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Theorem nzerooringczr 40178
Description: There is no zero object in the category of unital rings (at least in a universe which contains the zero ring and the ring of integers). Example 7.9 (3) in [Adamek] p. 103. (Contributed by AV, 18-Apr-2020.)
Hypotheses
Ref Expression
nzerooringczr.u  |-  ( ph  ->  U  e.  V )
nzerooringczr.c  |-  C  =  (RingCat `  U )
nzerooringczr.z  |-  ( ph  ->  Z  e.  ( Ring  \ NzRing ) )
nzerooringczr.e  |-  ( ph  ->  Z  e.  U )
nzerooringczr.i  |-  ( ph  ->ring  e.  U )
Assertion
Ref Expression
nzerooringczr  |-  ( ph  ->  (ZeroO `  C )  =  (/) )

Proof of Theorem nzerooringczr
Dummy variables  f  h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ax-1 6 . 2  |-  ( (ZeroO `  C )  =  (/)  ->  ( ph  ->  (ZeroO `  C )  =  (/) ) )
2 neq0 3744 . . 3  |-  ( -.  (ZeroO `  C )  =  (/)  <->  E. h  h  e.  (ZeroO `  C )
)
3 nzerooringczr.u . . . . . . . 8  |-  ( ph  ->  U  e.  V )
4 nzerooringczr.c . . . . . . . . 9  |-  C  =  (RingCat `  U )
54ringccat 40130 . . . . . . . 8  |-  ( U  e.  V  ->  C  e.  Cat )
63, 5syl 17 . . . . . . 7  |-  ( ph  ->  C  e.  Cat )
7 iszeroi 15916 . . . . . . 7  |-  ( ( C  e.  Cat  /\  h  e.  (ZeroO `  C
) )  ->  (
h  e.  ( Base `  C )  /\  (
h  e.  (InitO `  C )  /\  h  e.  (TermO `  C )
) ) )
86, 7sylan 474 . . . . . 6  |-  ( (
ph  /\  h  e.  (ZeroO `  C ) )  ->  ( h  e.  ( Base `  C
)  /\  ( h  e.  (InitO `  C )  /\  h  e.  (TermO `  C ) ) ) )
9 nzerooringczr.z . . . . . . . . 9  |-  ( ph  ->  Z  e.  ( Ring  \ NzRing ) )
10 nzerooringczr.e . . . . . . . . 9  |-  ( ph  ->  Z  e.  U )
113, 4, 9, 10zrtermoringc 40176 . . . . . . . 8  |-  ( ph  ->  Z  e.  (TermO `  C ) )
12 nzerooringczr.i . . . . . . . . . 10  |-  ( ph  ->ring  e.  U )
133, 12, 4irinitoringc 40175 . . . . . . . . 9  |-  ( ph  ->ring  e.  (InitO `  C )
)
146ad2antrr 733 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  h  e.  (InitO `  C )
)  /\ring  e.  (InitO `  C
) )  ->  C  e.  Cat )
15 simplr 763 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  h  e.  (InitO `  C )
)  /\ring  e.  (InitO `  C
) )  ->  h  e.  (InitO `  C )
)
16 simpr 463 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  h  e.  (InitO `  C )
)  /\ring  e.  (InitO `  C
) )  ->ring  e.  (InitO `  C
) )
1714, 15, 16initoeu1w 15919 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  h  e.  (InitO `  C )
)  /\ring  e.  (InitO `  C
) )  ->  h
(  ~=c𝑐  `  C )ring )
186ad2antrr 733 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ph  /\  h  e.  (TermO `  C )
)  /\  Z  e.  (TermO `  C ) )  ->  C  e.  Cat )
19 simpr 463 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ph  /\  h  e.  (TermO `  C )
)  /\  Z  e.  (TermO `  C ) )  ->  Z  e.  (TermO `  C ) )
20 simplr 763 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ph  /\  h  e.  (TermO `  C )
)  /\  Z  e.  (TermO `  C ) )  ->  h  e.  (TermO `  C ) )
2118, 19, 20termoeu1w 15926 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  h  e.  (TermO `  C )
)  /\  Z  e.  (TermO `  C ) )  ->  Z (  ~=c𝑐  `  C
) h )
22 cictr 15722 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( C  e.  Cat  /\  Z (  ~=c𝑐  `  C ) h  /\  h ( 
~=c𝑐  `  C )ring )  ->  Z (  ~=c𝑐  `  C )ring )
236, 22syl3an1 1302 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( (
ph  /\  Z (  ~=c𝑐  `  C ) h  /\  h (  ~=c𝑐  `  C )ring )  ->  Z (  ~=c𝑐  `  C
)ring )
24 eqid 2453 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  (  Iso  `  C )  =  (  Iso  `  C )
25 eqid 2453 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( Base `  C )  =  (
Base `  C )
269eldifad 3418 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( ph  ->  Z  e.  Ring )
2710, 26elind 3620 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ph  ->  Z  e.  ( U  i^i  Ring ) )
284, 25, 3ringcbas 40117 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ph  ->  ( Base `  C
)  =  ( U  i^i  Ring ) )
2927, 28eleqtrrd 2534 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ph  ->  Z  e.  ( Base `  C ) )
30 zringring 19054 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31  |-ring  e.  Ring
3130a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( ph  ->ring  e. 
Ring )
3212, 31elind 3620 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ph  ->ring  e.  ( U  i^i  Ring ) )
3332, 28eleqtrrd 2534 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ph  ->ring  e.  ( Base `  C
) )
3424, 25, 6, 29, 33cic 15716 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ph  ->  ( Z (  ~=c𝑐  `  C
)ring  <->  E. f  f  e.  ( Z (  Iso  `  C
)ring ) ) )
35 n0 3743 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( Z (  Iso  `  C
)ring )  =/=  (/)  <->  E. f 
f  e.  ( Z (  Iso  `  C
)ring ) )
36 eqid 2453 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( Hom  `  C )  =  ( Hom  `  C )
3725, 36, 24, 6, 29, 33isohom 15693 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ph  ->  ( Z (  Iso  `  C )ring )  C_  ( Z
( Hom  `  C )ring ) )
38 ssn0 3769 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32  |-  ( ( ( Z (  Iso  `  C )ring )  C_  ( Z
( Hom  `  C )ring )  /\  ( Z (  Iso  `  C )ring )  =/=  (/) )  ->  ( Z ( Hom  `  C
)ring )  =/=  (/) )
394, 25, 3, 36, 29, 33ringchom 40119 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34  |-  ( ph  ->  ( Z ( Hom  `  C )ring )  =  ( Z RingHom ℤring )
)
4039neeq1d 2685 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  |-  ( ph  ->  ( ( Z ( Hom  `  C )ring )  =/=  (/)  <->  ( Z RingHom ℤring )  =/=  (/) ) )
41 zringnzr 19063 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35  |-ring  e. NzRing
42 nrhmzr 39977 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35  |-  ( ( Z  e.  ( Ring  \ NzRing )  /\ring  e. NzRing )  ->  ( Z RingHom ℤring )  =  (/) )
439, 41, 42sylancl 669 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34  |-  ( ph  ->  ( Z RingHom ℤring )  =  (/) )
44 eqneqall 2636 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34  |-  ( ( Z RingHom ℤring )  =  (/)  ->  (
( Z RingHom ℤring )  =/=  (/)  ->  (ZeroO `  C )  =  (/) ) )
4543, 44syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  |-  ( ph  ->  ( ( Z RingHom ℤring )  =/=  (/)  ->  (ZeroO `  C )  =  (/) ) )
4640, 45sylbid 219 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32  |-  ( ph  ->  ( ( Z ( Hom  `  C )ring )  =/=  (/)  ->  (ZeroO `  C
)  =  (/) ) )
4738, 46syl5com 31 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31  |-  ( ( ( Z (  Iso  `  C )ring )  C_  ( Z
( Hom  `  C )ring )  /\  ( Z (  Iso  `  C )ring )  =/=  (/) )  ->  ( ph  ->  (ZeroO `  C
)  =  (/) ) )
4847expcom 437 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( ( Z (  Iso  `  C
)ring )  =/=  (/)  ->  (
( Z (  Iso  `  C )ring )  C_  ( Z
( Hom  `  C )ring )  ->  ( ph  ->  (ZeroO `  C )  =  (/) ) ) )
4948com13 83 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ph  ->  ( ( Z (  Iso  `  C )ring )  C_  ( Z ( Hom  `  C )ring )  ->  ( ( Z (  Iso  `  C
)ring )  =/=  (/)  ->  (ZeroO `  C )  =  (/) ) ) )
5037, 49mpd 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ph  ->  ( ( Z (  Iso  `  C )ring )  =/=  (/)  ->  (ZeroO `  C
)  =  (/) ) )
5135, 50syl5bir 222 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ph  ->  ( E. f  f  e.  ( Z (  Iso  `  C )ring )  ->  (ZeroO `  C )  =  (/) ) )
5234, 51sylbid 219 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ph  ->  ( Z (  ~=c𝑐  `  C
)ring 
->  (ZeroO `  C )  =  (/) ) )
53523ad2ant1 1030 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( (
ph  /\  Z (  ~=c𝑐  `  C ) h  /\  h (  ~=c𝑐  `  C )ring )  ->  ( Z ( 
~=c𝑐  `  C )ring 
->  (ZeroO `  C )  =  (/) ) )
5423, 53mpd 15 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( (
ph  /\  Z (  ~=c𝑐  `  C ) h  /\  h (  ~=c𝑐  `  C )ring )  ->  (ZeroO `  C
)  =  (/) )
55543exp 1208 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ph  ->  ( Z (  ~=c𝑐  `  C
) h  ->  (
h (  ~=c𝑐  `  C )ring  -> 
(ZeroO `  C )  =  (/) ) ) )
5655a1dd 47 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ph  ->  ( Z (  ~=c𝑐  `  C
) h  ->  (
h  e.  ( Base `  C )  ->  (
h (  ~=c𝑐  `  C )ring  -> 
(ZeroO `  C )  =  (/) ) ) ) )
5756ad2antrr 733 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  h  e.  (TermO `  C )
)  /\  Z  e.  (TermO `  C ) )  ->  ( Z ( 
~=c𝑐  `  C ) h  -> 
( h  e.  (
Base `  C )  ->  ( h (  ~=c𝑐  `  C
)ring 
->  (ZeroO `  C )  =  (/) ) ) ) )
5821, 57mpd 15 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  h  e.  (TermO `  C )
)  /\  Z  e.  (TermO `  C ) )  ->  ( h  e.  ( Base `  C
)  ->  ( h
(  ~=c𝑐  `  C )ring 
->  (ZeroO `  C )  =  (/) ) ) )
5958exp31 609 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( h  e.  (TermO `  C )  ->  ( Z  e.  (TermO `  C
)  ->  ( h  e.  ( Base `  C
)  ->  ( h
(  ~=c𝑐  `  C )ring 
->  (ZeroO `  C )  =  (/) ) ) ) ) )
6059com34 86 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( h  e.  (TermO `  C )  ->  (
h  e.  ( Base `  C )  ->  ( Z  e.  (TermO `  C
)  ->  ( h
(  ~=c𝑐  `  C )ring 
->  (ZeroO `  C )  =  (/) ) ) ) ) )
6160com25 94 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( h (  ~=c𝑐  `  C
)ring 
->  ( h  e.  (
Base `  C )  ->  ( Z  e.  (TermO `  C )  ->  (
h  e.  (TermO `  C )  ->  (ZeroO `  C )  =  (/) ) ) ) ) )
6261ad2antrr 733 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  h  e.  (InitO `  C )
)  /\ring  e.  (InitO `  C
) )  ->  (
h (  ~=c𝑐  `  C )ring  -> 
( h  e.  (
Base `  C )  ->  ( Z  e.  (TermO `  C )  ->  (
h  e.  (TermO `  C )  ->  (ZeroO `  C )  =  (/) ) ) ) ) )
6317, 62mpd 15 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  h  e.  (InitO `  C )
)  /\ring  e.  (InitO `  C
) )  ->  (
h  e.  ( Base `  C )  ->  ( Z  e.  (TermO `  C
)  ->  ( h  e.  (TermO `  C )  ->  (ZeroO `  C )  =  (/) ) ) ) )
6463ex 436 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  h  e.  (InitO `  C ) )  ->  (ring  e.  (InitO `  C
)  ->  ( h  e.  ( Base `  C
)  ->  ( Z  e.  (TermO `  C )  ->  ( h  e.  (TermO `  C )  ->  (ZeroO `  C )  =  (/) ) ) ) ) )
6564com25 94 . . . . . . . . . . . . 13  |-  ( (
ph  /\  h  e.  (InitO `  C ) )  ->  ( h  e.  (TermO `  C )  ->  ( h  e.  (
Base `  C )  ->  ( Z  e.  (TermO `  C )  ->  (ring  e.  (InitO `  C )  -> 
(ZeroO `  C )  =  (/) ) ) ) ) )
6665expimpd 608 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( h  e.  (InitO `  C )  /\  h  e.  (TermO `  C ) )  -> 
( h  e.  (
Base `  C )  ->  ( Z  e.  (TermO `  C )  ->  (ring  e.  (InitO `  C )  -> 
(ZeroO `  C )  =  (/) ) ) ) ) )
6766com23 81 . . . . . . . . . . 11  |-  ( ph  ->  ( h  e.  (
Base `  C )  ->  ( ( h  e.  (InitO `  C )  /\  h  e.  (TermO `  C ) )  -> 
( Z  e.  (TermO `  C )  ->  (ring  e.  (InitO `  C )  -> 
(ZeroO `  C )  =  (/) ) ) ) ) )
6867impd 433 . . . . . . . . . 10  |-  ( ph  ->  ( ( h  e.  ( Base `  C
)  /\  ( h  e.  (InitO `  C )  /\  h  e.  (TermO `  C ) ) )  ->  ( Z  e.  (TermO `  C )  ->  (ring  e.  (InitO `  C
)  ->  (ZeroO `  C
)  =  (/) ) ) ) )
6968com24 90 . . . . . . . . 9  |-  ( ph  ->  (ring  e.  (InitO `  C
)  ->  ( Z  e.  (TermO `  C )  ->  ( ( h  e.  ( Base `  C
)  /\  ( h  e.  (InitO `  C )  /\  h  e.  (TermO `  C ) ) )  ->  (ZeroO `  C
)  =  (/) ) ) ) )
7013, 69mpd 15 . . . . . . . 8  |-  ( ph  ->  ( Z  e.  (TermO `  C )  ->  (
( h  e.  (
Base `  C )  /\  ( h  e.  (InitO `  C )  /\  h  e.  (TermO `  C )
) )  ->  (ZeroO `  C )  =  (/) ) ) )
7111, 70mpd 15 . . . . . . 7  |-  ( ph  ->  ( ( h  e.  ( Base `  C
)  /\  ( h  e.  (InitO `  C )  /\  h  e.  (TermO `  C ) ) )  ->  (ZeroO `  C
)  =  (/) ) )
7271adantr 467 . . . . . 6  |-  ( (
ph  /\  h  e.  (ZeroO `  C ) )  ->  ( ( h  e.  ( Base `  C
)  /\  ( h  e.  (InitO `  C )  /\  h  e.  (TermO `  C ) ) )  ->  (ZeroO `  C
)  =  (/) ) )
738, 72mpd 15 . . . . 5  |-  ( (
ph  /\  h  e.  (ZeroO `  C ) )  ->  (ZeroO `  C
)  =  (/) )
7473expcom 437 . . . 4  |-  ( h  e.  (ZeroO `  C
)  ->  ( ph  ->  (ZeroO `  C )  =  (/) ) )
7574exlimiv 1778 . . 3  |-  ( E. h  h  e.  (ZeroO `  C )  ->  ( ph  ->  (ZeroO `  C
)  =  (/) ) )
762, 75sylbi 199 . 2  |-  ( -.  (ZeroO `  C )  =  (/)  ->  ( ph  ->  (ZeroO `  C )  =  (/) ) )
771, 76pm2.61i 168 1  |-  ( ph  ->  (ZeroO `  C )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 371    /\ w3a 986    = wceq 1446   E.wex 1665    e. wcel 1889    =/= wne 2624    \ cdif 3403    i^i cin 3405    C_ wss 3406   (/)c0 3733   class class class wbr 4405   ` cfv 5585  (class class class)co 6295   Basecbs 15133   Hom chom 15213   Catccat 15582    Iso ciso 15663    ~=c𝑐 ccic 15712  InitOcinito 15895  TermOctermo 15896  ZeroOczeroo 15897   Ringcrg 17792   RingHom crh 17952  NzRingcnzr 18493  ℤringzring 19051  RingCatcringc 40109
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-inf2 8151  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621  ax-addf 9623  ax-mulf 9624
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-fal 1452  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-int 4238  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-1st 6798  df-2nd 6799  df-supp 6920  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-oadd 7191  df-er 7368  df-map 7479  df-pm 7480  df-ixp 7528  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-card 8378  df-cda 8603  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-nn 10617  df-2 10675  df-3 10676  df-4 10677  df-5 10678  df-6 10679  df-7 10680  df-8 10681  df-9 10682  df-10 10683  df-n0 10877  df-z 10945  df-dec 11059  df-uz 11167  df-fz 11792  df-seq 12221  df-hash 12523  df-struct 15135  df-ndx 15136  df-slot 15137  df-base 15138  df-sets 15139  df-ress 15140  df-plusg 15215  df-mulr 15216  df-starv 15217  df-tset 15221  df-ple 15222  df-ds 15224  df-unif 15225  df-hom 15226  df-cco 15227  df-0g 15352  df-cat 15586  df-cid 15587  df-homf 15588  df-sect 15664  df-inv 15665  df-iso 15666  df-cic 15713  df-ssc 15727  df-resc 15728  df-subc 15729  df-inito 15898  df-termo 15899  df-zeroo 15900  df-estrc 16020  df-mgm 16500  df-sgrp 16539  df-mnd 16549  df-mhm 16594  df-grp 16685  df-minusg 16686  df-mulg 16688  df-subg 16826  df-ghm 16893  df-cmn 17444  df-mgp 17736  df-ur 17748  df-ring 17794  df-cring 17795  df-rnghom 17955  df-subrg 18018  df-nzr 18494  df-cnfld 18983  df-zring 19052  df-ringc 40111
This theorem is referenced by: (None)
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