHom  |`  ( R  X.  R ) )  Fn  ( R  X.  R ) )
2320, 22sylibr 217 . 2  |-  ( ph  ->  H  Fn  ( R  X.  R ) )
241, 5, 12, 23rescval2 15811 1  |-  ( ph  ->  ( C  |`cat  H )  =  ( ( Cs  R ) sSet  <. ( Hom  `  ndx ) ,  H >. ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    = wceq 1452    e. wcel 1904   _Vcvv 3031    i^i cin 3389    C_ wss 3390   <.cop 3965    X. cxp 4837    |` cres 4841    Fn wfn 5584   ` cfv 5589  (class class class)co 6308   ndxcnx 15196   sSet csts 15197   ↾s cress 15200   Hom chom 15279    |`cat cresc 15791   Ringcrg 17858   RingHom crh 18018  RngCatALTVcrngcALTV 40468
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-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-resc 15794  df-mhm 16660  df-ghm 16959  df-mgp 17802  df-ur 17814  df-ring 17860  df-rnghom 18021
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator