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Theorem rhmsscrnghm 40536
Description: The unital ring homomorphisms between unital rings (in a universe) are a subcategory subset of the non-unital ring homomorphisms between non-unital rings (in the same universe). (Contributed by AV, 1-Mar-2020.)
Hypotheses
Ref Expression
rhmsscrnghm.u  |-  ( ph  ->  U  e.  V )
rhmsscrnghm.r  |-  ( ph  ->  R  =  ( Ring 
i^i  U ) )
rhmsscrnghm.s  |-  ( ph  ->  S  =  (Rng  i^i  U ) )
Assertion
Ref Expression
rhmsscrnghm  |-  ( ph  ->  ( RingHom  |`  ( R  X.  R ) )  C_cat  ( RngHomo  |`  ( S  X.  S
) ) )

Proof of Theorem rhmsscrnghm
Dummy variables  h  x  y  r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ringrng 40387 . . . . . 6  |-  ( r  e.  Ring  ->  r  e. Rng )
21a1i 11 . . . . 5  |-  ( ph  ->  ( r  e.  Ring  -> 
r  e. Rng ) )
32ssrdv 3424 . . . 4  |-  ( ph  ->  Ring  C_ Rng )
4 ssrin 3648 . . . 4  |-  ( Ring  C_ Rng  ->  ( Ring  i^i  U )  C_  (Rng  i^i  U ) )
53, 4syl 17 . . 3  |-  ( ph  ->  ( Ring  i^i  U ) 
C_  (Rng  i^i  U
) )
6 rhmsscrnghm.r . . 3  |-  ( ph  ->  R  =  ( Ring 
i^i  U ) )
7 rhmsscrnghm.s . . 3  |-  ( ph  ->  S  =  (Rng  i^i  U ) )
85, 6, 73sstr4d 3461 . 2  |-  ( ph  ->  R  C_  S )
9 ovres 6455 . . . . . . 7  |-  ( ( x  e.  R  /\  y  e.  R )  ->  ( x ( RingHom  |`  ( R  X.  R ) ) y )  =  ( x RingHom  y ) )
109adantl 473 . . . . . 6  |-  ( (
ph  /\  ( x  e.  R  /\  y  e.  R ) )  -> 
( x ( RingHom  |`  ( R  X.  R ) ) y )  =  ( x RingHom  y ) )
1110eleq2d 2534 . . . . 5  |-  ( (
ph  /\  ( x  e.  R  /\  y  e.  R ) )  -> 
( h  e.  ( x ( RingHom  |`  ( R  X.  R ) ) y )  <->  h  e.  ( x RingHom  y ) ) )
12 rhmisrnghm 40428 . . . . . 6  |-  ( h  e.  ( x RingHom  y
)  ->  h  e.  ( x RngHomo  y ) )
138sseld 3417 . . . . . . . . . 10  |-  ( ph  ->  ( x  e.  R  ->  x  e.  S ) )
148sseld 3417 . . . . . . . . . 10  |-  ( ph  ->  ( y  e.  R  ->  y  e.  S ) )
1513, 14anim12d 572 . . . . . . . . 9  |-  ( ph  ->  ( ( x  e.  R  /\  y  e.  R )  ->  (
x  e.  S  /\  y  e.  S )
) )
1615imp 436 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  R  /\  y  e.  R ) )  -> 
( x  e.  S  /\  y  e.  S
) )
17 ovres 6455 . . . . . . . 8  |-  ( ( x  e.  S  /\  y  e.  S )  ->  ( x ( RngHomo  |`  ( S  X.  S ) ) y )  =  ( x RngHomo  y ) )
1816, 17syl 17 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  R  /\  y  e.  R ) )  -> 
( x ( RngHomo  |`  ( S  X.  S ) ) y )  =  ( x RngHomo  y ) )
1918eleq2d 2534 . . . . . 6  |-  ( (
ph  /\  ( x  e.  R  /\  y  e.  R ) )  -> 
( h  e.  ( x ( RngHomo  |`  ( S  X.  S ) ) y )  <->  h  e.  ( x RngHomo  y ) ) )
2012, 19syl5ibr 229 . . . . 5  |-  ( (
ph  /\  ( x  e.  R  /\  y  e.  R ) )  -> 
( h  e.  ( x RingHom  y )  ->  h  e.  ( x
( RngHomo  |`  ( S  X.  S ) ) y ) ) )
2111, 20sylbid 223 . . . 4  |-  ( (
ph  /\  ( x  e.  R  /\  y  e.  R ) )  -> 
( h  e.  ( x ( RingHom  |`  ( R  X.  R ) ) y )  ->  h  e.  ( x ( RngHomo  |`  ( S  X.  S ) ) y ) ) )
2221ssrdv 3424 . . 3  |-  ( (
ph  /\  ( x  e.  R  /\  y  e.  R ) )  -> 
( x ( RingHom  |`  ( R  X.  R ) ) y )  C_  (
x ( RngHomo  |`  ( S  X.  S ) ) y ) )
2322ralrimivva 2814 . 2  |-  ( ph  ->  A. x  e.  R  A. y  e.  R  ( x ( RingHom  |`  ( R  X.  R ) ) y )  C_  (
x ( RngHomo  |`  ( S  X.  S ) ) y ) )
24 inss1 3643 . . . . . 6  |-  ( Ring 
i^i  U )  C_  Ring
256, 24syl6eqss 3468 . . . . 5  |-  ( ph  ->  R  C_  Ring )
26 xpss12 4945 . . . . 5  |-  ( ( R  C_  Ring  /\  R  C_ 
Ring )  ->  ( R  X.  R )  C_  ( Ring  X.  Ring )
)
2725, 25, 26syl2anc 673 . . . 4  |-  ( ph  ->  ( R  X.  R
)  C_  ( Ring  X. 
Ring ) )
28 rhmfn 40426 . . . . 5  |- RingHom  Fn  ( Ring  X.  Ring )
29 fnssresb 5698 . . . . 5  |-  ( RingHom  Fn  ( Ring  X.  Ring )  ->  ( ( RingHom  |`  ( R  X.  R ) )  Fn  ( R  X.  R )  <->  ( R  X.  R )  C_  ( Ring  X.  Ring ) ) )
3028, 29mp1i 13 . . . 4  |-  ( ph  ->  ( ( RingHom  |`  ( R  X.  R ) )  Fn  ( R  X.  R )  <->  ( R  X.  R )  C_  ( Ring  X.  Ring ) ) )
3127, 30mpbird 240 . . 3  |-  ( ph  ->  ( RingHom  |`  ( R  X.  R ) )  Fn  ( R  X.  R
) )
32 inss1 3643 . . . . . 6  |-  (Rng  i^i  U )  C_ Rng
337, 32syl6eqss 3468 . . . . 5  |-  ( ph  ->  S  C_ Rng )
34 xpss12 4945 . . . . 5  |-  ( ( S  C_ Rng  /\  S  C_ Rng )  ->  ( S  X.  S )  C_  (Rng  X. Rng ) )
3533, 33, 34syl2anc 673 . . . 4  |-  ( ph  ->  ( S  X.  S
)  C_  (Rng  X. Rng ) )
36 rnghmfn 40398 . . . . 5  |- RngHomo  Fn  (Rng  X. Rng )
37 fnssresb 5698 . . . . 5  |-  ( RngHomo  Fn  (Rng  X. Rng )  ->  (
( RngHomo  |`  ( S  X.  S ) )  Fn  ( S  X.  S
)  <->  ( S  X.  S )  C_  (Rng  X. Rng ) ) )
3836, 37mp1i 13 . . . 4  |-  ( ph  ->  ( ( RngHomo  |`  ( S  X.  S ) )  Fn  ( S  X.  S )  <->  ( S  X.  S )  C_  (Rng  X. Rng ) ) )
3935, 38mpbird 240 . . 3  |-  ( ph  ->  ( RngHomo  |`  ( S  X.  S ) )  Fn  ( S  X.  S
) )
40 rhmsscrnghm.u . . . . 5  |-  ( ph  ->  U  e.  V )
41 incom 3616 . . . . . 6  |-  (Rng  i^i  U )  =  ( U  i^i Rng )
42 inex1g 4539 . . . . . 6  |-  ( U  e.  V  ->  ( U  i^i Rng )  e.  _V )
4341, 42syl5eqel 2553 . . . . 5  |-  ( U  e.  V  ->  (Rng  i^i  U )  e.  _V )
4440, 43syl 17 . . . 4  |-  ( ph  ->  (Rng  i^i  U )  e.  _V )
457, 44eqeltrd 2549 . . 3  |-  ( ph  ->  S  e.  _V )
4631, 39, 45isssc 15803 . 2  |-  ( ph  ->  ( ( RingHom  |`  ( R  X.  R ) ) 
C_cat  ( RngHomo  |`  ( S  X.  S ) )  <->  ( R  C_  S  /\  A. x  e.  R  A. y  e.  R  ( x
( RingHom  |`  ( R  X.  R ) ) y )  C_  ( x
( RngHomo  |`  ( S  X.  S ) ) y ) ) ) )
478, 23, 46mpbir2and 936 1  |-  ( ph  ->  ( RingHom  |`  ( R  X.  R ) )  C_cat  ( RngHomo  |`  ( S  X.  S
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904   A.wral 2756   _Vcvv 3031    i^i cin 3389    C_ wss 3390   class class class wbr 4395    X. cxp 4837    |` cres 4841    Fn wfn 5584  (class class class)co 6308    C_cat cssc 15790   Ringcrg 17858   RingHom crh 18018  Rngcrng 40382   RngHomo crngh 40393
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-rep 4508  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-fal 1458  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-rmo 2764  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-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-er 7381  df-map 7492  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  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-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-plusg 15281  df-0g 15418  df-ssc 15793  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-mhm 16660  df-grp 16751  df-minusg 16752  df-ghm 16959  df-cmn 17510  df-abl 17511  df-mgp 17802  df-ur 17814  df-ring 17860  df-rnghom 18021  df-mgmhm 40287  df-rng0 40383  df-rnghomo 40395
This theorem is referenced by:  rhmsubcrngc  40539  rhmsubc  40600  rhmsubcALTV  40619
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