Theorem rngcifuestrc 41997

Description: The "inclusion functor" from the category of non-unital rings into the category of extensible structures. (Contributed by AV, 30-Mar-2020.)

Hypotheses

Ref	Expression
rngcifuestrc.r	|- R = (RngCat ` U )
rngcifuestrc.e	|- E = (ExtStrCat ` U )
rngcifuestrc.b	|- B = ( Base ` R )
rngcifuestrc.u	|- ( ph -> U e. V )
rngcifuestrc.f	|- ( ph -> F = ( _I |` B ) )
rngcifuestrc.g	|- ( ph -> G = ( x e. B , y e. B |-> ( _I |` ( x RngHomo y ) ) ) )

Assertion

Ref	Expression
rngcifuestrc	|- ( ph -> F ( R Func E ) G )

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

Allowed substitution hints:   B( x, y)   E( x, y)   F( x, y)   G( x, y)   V( x, y)

Proof of Theorem rngcifuestrc

Step	Hyp	Ref	Expression
1		eqid 2622	. . . . 5 |- (ExtStrCat ` U ) = (ExtStrCat ` U )
2		rngcifuestrc.u	. . . . 5 |- ( ph -> U e. V )
3		rngcifuestrc.r	. . . . . . 7 |- R = (RngCat ` U )
4		rngcifuestrc.b	. . . . . . 7 |- B = ( Base ` R )
5	3, 4, 2	rngcbas 41965	. . . . . 6 |- ( ph -> B = ( U i^i Rng ) )
6		incom 3805	. . . . . 6 |- ( U i^i Rng ) = (Rng i^i U )
7	5, 6	syl6eq 2672	. . . . 5 |- ( ph -> B = (Rng i^i U ) )
8		eqid 2622	. . . . . 6 |- ( Hom ` R ) = ( Hom ` R )
9	3, 4, 2, 8	rngchomfval 41966	. . . . 5 |- ( ph -> ( Hom ` R ) = ( RngHomo |` ( B X. B ) ) )
10	1, 2, 7, 9	rnghmsubcsetc 41977	. . . 4 |- ( ph -> ( Hom ` R ) e. (Subcat ` (ExtStrCat ` U ) ) )
11	10	idi 2	. . 3 |- ( ph -> ( Hom ` R ) e. (Subcat ` (ExtStrCat ` U ) ) )
12		eqid 2622	. . 3 |- ( (ExtStrCat ` U ) |`cat ( Hom ` R ) ) = ( (ExtStrCat ` U ) |`cat ( Hom ` R ) )
13		eqid 2622	. . 3 |- ( Base ` ( (ExtStrCat ` U ) |`cat ( Hom ` R ) ) ) = ( Base ` ( (ExtStrCat ` U ) |`cat ( Hom ` R ) ) )
14		rngcifuestrc.f	. . . 4 |- ( ph -> F = ( _I |` B ) )
15	3, 2, 5, 9	rngcval 41962	. . . . . . 7 |- ( ph -> R = ( (ExtStrCat ` U ) |`cat ( Hom ` R ) ) )
16	15	fveq2d 6195	. . . . . 6 |- ( ph -> ( Base ` R ) = ( Base ` ( (ExtStrCat ` U ) |`cat ( Hom ` R ) ) ) )
17	4, 16	syl5eq 2668	. . . . 5 |- ( ph -> B = ( Base ` ( (ExtStrCat ` U ) |`cat ( Hom ` R ) ) ) )
18	17	reseq2d 5396	. . . 4 |- ( ph -> ( _I |` B ) = ( _I |` ( Base ` ( (ExtStrCat ` U ) |`cat ( Hom ` R ) ) ) ) )
19	14, 18	eqtrd 2656	. . 3 |- ( ph -> F = ( _I |` ( Base ` ( (ExtStrCat ` U ) |`cat ( Hom ` R ) ) ) ) )
20		rngcifuestrc.g	. . . 4 |- ( ph -> G = ( x e. B , y e. B |-> ( _I |` ( x RngHomo y ) ) ) )
21	17	adantr 481	. . . . 5 |- ( ( ph /\ x e. B ) -> B = ( Base ` ( (ExtStrCat ` U ) |`cat ( Hom ` R ) ) ) )
22	9	oveqdr 6674	. . . . . . 7 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( Hom ` R ) y ) = ( x ( RngHomo |` ( B X. B ) ) y ) )
23		ovres 6800	. . . . . . . 8 |- ( ( x e. B /\ y e. B ) -> ( x ( RngHomo |` ( B X. B ) ) y ) = ( x RngHomo y ) )
24	23	adantl 482	. . . . . . 7 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( RngHomo |` ( B X. B ) ) y ) = ( x RngHomo y ) )
25	22, 24	eqtr2d 2657	. . . . . 6 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x RngHomo y ) = ( x ( Hom ` R ) y ) )
26	25	reseq2d 5396	. . . . 5 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( _I |` ( x RngHomo y ) ) = ( _I |` ( x ( Hom ` R ) y ) ) )
27	17, 21, 26	mpt2eq123dva 6716	. . . 4 |- ( ph -> ( x e. B , y e. B |-> ( _I |` ( x RngHomo y ) ) ) = ( x e. ( Base ` ( (ExtStrCat ` U ) |`cat ( Hom ` R ) ) ) , y e. ( Base ` ( (ExtStrCat ` U ) |`cat ( Hom ` R ) ) ) |-> ( _I |` ( x ( Hom ` R ) y ) ) ) )
28	20, 27	eqtrd 2656	. . 3 |- ( ph -> G = ( x e. ( Base ` ( (ExtStrCat ` U ) |`cat ( Hom ` R ) ) ) , y e. ( Base ` ( (ExtStrCat ` U ) |`cat ( Hom ` R ) ) ) |-> ( _I |` ( x ( Hom ` R ) y ) ) ) )
29	11, 12, 13, 19, 28	inclfusubc 41867	. 2 |- ( ph -> F ( ( (ExtStrCat ` U ) |`cat ( Hom ` R ) ) Func (ExtStrCat ` U ) ) G )
30		rngcifuestrc.e	. . . . 5 |- E = (ExtStrCat ` U )
31	30	a1i 11	. . . 4 |- ( ph -> E = (ExtStrCat ` U ) )
32	15, 31	oveq12d 6668	. . 3 |- ( ph -> ( R Func E ) = ( ( (ExtStrCat ` U ) |`cat ( Hom ` R ) ) Func (ExtStrCat ` U ) ) )
33	32	breqd 4664	. 2 |- ( ph -> ( F ( R Func E ) G <-> F ( ( (ExtStrCat ` U ) |`cat ( Hom ` R ) ) Func (ExtStrCat ` U ) ) G ) )
34	29, 33	mpbird 247	1 |- ( ph -> F ( R Func E ) G )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   i^i cin 3573   class class class wbr 4653   _I cid 5023   X. cxp 5112   |` cres 5116   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   Basecbs 15857   Hom chom 15952   |`_cat cresc 16468  Subcatcsubc 16469   Func cfunc 16514  ExtStrCatcestrc 16762  Rngcrng 41874   RngHomo crngh 41885  RngCatcrngc 41957

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  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-hom 15966  df-cco 15967  df-0g 16102  df-cat 16329  df-cid 16330  df-homf 16331  df-ssc 16470  df-resc 16471  df-subc 16472  df-func 16518  df-idfu 16519  df-full 16564  df-fth 16565  df-estrc 16763  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-grp 17425  df-ghm 17658  df-abl 18196  df-mgp 18490  df-mgmhm 41779  df-rng0 41875  df-rnghomo 41887  df-rngc 41959

This theorem is referenced by:  funcrngcsetcALT  41999