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Theorem fldhmsubcALTV 40377
Description: According to df-subc 15765, the subcategories  (Subcat `  C ) of a category  C are subsets of the homomorphisms of  C ( see subcssc 15793 and subcss2 15796). Therefore, the set of field homomorphisms is a "subcategory" of the category of division rings. (Contributed by AV, 20-Feb-2020.) (New usage is discouraged.)
Hypotheses
Ref Expression
drhmsubcALTV.c  |-  C  =  ( U  i^i  DivRing )
drhmsubcALTV.j  |-  J  =  ( r  e.  C ,  s  e.  C  |->  ( r RingHom  s ) )
fldhmsubcALTV.d  |-  D  =  ( U  i^i Field )
fldhmsubcALTV.f  |-  F  =  ( r  e.  D ,  s  e.  D  |->  ( r RingHom  s ) )
Assertion
Ref Expression
fldhmsubcALTV  |-  ( U  e.  V  ->  F  e.  (Subcat `  ( (RingCatALTV `  U )  |`cat  J )
) )
Distinct variable groups:    C, r,
s    U, r, s    V, r, s    D, r, s
Allowed substitution hints:    F( s, r)    J( s, r)

Proof of Theorem fldhmsubcALTV
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elin 3628 . . . . . . 7  |-  ( r  e.  ( DivRing  i^i  CRing )  <-> 
( r  e.  DivRing  /\  r  e.  CRing ) )
21simprbi 470 . . . . . 6  |-  ( r  e.  ( DivRing  i^i  CRing )  ->  r  e.  CRing )
3 crngring 17839 . . . . . 6  |-  ( r  e.  CRing  ->  r  e.  Ring )
42, 3syl 17 . . . . 5  |-  ( r  e.  ( DivRing  i^i  CRing )  ->  r  e.  Ring )
5 df-field 18026 . . . . 5  |- Field  =  (
DivRing  i^i  CRing )
64, 5eleq2s 2557 . . . 4  |-  ( r  e. Field  ->  r  e.  Ring )
76rgen 2758 . . 3  |-  A. r  e. Field  r  e.  Ring
8 fldhmsubcALTV.d . . 3  |-  D  =  ( U  i^i Field )
9 fldhmsubcALTV.f . . 3  |-  F  =  ( r  e.  D ,  s  e.  D  |->  ( r RingHom  s ) )
107, 8, 9srhmsubcALTV 40369 . 2  |-  ( U  e.  V  ->  F  e.  (Subcat `  (RingCatALTV `  U
) ) )
11 inss1 3663 . . . . . . 7  |-  ( DivRing  i^i  CRing )  C_  DivRing
125, 11eqsstri 3473 . . . . . 6  |- Field  C_  DivRing
13 sslin 3669 . . . . . 6  |-  (Field  C_  DivRing  ->  ( U  i^i Field )  C_  ( U  i^i  DivRing ) )
1412, 13ax-mp 5 . . . . 5  |-  ( U  i^i Field )  C_  ( U  i^i  DivRing )
1514a1i 11 . . . 4  |-  ( U  e.  V  ->  ( U  i^i Field )  C_  ( U  i^i  DivRing ) )
16 drhmsubcALTV.c . . . . 5  |-  C  =  ( U  i^i  DivRing )
178, 16sseq12i 3469 . . . 4  |-  ( D 
C_  C  <->  ( U  i^i Field )  C_  ( U  i^i 
DivRing ) )
1815, 17sylibr 217 . . 3  |-  ( U  e.  V  ->  D  C_  C )
19 ssid 3462 . . . . . 6  |-  ( x RingHom 
y )  C_  (
x RingHom  y )
2019a1i 11 . . . . 5  |-  ( ( U  e.  V  /\  ( x  e.  D  /\  y  e.  D
) )  ->  (
x RingHom  y )  C_  (
x RingHom  y ) )
219a1i 11 . . . . . 6  |-  ( ( U  e.  V  /\  ( x  e.  D  /\  y  e.  D
) )  ->  F  =  ( r  e.  D ,  s  e.  D  |->  ( r RingHom  s
) ) )
22 oveq12 6323 . . . . . . 7  |-  ( ( r  =  x  /\  s  =  y )  ->  ( r RingHom  s )  =  ( x RingHom  y
) )
2322adantl 472 . . . . . 6  |-  ( ( ( U  e.  V  /\  ( x  e.  D  /\  y  e.  D
) )  /\  (
r  =  x  /\  s  =  y )
)  ->  ( r RingHom  s )  =  ( x RingHom 
y ) )
24 simprl 769 . . . . . 6  |-  ( ( U  e.  V  /\  ( x  e.  D  /\  y  e.  D
) )  ->  x  e.  D )
25 simpr 467 . . . . . . 7  |-  ( ( x  e.  D  /\  y  e.  D )  ->  y  e.  D )
2625adantl 472 . . . . . 6  |-  ( ( U  e.  V  /\  ( x  e.  D  /\  y  e.  D
) )  ->  y  e.  D )
27 ovex 6342 . . . . . . 7  |-  ( x RingHom 
y )  e.  _V
2827a1i 11 . . . . . 6  |-  ( ( U  e.  V  /\  ( x  e.  D  /\  y  e.  D
) )  ->  (
x RingHom  y )  e.  _V )
2921, 23, 24, 26, 28ovmpt2d 6450 . . . . 5  |-  ( ( U  e.  V  /\  ( x  e.  D  /\  y  e.  D
) )  ->  (
x F y )  =  ( x RingHom  y
) )
30 drhmsubcALTV.j . . . . . . 7  |-  J  =  ( r  e.  C ,  s  e.  C  |->  ( r RingHom  s ) )
3130a1i 11 . . . . . 6  |-  ( ( U  e.  V  /\  ( x  e.  D  /\  y  e.  D
) )  ->  J  =  ( r  e.  C ,  s  e.  C  |->  ( r RingHom  s
) ) )
3214, 17mpbir 214 . . . . . . . 8  |-  D  C_  C
3332sseli 3439 . . . . . . 7  |-  ( x  e.  D  ->  x  e.  C )
3433ad2antrl 739 . . . . . 6  |-  ( ( U  e.  V  /\  ( x  e.  D  /\  y  e.  D
) )  ->  x  e.  C )
3532sseli 3439 . . . . . . . 8  |-  ( y  e.  D  ->  y  e.  C )
3635adantl 472 . . . . . . 7  |-  ( ( x  e.  D  /\  y  e.  D )  ->  y  e.  C )
3736adantl 472 . . . . . 6  |-  ( ( U  e.  V  /\  ( x  e.  D  /\  y  e.  D
) )  ->  y  e.  C )
3831, 23, 34, 37, 28ovmpt2d 6450 . . . . 5  |-  ( ( U  e.  V  /\  ( x  e.  D  /\  y  e.  D
) )  ->  (
x J y )  =  ( x RingHom  y
) )
3920, 29, 383sstr4d 3486 . . . 4  |-  ( ( U  e.  V  /\  ( x  e.  D  /\  y  e.  D
) )  ->  (
x F y ) 
C_  ( x J y ) )
4039ralrimivva 2820 . . 3  |-  ( U  e.  V  ->  A. x  e.  D  A. y  e.  D  ( x F y )  C_  ( x J y ) )
41 ovex 6342 . . . . . 6  |-  ( r RingHom 
s )  e.  _V
429, 41fnmpt2i 6888 . . . . 5  |-  F  Fn  ( D  X.  D
)
4342a1i 11 . . . 4  |-  ( U  e.  V  ->  F  Fn  ( D  X.  D
) )
4430, 41fnmpt2i 6888 . . . . 5  |-  J  Fn  ( C  X.  C
)
4544a1i 11 . . . 4  |-  ( U  e.  V  ->  J  Fn  ( C  X.  C
) )
46 inex1g 4559 . . . . 5  |-  ( U  e.  V  ->  ( U  i^i  DivRing )  e.  _V )
4716, 46syl5eqel 2543 . . . 4  |-  ( U  e.  V  ->  C  e.  _V )
4843, 45, 47isssc 15773 . . 3  |-  ( U  e.  V  ->  ( F  C_cat  J  <->  ( D  C_  C  /\  A. x  e.  D  A. y  e.  D  ( x F y )  C_  (
x J y ) ) ) )
4918, 40, 48mpbir2and 938 . 2  |-  ( U  e.  V  ->  F  C_cat  J )
5016, 30drhmsubcALTV 40373 . . 3  |-  ( U  e.  V  ->  J  e.  (Subcat `  (RingCatALTV `  U
) ) )
51 eqid 2461 . . . 4  |-  ( (RingCatALTV `  U )  |`cat  J )  =  ( (RingCatALTV `  U
)  |`cat  J )
5251subsubc 15806 . . 3  |-  ( J  e.  (Subcat `  (RingCatALTV `  U ) )  -> 
( F  e.  (Subcat `  ( (RingCatALTV `  U )  |`cat  J
) )  <->  ( F  e.  (Subcat `  (RingCatALTV `  U
) )  /\  F  C_cat  J ) ) )
5350, 52syl 17 . 2  |-  ( U  e.  V  ->  ( F  e.  (Subcat `  (
(RingCatALTV `  U )  |`cat  J
) )  <->  ( F  e.  (Subcat `  (RingCatALTV `  U
) )  /\  F  C_cat  J ) ) )
5410, 49, 53mpbir2and 938 1  |-  ( U  e.  V  ->  F  e.  (Subcat `  ( (RingCatALTV `  U )  |`cat  J )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375    = wceq 1454    e. wcel 1897   A.wral 2748   _Vcvv 3056    i^i cin 3414    C_ wss 3415   class class class wbr 4415    X. cxp 4850    Fn wfn 5595   ` cfv 5600  (class class class)co 6314    |-> cmpt2 6316    C_cat cssc 15760    |`cat cresc 15761  Subcatcsubc 15762   Ringcrg 17828   CRingccrg 17829   RingHom crh 17988   DivRingcdr 18023  Fieldcfield 18024  RingCatALTVcringcALTV 40278
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-rep 4528  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609  ax-cnex 9620  ax-resscn 9621  ax-1cn 9622  ax-icn 9623  ax-addcl 9624  ax-addrcl 9625  ax-mulcl 9626  ax-mulrcl 9627  ax-mulcom 9628  ax-addass 9629  ax-mulass 9630  ax-distr 9631  ax-i2m1 9632  ax-1ne0 9633  ax-1rid 9634  ax-rnegex 9635  ax-rrecex 9636  ax-cnre 9637  ax-pre-lttri 9638  ax-pre-lttrn 9639  ax-pre-ltadd 9640  ax-pre-mulgt0 9641
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1457  df-fal 1460  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-nel 2635  df-ral 2753  df-rex 2754  df-reu 2755  df-rmo 2756  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-pss 3431  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-tp 3984  df-op 3986  df-uni 4212  df-int 4248  df-iun 4293  df-br 4416  df-opab 4475  df-mpt 4476  df-tr 4511  df-eprel 4763  df-id 4767  df-po 4773  df-so 4774  df-fr 4811  df-we 4813  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-pred 5398  df-ord 5444  df-on 5445  df-lim 5446  df-suc 5447  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-riota 6276  df-ov 6317  df-oprab 6318  df-mpt2 6319  df-om 6719  df-1st 6819  df-2nd 6820  df-wrecs 7053  df-recs 7115  df-rdg 7153  df-1o 7207  df-oadd 7211  df-er 7388  df-map 7499  df-pm 7500  df-ixp 7548  df-en 7595  df-dom 7596  df-sdom 7597  df-fin 7598  df-pnf 9702  df-mnf 9703  df-xr 9704  df-ltxr 9705  df-le 9706  df-sub 9887  df-neg 9888  df-nn 10637  df-2 10695  df-3 10696  df-4 10697  df-5 10698  df-6 10699  df-7 10700  df-8 10701  df-9 10702  df-10 10703  df-n0 10898  df-z 10966  df-dec 11080  df-uz 11188  df-fz 11813  df-struct 15171  df-ndx 15172  df-slot 15173  df-base 15174  df-sets 15175  df-ress 15176  df-plusg 15251  df-hom 15262  df-cco 15263  df-0g 15388  df-cat 15622  df-cid 15623  df-homf 15624  df-ssc 15763  df-resc 15764  df-subc 15765  df-mgm 16536  df-sgrp 16575  df-mnd 16585  df-mhm 16630  df-grp 16721  df-ghm 16929  df-mgp 17772  df-ur 17784  df-ring 17830  df-cring 17831  df-rnghom 17991  df-drng 18025  df-field 18026  df-ringcALTV 40280
This theorem is referenced by: (None)
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