Theorem rnghmsscmap 41974

Description: The non-unital ring homomorphisms between non-unital rings (in a universe) are a subcategory subset of the mappings between base sets of extensible structures (in the same universe). (Contributed by AV, 9-Mar-2020.)

Hypotheses

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

Assertion

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

Distinct variable groups:   x, R, y   x, U, y   ph, x, y

Allowed substitution hints:   V( x, y)

Proof of Theorem rnghmsscmap

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

Step	Hyp	Ref	Expression
1		rnghmsscmap.r	. . 3 |- ( ph -> R = (Rng i^i U ) )
2		inss2 3834	. . 3 |- (Rng i^i U ) C_ U
3	1, 2	syl6eqss 3655	. 2 |- ( ph -> R C_ U )
4		eqid 2622	. . . . . . 7 |- ( Base ` a ) = ( Base ` a )
5		eqid 2622	. . . . . . 7 |- ( Base ` b ) = ( Base ` b )
6	4, 5	rnghmf 41899	. . . . . 6 |- ( h e. ( a RngHomo b ) -> h : ( Base ` a ) --> ( Base ` b ) )
7		simpr 477	. . . . . . . 8 |- ( ( ( ph /\ ( a e. R /\ b e. R ) ) /\ h : ( Base ` a ) --> ( Base ` b ) ) -> h : ( Base ` a ) --> ( Base ` b ) )
8		fvex 6201	. . . . . . . . . 10 |- ( Base ` b ) e. _V
9		fvex 6201	. . . . . . . . . 10 |- ( Base ` a ) e. _V
10	8, 9	pm3.2i 471	. . . . . . . . 9 |- ( ( Base ` b ) e. _V /\ ( Base ` a ) e. _V )
11		elmapg 7870	. . . . . . . . 9 |- ( ( ( Base ` b ) e. _V /\ ( Base ` a ) e. _V ) -> ( h e. ( ( Base ` b ) ^m ( Base ` a ) ) <-> h : ( Base ` a ) --> ( Base ` b ) ) )
12	10, 11	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 ) ) )
13	7, 12	mpbird 247	. . . . . . 7 |- ( ( ( ph /\ ( a e. R /\ b e. R ) ) /\ h : ( Base ` a ) --> ( Base ` b ) ) -> h e. ( ( Base ` b ) ^m ( Base ` a ) ) )
14	13	ex 450	. . . . . 6 |- ( ( ph /\ ( a e. R /\ b e. R ) ) -> ( h : ( Base ` a ) --> ( Base ` b ) -> h e. ( ( Base ` b ) ^m ( Base ` a ) ) ) )
15	6, 14	syl5 34	. . . . 5 |- ( ( ph /\ ( a e. R /\ b e. R ) ) -> ( h e. ( a RngHomo b ) -> h e. ( ( Base ` b ) ^m ( Base ` a ) ) ) )
16	15	ssrdv 3609	. . . 4 |- ( ( ph /\ ( a e. R /\ b e. R ) ) -> ( a RngHomo b ) C_ ( ( Base ` b ) ^m ( Base ` a ) ) )
17		ovres 6800	. . . . 5 |- ( ( a e. R /\ b e. R ) -> ( a ( RngHomo |` ( R X. R ) ) b ) = ( a RngHomo b ) )
18	17	adantl 482	. . . 4 |- ( ( ph /\ ( a e. R /\ b e. R ) ) -> ( a ( RngHomo |` ( R X. R ) ) b ) = ( a RngHomo b ) )
19		eqidd 2623	. . . . 5 |- ( ( ph /\ ( a e. R /\ b e. R ) ) -> ( x e. U , y e. U |-> ( ( Base ` y ) ^m ( Base ` x ) ) ) = ( x e. U , y e. U |-> ( ( Base ` y ) ^m ( Base ` x ) ) ) )
20		fveq2 6191	. . . . . . 7 |- ( y = b -> ( Base ` y ) = ( Base ` b ) )
21		fveq2 6191	. . . . . . 7 |- ( x = a -> ( Base ` x ) = ( Base ` a ) )
22	20, 21	oveqan12rd 6670	. . . . . 6 |- ( ( x = a /\ y = b ) -> ( ( Base ` y ) ^m ( Base ` x ) ) = ( ( Base ` b ) ^m ( Base ` a ) ) )
23	22	adantl 482	. . . . 5 |- ( ( ( ph /\ ( a e. R /\ b e. R ) ) /\ ( x = a /\ y = b ) ) -> ( ( Base ` y ) ^m ( Base ` x ) ) = ( ( Base ` b ) ^m ( Base ` a ) ) )
24	3	sseld 3602	. . . . . . . 8 |- ( ph -> ( a e. R -> a e. U ) )
25	24	com12 32	. . . . . . 7 |- ( a e. R -> ( ph -> a e. U ) )
26	25	adantr 481	. . . . . 6 |- ( ( a e. R /\ b e. R ) -> ( ph -> a e. U ) )
27	26	impcom 446	. . . . 5 |- ( ( ph /\ ( a e. R /\ b e. R ) ) -> a e. U )
28	3	sseld 3602	. . . . . . . 8 |- ( ph -> ( b e. R -> b e. U ) )
29	28	com12 32	. . . . . . 7 |- ( b e. R -> ( ph -> b e. U ) )
30	29	adantl 482	. . . . . 6 |- ( ( a e. R /\ b e. R ) -> ( ph -> b e. U ) )
31	30	impcom 446	. . . . 5 |- ( ( ph /\ ( a e. R /\ b e. R ) ) -> b e. U )
32		ovexd 6680	. . . . 5 |- ( ( ph /\ ( a e. R /\ b e. R ) ) -> ( ( Base ` b ) ^m ( Base ` a ) ) e. _V )
33	19, 23, 27, 31, 32	ovmpt2d 6788	. . . 4 |- ( ( ph /\ ( a e. R /\ b e. R ) ) -> ( a ( x e. U , y e. U |-> ( ( Base ` y ) ^m ( Base ` x ) ) ) b ) = ( ( Base ` b ) ^m ( Base ` a ) ) )
34	16, 18, 33	3sstr4d 3648	. . 3 |- ( ( ph /\ ( a e. R /\ b e. R ) ) -> ( a ( RngHomo |` ( R X. R ) ) b ) C_ ( a ( x e. U , y e. U |-> ( ( Base ` y ) ^m ( Base ` x ) ) ) b ) )
35	34	ralrimivva 2971	. 2 |- ( ph -> A. a e. R A. b e. R ( a ( RngHomo |` ( R X. R ) ) b ) C_ ( a ( x e. U , y e. U |-> ( ( Base ` y ) ^m ( Base ` x ) ) ) b ) )
36		rnghmfn 41890	. . . . 5 |- RngHomo Fn (Rng X. Rng )
37	36	a1i 11	. . . 4 |- ( ph -> RngHomo Fn (Rng X. Rng ) )
38		inss1 3833	. . . . . 6 |- (Rng i^i U ) C_ Rng
39	1, 38	syl6eqss 3655	. . . . 5 |- ( ph -> R C_ Rng )
40		xpss12 5225	. . . . 5 |- ( ( R C_ Rng /\ R C_ Rng ) -> ( R X. R ) C_ (Rng X. Rng ) )
41	39, 39, 40	syl2anc 693	. . . 4 |- ( ph -> ( R X. R ) C_ (Rng X. Rng ) )
42		fnssres 6004	. . . 4 |- ( ( RngHomo Fn (Rng X. Rng ) /\ ( R X. R ) C_ (Rng X. Rng ) ) -> ( RngHomo |` ( R X. R ) ) Fn ( R X. R ) )
43	37, 41, 42	syl2anc 693	. . 3 |- ( ph -> ( RngHomo |` ( R X. R ) ) Fn ( R X. R ) )
44		eqid 2622	. . . . 5 |- ( x e. U , y e. U |-> ( ( Base ` y ) ^m ( Base ` x ) ) ) = ( x e. U , y e. U |-> ( ( Base ` y ) ^m ( Base ` x ) ) )
45		ovex 6678	. . . . 5 |- ( ( Base ` y ) ^m ( Base ` x ) ) e. _V
46	44, 45	fnmpt2i 7239	. . . 4 |- ( x e. U , y e. U |-> ( ( Base ` y ) ^m ( Base ` x ) ) ) Fn ( U X. U )
47	46	a1i 11	. . 3 |- ( ph -> ( x e. U , y e. U |-> ( ( Base ` y ) ^m ( Base ` x ) ) ) Fn ( U X. U ) )
48		rnghmsscmap.u	. . . 4 |- ( ph -> U e. V )
49		elex 3212	. . . 4 |- ( U e. V -> U e. _V )
50	48, 49	syl 17	. . 3 |- ( ph -> U e. _V )
51	43, 47, 50	isssc 16480	. 2 |- ( ph -> ( ( RngHomo |` ( R X. R ) ) C_cat ( x e. U , y e. U |-> ( ( Base ` y ) ^m ( Base ` x ) ) ) <-> ( R C_ U /\ A. a e. R A. b e. R ( a ( RngHomo |` ( R X. R ) ) b ) C_ ( a ( x e. U , y e. U |-> ( ( Base ` y ) ^m ( Base ` x ) ) ) b ) ) ) )
52	3, 35, 51	mpbir2and 957	1 |- ( ph -> ( RngHomo |` ( R X. R ) ) C_cat ( x e. U , y e. U |-> ( ( Base ` y ) ^m ( Base ` x ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   i^i cin 3573   C_ wss 3574   class class class wbr 4653   X. cxp 5112   |` cres 5116   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   ^m cmap 7857   Basecbs 15857   C__cat cssc 16467  Rngcrng 41874   RngHomo crngh 41885

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859  df-ixp 7909  df-ssc 16470  df-ghm 17658  df-abl 18196  df-rng0 41875  df-rnghomo 41887

This theorem is referenced by:  rnghmsubcsetc  41977