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Theorem rngcbasALTV 40493
Description: Set of objects of the category of non-unital rings (in a universe). (New usage is discouraged.) (Contributed by AV, 27-Feb-2020.)
Hypotheses
Ref Expression
rngcbasALTV.c  |-  C  =  (RngCatALTV `  U )
rngcbasALTV.b  |-  B  =  ( Base `  C
)
rngcbasALTV.u  |-  ( ph  ->  U  e.  V )
Assertion
Ref Expression
rngcbasALTV  |-  ( ph  ->  B  =  ( U  i^i Rng ) )

Proof of Theorem rngcbasALTV
Dummy variables  f 
g  v  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rngcbasALTV.c . . 3  |-  C  =  (RngCatALTV `  U )
2 rngcbasALTV.u . . 3  |-  ( ph  ->  U  e.  V )
3 eqidd 2472 . . 3  |-  ( ph  ->  ( U  i^i Rng )  =  ( U  i^i Rng ) )
4 eqidd 2472 . . 3  |-  ( ph  ->  ( x  e.  ( U  i^i Rng ) , 
y  e.  ( U  i^i Rng )  |->  ( x RngHomo 
y ) )  =  ( x  e.  ( U  i^i Rng ) , 
y  e.  ( U  i^i Rng )  |->  ( x RngHomo 
y ) ) )
5 eqidd 2472 . . 3  |-  ( ph  ->  ( v  e.  ( ( U  i^i Rng )  X.  ( U  i^i Rng )
) ,  z  e.  ( U  i^i Rng )  |->  ( f  e.  ( ( 2nd `  v
) RngHomo  z ) ,  g  e.  ( ( 1st `  v ) RngHomo  ( 2nd `  v ) )  |->  ( f  o.  g ) ) )  =  ( v  e.  ( ( U  i^i Rng )  X.  ( U  i^i Rng ) ) ,  z  e.  ( U  i^i Rng )  |->  ( f  e.  ( ( 2nd `  v ) RngHomo  z ) ,  g  e.  ( ( 1st `  v
) RngHomo  ( 2nd `  v
) )  |->  ( f  o.  g ) ) ) )
61, 2, 3, 4, 5rngcvalALTV 40471 . 2  |-  ( ph  ->  C  =  { <. (
Base `  ndx ) ,  ( U  i^i Rng ) >. ,  <. ( Hom  `  ndx ) ,  ( x  e.  ( U  i^i Rng ) ,  y  e.  ( U  i^i Rng )  |->  ( x RngHomo 
y ) ) >. ,  <. (comp `  ndx ) ,  ( v  e.  ( ( U  i^i Rng )  X.  ( U  i^i Rng ) ) ,  z  e.  ( U  i^i Rng )  |->  ( f  e.  ( ( 2nd `  v
) RngHomo  z ) ,  g  e.  ( ( 1st `  v ) RngHomo  ( 2nd `  v ) )  |->  ( f  o.  g ) ) ) >. } )
7 catstr 15940 . 2  |-  { <. (
Base `  ndx ) ,  ( U  i^i Rng ) >. ,  <. ( Hom  `  ndx ) ,  ( x  e.  ( U  i^i Rng ) ,  y  e.  ( U  i^i Rng )  |->  ( x RngHomo 
y ) ) >. ,  <. (comp `  ndx ) ,  ( v  e.  ( ( U  i^i Rng )  X.  ( U  i^i Rng ) ) ,  z  e.  ( U  i^i Rng )  |->  ( f  e.  ( ( 2nd `  v
) RngHomo  z ) ,  g  e.  ( ( 1st `  v ) RngHomo  ( 2nd `  v ) )  |->  ( f  o.  g ) ) ) >. } Struct  <. 1 , ; 1 5 >.
8 baseid 15247 . 2  |-  Base  = Slot  ( Base `  ndx )
9 snsstp1 4114 . 2  |-  { <. (
Base `  ndx ) ,  ( U  i^i Rng ) >. }  C_  { <. ( Base `  ndx ) ,  ( U  i^i Rng ) >. ,  <. ( Hom  `  ndx ) ,  ( x  e.  ( U  i^i Rng ) ,  y  e.  ( U  i^i Rng )  |->  ( x RngHomo 
y ) ) >. ,  <. (comp `  ndx ) ,  ( v  e.  ( ( U  i^i Rng )  X.  ( U  i^i Rng ) ) ,  z  e.  ( U  i^i Rng )  |->  ( f  e.  ( ( 2nd `  v
) RngHomo  z ) ,  g  e.  ( ( 1st `  v ) RngHomo  ( 2nd `  v ) )  |->  ( f  o.  g ) ) ) >. }
10 inex1g 4539 . . 3  |-  ( U  e.  V  ->  ( U  i^i Rng )  e.  _V )
112, 10syl 17 . 2  |-  ( ph  ->  ( U  i^i Rng )  e.  _V )
12 rngcbasALTV.b . 2  |-  B  =  ( Base `  C
)
136, 7, 8, 9, 11, 12strfv3 15236 1  |-  ( ph  ->  B  =  ( U  i^i Rng ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1452    e. wcel 1904   _Vcvv 3031    i^i cin 3389   {ctp 3963   <.cop 3965    X. cxp 4837    o. ccom 4843   ` cfv 5589  (class class class)co 6308    |-> cmpt2 6310   1stc1st 6810   2ndc2nd 6811   1c1 9558   5c5 10684  ;cdc 11074   ndxcnx 15196   Basecbs 15199   Hom chom 15279  compcco 15280  Rngcrng 40382   RngHomo crngh 40393  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-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-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-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-hom 15292  df-cco 15293  df-rngcALTV 40470
This theorem is referenced by:  rngchomfvalALTV  40494  rngccofvalALTV  40497  rngccatidALTV  40499  rngchomrnghmresALTV  40506  rhmsubcALTVlem3  40617  rhmsubcALTVlem4  40618
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