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Theorem rngcbas 32875
Description: Set of objects of the category of non-unital rings (in a universe). (Contributed by AV, 27-Feb-2020.) (Revised by AV, 8-Mar-2020.)
Hypotheses
Ref Expression
rngcbas.c  |-  C  =  (RngCat `  U )
rngcbas.b  |-  B  =  ( Base `  C
)
rngcbas.u  |-  ( ph  ->  U  e.  V )
Assertion
Ref Expression
rngcbas  |-  ( ph  ->  B  =  ( U  i^i Rng ) )

Proof of Theorem rngcbas
StepHypRef Expression
1 rngcbas.c . . . 4  |-  C  =  (RngCat `  U )
2 rngcbas.u . . . 4  |-  ( ph  ->  U  e.  V )
3 eqidd 2458 . . . 4  |-  ( ph  ->  ( U  i^i Rng )  =  ( U  i^i Rng ) )
4 eqidd 2458 . . . 4  |-  ( ph  ->  ( RngHomo  |`  ( ( U  i^i Rng )  X.  ( U  i^i Rng ) ) )  =  ( RngHomo  |`  (
( U  i^i Rng )  X.  ( U  i^i Rng )
) ) )
51, 2, 3, 4rngcval 32872 . . 3  |-  ( ph  ->  C  =  ( (ExtStrCat `  U )  |`cat  ( RngHomo  |`  (
( U  i^i Rng )  X.  ( U  i^i Rng )
) ) ) )
65fveq2d 5876 . 2  |-  ( ph  ->  ( Base `  C
)  =  ( Base `  ( (ExtStrCat `  U
)  |`cat  ( RngHomo  |`  ( ( U  i^i Rng )  X.  ( U  i^i Rng ) ) ) ) ) )
7 rngcbas.b . . 3  |-  B  =  ( Base `  C
)
87a1i 11 . 2  |-  ( ph  ->  B  =  ( Base `  C ) )
9 eqid 2457 . . 3  |-  ( (ExtStrCat `  U )  |`cat  ( RngHomo  |`  (
( U  i^i Rng )  X.  ( U  i^i Rng )
) ) )  =  ( (ExtStrCat `  U
)  |`cat  ( RngHomo  |`  ( ( U  i^i Rng )  X.  ( U  i^i Rng ) ) ) )
10 eqid 2457 . . 3  |-  ( Base `  (ExtStrCat `  U )
)  =  ( Base `  (ExtStrCat `  U )
)
11 fvex 5882 . . . 4  |-  (ExtStrCat `  U
)  e.  _V
1211a1i 11 . . 3  |-  ( ph  ->  (ExtStrCat `  U )  e.  _V )
133, 4rnghmresfn 32873 . . 3  |-  ( ph  ->  ( RngHomo  |`  ( ( U  i^i Rng )  X.  ( U  i^i Rng ) ) )  Fn  ( ( U  i^i Rng )  X.  ( U  i^i Rng ) ) )
14 inss1 3714 . . . 4  |-  ( U  i^i Rng )  C_  U
15 eqid 2457 . . . . 5  |-  (ExtStrCat `  U
)  =  (ExtStrCat `  U
)
1615, 2estrcbas 15520 . . . 4  |-  ( ph  ->  U  =  ( Base `  (ExtStrCat `  U )
) )
1714, 16syl5sseq 3547 . . 3  |-  ( ph  ->  ( U  i^i Rng )  C_  ( Base `  (ExtStrCat `  U ) ) )
189, 10, 12, 13, 17rescbas 15244 . 2  |-  ( ph  ->  ( U  i^i Rng )  =  ( Base `  (
(ExtStrCat `  U )  |`cat  ( RngHomo  |`  ( ( U  i^i Rng )  X.  ( U  i^i Rng ) ) ) ) ) )
196, 8, 183eqtr4d 2508 1  |-  ( ph  ->  B  =  ( U  i^i Rng ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1395    e. wcel 1819   _Vcvv 3109    i^i cin 3470    X. cxp 5006    |` cres 5010   ` cfv 5594  (class class class)co 6296   Basecbs 14643    |`cat cresc 15223  ExtStrCatcestrc 15517  Rngcrng 32782   RngHomo crngh 32793  RngCatcrngc 32867
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-fz 11698  df-struct 14645  df-ndx 14646  df-slot 14647  df-base 14648  df-sets 14649  df-ress 14650  df-hom 14735  df-cco 14736  df-resc 15226  df-estrc 15518  df-rnghomo 32795  df-rngc 32869
This theorem is referenced by:  rngchomfval  32876  rngchomfeqhom  32879  rngccofval  32880  rnghmsubcsetclem1  32885  rngcid  32889  rngcsect  32890  rngcifuestrc  32907  funcrngcsetc  32908  funcrngcsetcALT  32909  zrinitorngc  32910  zrtermorngc  32911  zrzeroorngc  32912  rhmsubcrngclem1  32937  rhmsubcrngc  32939  rhmsubclem3  32998  rhmsubc  33000
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