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Theorem rhmeql 15898
Description: The equalizer of two ring homomorphisms is a subring. (Contributed by Stefan O'Rear, 7-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.)
Assertion
Ref Expression
rhmeql  |-  ( ( F  e.  ( S RingHom  T )  /\  G  e.  ( S RingHom  T )
)  ->  dom  ( F  i^i  G )  e.  (SubRing `  S )
)

Proof of Theorem rhmeql
StepHypRef Expression
1 rhmghm 15826 . . 3  |-  ( F  e.  ( S RingHom  T
)  ->  F  e.  ( S  GrpHom  T ) )
2 rhmghm 15826 . . 3  |-  ( G  e.  ( S RingHom  T
)  ->  G  e.  ( S  GrpHom  T ) )
3 ghmeql 15028 . . 3  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  G  e.  ( S  GrpHom  T ) )  ->  dom  ( F  i^i  G )  e.  (SubGrp `  S )
)
41, 2, 3syl2an 464 . 2  |-  ( ( F  e.  ( S RingHom  T )  /\  G  e.  ( S RingHom  T )
)  ->  dom  ( F  i^i  G )  e.  (SubGrp `  S )
)
5 eqid 2436 . . . 4  |-  (mulGrp `  S )  =  (mulGrp `  S )
6 eqid 2436 . . . 4  |-  (mulGrp `  T )  =  (mulGrp `  T )
75, 6rhmmhm 15825 . . 3  |-  ( F  e.  ( S RingHom  T
)  ->  F  e.  ( (mulGrp `  S ) MndHom  (mulGrp `  T ) ) )
85, 6rhmmhm 15825 . . 3  |-  ( G  e.  ( S RingHom  T
)  ->  G  e.  ( (mulGrp `  S ) MndHom  (mulGrp `  T ) ) )
9 mhmeql 14764 . . 3  |-  ( ( F  e.  ( (mulGrp `  S ) MndHom  (mulGrp `  T ) )  /\  G  e.  ( (mulGrp `  S ) MndHom  (mulGrp `  T ) ) )  ->  dom  ( F  i^i  G )  e.  (SubMnd `  (mulGrp `  S )
) )
107, 8, 9syl2an 464 . 2  |-  ( ( F  e.  ( S RingHom  T )  /\  G  e.  ( S RingHom  T )
)  ->  dom  ( F  i^i  G )  e.  (SubMnd `  (mulGrp `  S
) ) )
11 rhmrcl1 15822 . . . 4  |-  ( F  e.  ( S RingHom  T
)  ->  S  e.  Ring )
1211adantr 452 . . 3  |-  ( ( F  e.  ( S RingHom  T )  /\  G  e.  ( S RingHom  T )
)  ->  S  e.  Ring )
135issubrg3 15896 . . 3  |-  ( S  e.  Ring  ->  ( dom  ( F  i^i  G
)  e.  (SubRing `  S
)  <->  ( dom  ( F  i^i  G )  e.  (SubGrp `  S )  /\  dom  ( F  i^i  G )  e.  (SubMnd `  (mulGrp `  S ) ) ) ) )
1412, 13syl 16 . 2  |-  ( ( F  e.  ( S RingHom  T )  /\  G  e.  ( S RingHom  T )
)  ->  ( dom  ( F  i^i  G )  e.  (SubRing `  S
)  <->  ( dom  ( F  i^i  G )  e.  (SubGrp `  S )  /\  dom  ( F  i^i  G )  e.  (SubMnd `  (mulGrp `  S ) ) ) ) )
154, 10, 14mpbir2and 889 1  |-  ( ( F  e.  ( S RingHom  T )  /\  G  e.  ( S RingHom  T )
)  ->  dom  ( F  i^i  G )  e.  (SubRing `  S )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    e. wcel 1725    i^i cin 3319   dom cdm 4878   ` cfv 5454  (class class class)co 6081   MndHom cmhm 14736  SubMndcsubmnd 14737  SubGrpcsubg 14938    GrpHom cghm 15003  mulGrpcmgp 15648   Ringcrg 15660   RingHom crh 15817  SubRingcsubrg 15864
This theorem is referenced by:  evlseu  19937
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-3 10059  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-mulr 13543  df-0g 13727  df-mnd 14690  df-mhm 14738  df-submnd 14739  df-grp 14812  df-minusg 14813  df-subg 14941  df-ghm 15004  df-mgp 15649  df-rng 15663  df-ur 15665  df-rnghom 15819  df-subrg 15866
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