Theorem ringcid 42025

Description: The identity arrow in the category of unital rings is the identity function. (Contributed by AV, 14-Feb-2020.) (Revised by AV, 10-Mar-2020.)

Hypotheses

Ref	Expression
ringccat.c	|- C = (RingCat ` U )
ringcid.b	|- B = ( Base ` C )
ringcid.o	|- .1. = ( Id ` C )
ringcid.u	|- ( ph -> U e. V )
ringcid.x	|- ( ph -> X e. B )
ringcid.s	|- S = ( Base ` X )

Assertion

Ref	Expression
ringcid	|- ( ph -> ( .1. ` X ) = ( _I |` S ) )

Proof of Theorem ringcid

Step	Hyp	Ref	Expression
1		ringcid.o	. . . 4 |- .1. = ( Id ` C )
2		ringccat.c	. . . . . 6 |- C = (RingCat ` U )
3		ringcid.u	. . . . . 6 |- ( ph -> U e. V )
4		eqidd 2623	. . . . . 6 |- ( ph -> ( U i^i Ring ) = ( U i^i Ring ) )
5		eqidd 2623	. . . . . 6 |- ( ph -> ( RingHom |` ( ( U i^i Ring ) X. ( U i^i Ring ) ) ) = ( RingHom |` ( ( U i^i Ring ) X. ( U i^i Ring ) ) ) )
6	2, 3, 4, 5	ringcval 42008	. . . . 5 |- ( ph -> C = ( (ExtStrCat ` U ) |`cat ( RingHom |` ( ( U i^i Ring ) X. ( U i^i Ring ) ) ) ) )
7	6	fveq2d 6195	. . . 4 |- ( ph -> ( Id ` C ) = ( Id ` ( (ExtStrCat ` U ) |`cat ( RingHom |` ( ( U i^i Ring ) X. ( U i^i Ring ) ) ) ) ) )
8	1, 7	syl5eq 2668	. . 3 |- ( ph -> .1. = ( Id ` ( (ExtStrCat ` U ) |`cat ( RingHom |` ( ( U i^i Ring ) X. ( U i^i Ring ) ) ) ) ) )
9	8	fveq1d 6193	. 2 |- ( ph -> ( .1. ` X ) = ( ( Id ` ( (ExtStrCat ` U ) |`cat ( RingHom |` ( ( U i^i Ring ) X. ( U i^i Ring ) ) ) ) ) ` X ) )
10		eqid 2622	. . 3 |- ( (ExtStrCat ` U ) |`cat ( RingHom |` ( ( U i^i Ring ) X. ( U i^i Ring ) ) ) ) = ( (ExtStrCat ` U ) |`cat ( RingHom |` ( ( U i^i Ring ) X. ( U i^i Ring ) ) ) )
11		eqid 2622	. . . 4 |- (ExtStrCat ` U ) = (ExtStrCat ` U )
12		incom 3805	. . . . 5 |- ( U i^i Ring ) = ( Ring i^i U )
13	12	a1i 11	. . . 4 |- ( ph -> ( U i^i Ring ) = ( Ring i^i U ) )
14	11, 3, 13, 5	rhmsubcsetc 42023	. . 3 |- ( ph -> ( RingHom |` ( ( U i^i Ring ) X. ( U i^i Ring ) ) ) e. (Subcat ` (ExtStrCat ` U ) ) )
15	4, 5	rhmresfn 42009	. . 3 |- ( ph -> ( RingHom |` ( ( U i^i Ring ) X. ( U i^i Ring ) ) ) Fn ( ( U i^i Ring ) X. ( U i^i Ring ) ) )
16		eqid 2622	. . 3 |- ( Id ` (ExtStrCat ` U ) ) = ( Id ` (ExtStrCat ` U ) )
17		ringcid.x	. . . 4 |- ( ph -> X e. B )
18		ringcid.b	. . . . . 6 |- B = ( Base ` C )
19	2, 18, 3	ringcbas 42011	. . . . 5 |- ( ph -> B = ( U i^i Ring ) )
20	19	eleq2d 2687	. . . 4 |- ( ph -> ( X e. B <-> X e. ( U i^i Ring ) ) )
21	17, 20	mpbid 222	. . 3 |- ( ph -> X e. ( U i^i Ring ) )
22	10, 14, 15, 16, 21	subcid 16507	. 2 |- ( ph -> ( ( Id ` (ExtStrCat ` U ) ) ` X ) = ( ( Id ` ( (ExtStrCat ` U ) |`cat ( RingHom |` ( ( U i^i Ring ) X. ( U i^i Ring ) ) ) ) ) ` X ) )
23		elinel1 3799	. . . . . 6 |- ( X e. ( U i^i Ring ) -> X e. U )
24	20, 23	syl6bi 243	. . . . 5 |- ( ph -> ( X e. B -> X e. U ) )
25	17, 24	mpd 15	. . . 4 |- ( ph -> X e. U )
26	11, 16, 3, 25	estrcid 16774	. . 3 |- ( ph -> ( ( Id ` (ExtStrCat ` U ) ) ` X ) = ( _I |` ( Base ` X ) ) )
27		ringcid.s	. . . . . 6 |- S = ( Base ` X )
28	27	eqcomi 2631	. . . . 5 |- ( Base ` X ) = S
29	28	a1i 11	. . . 4 |- ( ph -> ( Base ` X ) = S )
30	29	reseq2d 5396	. . 3 |- ( ph -> ( _I |` ( Base ` X ) ) = ( _I |` S ) )
31	26, 30	eqtrd 2656	. 2 |- ( ph -> ( ( Id ` (ExtStrCat ` U ) ) ` X ) = ( _I |` S ) )
32	9, 22, 31	3eqtr2d 2662	1 |- ( ph -> ( .1. ` X ) = ( _I |` S ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   i^i cin 3573   _I cid 5023   X. cxp 5112   |` cres 5116   ` cfv 5888  (class class class)co 6650   Basecbs 15857   Idccid 16326   |`_cat cresc 16468  ExtStrCatcestrc 16762   Ringcrg 18547   RingHom crh 18712  RingCatcringc 42003

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-estrc 16763  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-grp 17425  df-ghm 17658  df-mgp 18490  df-ur 18502  df-ring 18549  df-rnghom 18715  df-ringc 42005

This theorem is referenced by:  ringcsect  42031  funcringcsetcALTV2lem7  42042  srhmsubc  42076