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Theorem rhmsscrnghm 33021
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 32872 . . . . . 6  |-  ( r  e.  Ring  ->  r  e. Rng )
21a1i 11 . . . . 5  |-  ( ph  ->  ( r  e.  Ring  -> 
r  e. Rng ) )
32ssrdv 3505 . . . 4  |-  ( ph  ->  Ring  C_ Rng )
4 ssrin 3719 . . . 4  |-  ( Ring  C_ Rng  ->  ( Ring  i^i  U )  C_  (Rng  i^i  U ) )
53, 4syl 16 . . 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 3542 . 2  |-  ( ph  ->  R  C_  S )
9 ovres 6441 . . . . . . 7  |-  ( ( x  e.  R  /\  y  e.  R )  ->  ( x ( RingHom  |`  ( R  X.  R ) ) y )  =  ( x RingHom  y ) )
109adantl 466 . . . . . 6  |-  ( (
ph  /\  ( x  e.  R  /\  y  e.  R ) )  -> 
( x ( RingHom  |`  ( R  X.  R ) ) y )  =  ( x RingHom  y ) )
1110eleq2d 2527 . . . . 5  |-  ( (
ph  /\  ( x  e.  R  /\  y  e.  R ) )  -> 
( h  e.  ( x ( RingHom  |`  ( R  X.  R ) ) y )  <->  h  e.  ( x RingHom  y ) ) )
12 rhmisrnghm 32913 . . . . . 6  |-  ( h  e.  ( x RingHom  y
)  ->  h  e.  ( x RngHomo  y ) )
138sseld 3498 . . . . . . . . . 10  |-  ( ph  ->  ( x  e.  R  ->  x  e.  S ) )
148sseld 3498 . . . . . . . . . 10  |-  ( ph  ->  ( y  e.  R  ->  y  e.  S ) )
1513, 14anim12d 563 . . . . . . . . 9  |-  ( ph  ->  ( ( x  e.  R  /\  y  e.  R )  ->  (
x  e.  S  /\  y  e.  S )
) )
1615imp 429 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  R  /\  y  e.  R ) )  -> 
( x  e.  S  /\  y  e.  S
) )
17 ovres 6441 . . . . . . . 8  |-  ( ( x  e.  S  /\  y  e.  S )  ->  ( x ( RngHomo  |`  ( S  X.  S ) ) y )  =  ( x RngHomo  y ) )
1816, 17syl 16 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  R  /\  y  e.  R ) )  -> 
( x ( RngHomo  |`  ( S  X.  S ) ) y )  =  ( x RngHomo  y ) )
1918eleq2d 2527 . . . . . 6  |-  ( (
ph  /\  ( x  e.  R  /\  y  e.  R ) )  -> 
( h  e.  ( x ( RngHomo  |`  ( S  X.  S ) ) y )  <->  h  e.  ( x RngHomo  y ) ) )
2012, 19syl5ibr 221 . . . . 5  |-  ( (
ph  /\  ( x  e.  R  /\  y  e.  R ) )  -> 
( h  e.  ( x RingHom  y )  ->  h  e.  ( x
( RngHomo  |`  ( S  X.  S ) ) y ) ) )
2111, 20sylbid 215 . . . 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 3505 . . 3  |-  ( (
ph  /\  ( x  e.  R  /\  y  e.  R ) )  -> 
( x ( RingHom  |`  ( R  X.  R ) ) y )  C_  (
x ( RngHomo  |`  ( S  X.  S ) ) y ) )
2322ralrimivva 2878 . 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 3714 . . . . . 6  |-  ( Ring 
i^i  U )  C_  Ring
256, 24syl6eqss 3549 . . . . 5  |-  ( ph  ->  R  C_  Ring )
26 xpss12 5117 . . . . 5  |-  ( ( R  C_  Ring  /\  R  C_ 
Ring )  ->  ( R  X.  R )  C_  ( Ring  X.  Ring )
)
2725, 25, 26syl2anc 661 . . . 4  |-  ( ph  ->  ( R  X.  R
)  C_  ( Ring  X. 
Ring ) )
28 rhmfn 32911 . . . . 5  |- RingHom  Fn  ( Ring  X.  Ring )
29 fnssresb 5699 . . . . 5  |-  ( RingHom  Fn  ( Ring  X.  Ring )  ->  ( ( RingHom  |`  ( R  X.  R ) )  Fn  ( R  X.  R )  <->  ( R  X.  R )  C_  ( Ring  X.  Ring ) ) )
3028, 29mp1i 12 . . . 4  |-  ( ph  ->  ( ( RingHom  |`  ( R  X.  R ) )  Fn  ( R  X.  R )  <->  ( R  X.  R )  C_  ( Ring  X.  Ring ) ) )
3127, 30mpbird 232 . . 3  |-  ( ph  ->  ( RingHom  |`  ( R  X.  R ) )  Fn  ( R  X.  R
) )
32 inss1 3714 . . . . . 6  |-  (Rng  i^i  U )  C_ Rng
337, 32syl6eqss 3549 . . . . 5  |-  ( ph  ->  S  C_ Rng )
34 xpss12 5117 . . . . 5  |-  ( ( S  C_ Rng  /\  S  C_ Rng )  ->  ( S  X.  S )  C_  (Rng  X. Rng ) )
3533, 33, 34syl2anc 661 . . . 4  |-  ( ph  ->  ( S  X.  S
)  C_  (Rng  X. Rng ) )
36 rnghmfn 32883 . . . . 5  |- RngHomo  Fn  (Rng  X. Rng )
37 fnssresb 5699 . . . . 5  |-  ( RngHomo  Fn  (Rng  X. Rng )  ->  (
( RngHomo  |`  ( S  X.  S ) )  Fn  ( S  X.  S
)  <->  ( S  X.  S )  C_  (Rng  X. Rng ) ) )
3836, 37mp1i 12 . . . 4  |-  ( ph  ->  ( ( RngHomo  |`  ( S  X.  S ) )  Fn  ( S  X.  S )  <->  ( S  X.  S )  C_  (Rng  X. Rng ) ) )
3935, 38mpbird 232 . . 3  |-  ( ph  ->  ( RngHomo  |`  ( S  X.  S ) )  Fn  ( S  X.  S
) )
40 rhmsscrnghm.u . . . . 5  |-  ( ph  ->  U  e.  V )
41 incom 3687 . . . . . 6  |-  (Rng  i^i  U )  =  ( U  i^i Rng )
42 inex1g 4599 . . . . . 6  |-  ( U  e.  V  ->  ( U  i^i Rng )  e.  _V )
4341, 42syl5eqel 2549 . . . . 5  |-  ( U  e.  V  ->  (Rng  i^i  U )  e.  _V )
4440, 43syl 16 . . . 4  |-  ( ph  ->  (Rng  i^i  U )  e.  _V )
457, 44eqeltrd 2545 . . 3  |-  ( ph  ->  S  e.  _V )
4631, 39, 45isssc 15327 . 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 922 1  |-  ( ph  ->  ( RingHom  |`  ( R  X.  R ) )  C_cat  ( RngHomo  |`  ( S  X.  S
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   A.wral 2807   _Vcvv 3109    i^i cin 3470    C_ wss 3471   class class class wbr 4456    X. cxp 5006    |` cres 5010    Fn wfn 5589  (class class class)co 6296    C_cat cssc 15314   Ringcrg 17416   RingHom crh 17579  Rngcrng 32867   RngHomo crngh 32878
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-rmo 2815  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-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-er 7329  df-map 7440  df-ixp 7489  df-en 7536  df-dom 7537  df-sdom 7538  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-ndx 14738  df-slot 14739  df-base 14740  df-sets 14741  df-plusg 14816  df-0g 14950  df-ssc 15317  df-mgm 16090  df-sgrp 16129  df-mnd 16139  df-mhm 16184  df-grp 16275  df-minusg 16276  df-ghm 16483  df-cmn 17018  df-abl 17019  df-mgp 17360  df-ur 17372  df-ring 17418  df-rnghom 17582  df-mgmhm 32772  df-rng0 32868  df-rnghomo 32880
This theorem is referenced by:  rhmsubcrngc  33024  rhmsubc  33085  rhmsubcOLD  33104
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