Theorem rnghmsscmap2 40483

Description: The non-unital ring homomorphisms between non-unital rings (in a universe) are a subcategory subset of the mappings between base sets of non-unital rings (in the same universe). (Contributed by AV, 6-Mar-2020.)

Hypotheses

Ref	Expression
rnghmsscmap.u	|- ( ph -> U e. V )
rnghmsscmap.r	|- ( ph -> R = (Rng i^i U ) )

Assertion

Ref	Expression
rnghmsscmap2	|- ( ph -> ( RngHomo |` ( R X. R ) ) C_cat ( x e. R , y e. R |-> ( ( Base ` y ) ^m ( Base ` x ) ) ) )

Distinct variable group:   x, R, y

Allowed substitution hints:   ph( x, y)   U( x, y)   V( x, y)

Proof of Theorem rnghmsscmap2

Dummy variables a b h are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssid 3437	. . 3 |- R C_ R
2	1	a1i 11	. 2 |- ( ph -> R C_ R )
3		eqid 2471	. . . . . . 7 |- ( Base ` a ) = ( Base ` a )
4		eqid 2471	. . . . . . 7 |- ( Base ` b ) = ( Base ` b )
5	3, 4	rnghmf 40407	. . . . . 6 |- ( h e. ( a RngHomo b ) -> h : ( Base ` a ) --> ( Base ` b ) )
6		simpr 468	. . . . . . . 8 |- ( ( ( ph /\ ( a e. R /\ b e. R ) ) /\ h : ( Base ` a ) --> ( Base ` b ) ) -> h : ( Base ` a ) --> ( Base ` b ) )
7		fvex 5889	. . . . . . . . . 10 |- ( Base ` b ) e. _V
8		fvex 5889	. . . . . . . . . 10 |- ( Base ` a ) e. _V
9	7, 8	pm3.2i 462	. . . . . . . . 9 |- ( ( Base ` b ) e. _V /\ ( Base ` a ) e. _V )
10		elmapg 7503	. . . . . . . . 9 |- ( ( ( Base ` b ) e. _V /\ ( Base ` a ) e. _V ) -> ( h e. ( ( Base ` b ) ^m ( Base ` a ) ) <-> h : ( Base ` a ) --> ( Base ` b ) ) )
11	9, 10	mp1i 13	. . . . . . . 8 |- ( ( ( ph /\ ( a e. R /\ b e. R ) ) /\ h : ( Base ` a ) --> ( Base ` b ) ) -> ( h e. ( ( Base ` b ) ^m ( Base ` a ) ) <-> h : ( Base ` a ) --> ( Base ` b ) ) )
12	6, 11	mpbird 240	. . . . . . 7 |- ( ( ( ph /\ ( a e. R /\ b e. R ) ) /\ h : ( Base ` a ) --> ( Base ` b ) ) -> h e. ( ( Base ` b ) ^m ( Base ` a ) ) )
13	12	ex 441	. . . . . 6 |- ( ( ph /\ ( a e. R /\ b e. R ) ) -> ( h : ( Base ` a ) --> ( Base ` b ) -> h e. ( ( Base ` b ) ^m ( Base ` a ) ) ) )
14	5, 13	syl5 32	. . . . 5 |- ( ( ph /\ ( a e. R /\ b e. R ) ) -> ( h e. ( a RngHomo b ) -> h e. ( ( Base ` b ) ^m ( Base ` a ) ) ) )
15	14	ssrdv 3424	. . . 4 |- ( ( ph /\ ( a e. R /\ b e. R ) ) -> ( a RngHomo b ) C_ ( ( Base ` b ) ^m ( Base ` a ) ) )
16		ovres 6455	. . . . 5 |- ( ( a e. R /\ b e. R ) -> ( a ( RngHomo |` ( R X. R ) ) b ) = ( a RngHomo b ) )
17	16	adantl 473	. . . 4 |- ( ( ph /\ ( a e. R /\ b e. R ) ) -> ( a ( RngHomo |` ( R X. R ) ) b ) = ( a RngHomo b ) )
18		eqidd 2472	. . . . . 6 |- ( ( a e. R /\ b e. R ) -> ( x e. R , y e. R |-> ( ( Base ` y ) ^m ( Base ` x ) ) ) = ( x e. R , y e. R |-> ( ( Base ` y ) ^m ( Base ` x ) ) ) )
19		fveq2 5879	. . . . . . . 8 |- ( y = b -> ( Base ` y ) = ( Base ` b ) )
20		fveq2 5879	. . . . . . . 8 |- ( x = a -> ( Base ` x ) = ( Base ` a ) )
21	19, 20	oveqan12rd 6328	. . . . . . 7 |- ( ( x = a /\ y = b ) -> ( ( Base ` y ) ^m ( Base ` x ) ) = ( ( Base ` b ) ^m ( Base ` a ) ) )
22	21	adantl 473	. . . . . 6 |- ( ( ( a e. R /\ b e. R ) /\ ( x = a /\ y = b ) ) -> ( ( Base ` y ) ^m ( Base ` x ) ) = ( ( Base ` b ) ^m ( Base ` a ) ) )
23		simpl 464	. . . . . 6 |- ( ( a e. R /\ b e. R ) -> a e. R )
24		simpr 468	. . . . . 6 |- ( ( a e. R /\ b e. R ) -> b e. R )
25		ovex 6336	. . . . . . 7 |- ( ( Base ` b ) ^m ( Base ` a ) ) e. _V
26	25	a1i 11	. . . . . 6 |- ( ( a e. R /\ b e. R ) -> ( ( Base ` b ) ^m ( Base ` a ) ) e. _V )
27	18, 22, 23, 24, 26	ovmpt2d 6443	. . . . 5 |- ( ( a e. R /\ b e. R ) -> ( a ( x e. R , y e. R |-> ( ( Base ` y ) ^m ( Base ` x ) ) ) b ) = ( ( Base ` b ) ^m ( Base ` a ) ) )
28	27	adantl 473	. . . 4 |- ( ( ph /\ ( a e. R /\ b e. R ) ) -> ( a ( x e. R , y e. R |-> ( ( Base ` y ) ^m ( Base ` x ) ) ) b ) = ( ( Base ` b ) ^m ( Base ` a ) ) )
29	15, 17, 28	3sstr4d 3461	. . 3 |- ( ( ph /\ ( a e. R /\ b e. R ) ) -> ( a ( RngHomo |` ( R X. R ) ) b ) C_ ( a ( x e. R , y e. R |-> ( ( Base ` y ) ^m ( Base ` x ) ) ) b ) )
30	29	ralrimivva 2814	. 2 |- ( ph -> A. a e. R A. b e. R ( a ( RngHomo |` ( R X. R ) ) b ) C_ ( a ( x e. R , y e. R |-> ( ( Base ` y ) ^m ( Base ` x ) ) ) b ) )
31		rnghmfn 40398	. . . . 5 |- RngHomo Fn (Rng X. Rng )
32	31	a1i 11	. . . 4 |- ( ph -> RngHomo Fn (Rng X. Rng ) )
33		rnghmsscmap.r	. . . . . 6 |- ( ph -> R = (Rng i^i U ) )
34		inss1 3643	. . . . . 6 |- (Rng i^i U ) C_ Rng
35	33, 34	syl6eqss 3468	. . . . 5 |- ( ph -> R C_ Rng )
36		xpss12 4945	. . . . 5 |- ( ( R C_ Rng /\ R C_ Rng ) -> ( R X. R ) C_ (Rng X. Rng ) )
37	35, 35, 36	syl2anc 673	. . . 4 |- ( ph -> ( R X. R ) C_ (Rng X. Rng ) )
38		fnssres 5699	. . . 4 |- ( ( RngHomo Fn (Rng X. Rng ) /\ ( R X. R ) C_ (Rng X. Rng ) ) -> ( RngHomo |` ( R X. R ) ) Fn ( R X. R ) )
39	32, 37, 38	syl2anc 673	. . 3 |- ( ph -> ( RngHomo |` ( R X. R ) ) Fn ( R X. R ) )
40		eqid 2471	. . . . 5 |- ( x e. R , y e. R |-> ( ( Base ` y ) ^m ( Base ` x ) ) ) = ( x e. R , y e. R |-> ( ( Base ` y ) ^m ( Base ` x ) ) )
41		ovex 6336	. . . . 5 |- ( ( Base ` y ) ^m ( Base ` x ) ) e. _V
42	40, 41	fnmpt2i 6881	. . . 4 |- ( x e. R , y e. R |-> ( ( Base ` y ) ^m ( Base ` x ) ) ) Fn ( R X. R )
43	42	a1i 11	. . 3 |- ( ph -> ( x e. R , y e. R |-> ( ( Base ` y ) ^m ( Base ` x ) ) ) Fn ( R X. R ) )
44		incom 3616	. . . . 5 |- (Rng i^i U ) = ( U i^i Rng )
45		rnghmsscmap.u	. . . . . 6 |- ( ph -> U e. V )
46		inex1g 4539	. . . . . 6 |- ( U e. V -> ( U i^i Rng ) e. _V )
47	45, 46	syl 17	. . . . 5 |- ( ph -> ( U i^i Rng ) e. _V )
48	44, 47	syl5eqel 2553	. . . 4 |- ( ph -> (Rng i^i U ) e. _V )
49	33, 48	eqeltrd 2549	. . 3 |- ( ph -> R e. _V )
50	39, 43, 49	isssc 15803	. 2 |- ( ph -> ( ( RngHomo |` ( R X. R ) ) C_cat ( x e. R , y e. R |-> ( ( Base ` y ) ^m ( Base ` x ) ) ) <-> ( R C_ R /\ A. a e. R A. b e. R ( a ( RngHomo |` ( R X. R ) ) b ) C_ ( a ( x e. R , y e. R |-> ( ( Base ` y ) ^m ( Base ` x ) ) ) b ) ) ) )
51	2, 30, 50	mpbir2and 936	1 |- ( ph -> ( RngHomo |` ( R X. R ) ) C_cat ( x e. R , y e. R |-> ( ( Base ` y ) ^m ( Base ` x ) ) ) )

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   -->wf 5585   ` cfv 5589  (class class class)co 6308   |-> cmpt2 6310   ^m cmap 7490   Basecbs 15199   C__cat cssc 15790  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

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  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-ral 2761  df-rex 2762  df-reu 2763  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-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  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-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-1st 6812  df-2nd 6813  df-map 7492  df-ixp 7541  df-ssc 15793  df-ghm 16959  df-abl 17511  df-rng0 40383  df-rnghomo 40395

This theorem is referenced by: (None)