Theorem rngccoALTV 41988

Description: Composition in the category of non-unital rings. (New usage is discouraged.) (Contributed by AV, 27-Feb-2020.)

Hypotheses

Ref	Expression
rngcbasALTV.c	|- C = (RngCatALTV ` U )
rngcbasALTV.b	|- B = ( Base ` C )
rngcbasALTV.u	|- ( ph -> U e. V )
rngccofvalALTV.o	|- .x. = (comp ` C )
rngccoALTV.x	|- ( ph -> X e. B )
rngccoALTV.y	|- ( ph -> Y e. B )
rngccoALTV.z	|- ( ph -> Z e. B )
rngccoALTV.f	|- ( ph -> F e. ( X RngHomo Y ) )
rngccoALTV.g	|- ( ph -> G e. ( Y RngHomo Z ) )

Assertion

Ref	Expression
rngccoALTV	|- ( ph -> ( G ( <. X , Y >. .x. Z ) F ) = ( G o. F ) )

Proof of Theorem rngccoALTV

Dummy variables f g v z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rngcbasALTV.c	. . . 4 |- C = (RngCatALTV ` U )
2		rngcbasALTV.b	. . . 4 |- B = ( Base ` C )
3		rngcbasALTV.u	. . . 4 |- ( ph -> U e. V )
4		rngccofvalALTV.o	. . . 4 |- .x. = (comp ` C )
5	1, 2, 3, 4	rngccofvalALTV 41987	. . 3 |- ( ph -> .x. = ( v e. ( B X. B ) , z e. B |-> ( g e. ( ( 2nd ` v ) RngHomo z ) , f e. ( ( 1st ` v ) RngHomo ( 2nd ` v ) ) |-> ( g o. f ) ) ) )
6		simprl 794	. . . . . . 7 |- ( ( ph /\ ( v = <. X , Y >. /\ z = Z ) ) -> v = <. X , Y >. )
7	6	fveq2d 6195	. . . . . 6 |- ( ( ph /\ ( v = <. X , Y >. /\ z = Z ) ) -> ( 2nd ` v ) = ( 2nd ` <. X , Y >. ) )
8		rngccoALTV.x	. . . . . . . 8 |- ( ph -> X e. B )
9		rngccoALTV.y	. . . . . . . 8 |- ( ph -> Y e. B )
10		op2ndg 7181	. . . . . . . 8 |- ( ( X e. B /\ Y e. B ) -> ( 2nd ` <. X , Y >. ) = Y )
11	8, 9, 10	syl2anc 693	. . . . . . 7 |- ( ph -> ( 2nd ` <. X , Y >. ) = Y )
12	11	adantr 481	. . . . . 6 |- ( ( ph /\ ( v = <. X , Y >. /\ z = Z ) ) -> ( 2nd ` <. X , Y >. ) = Y )
13	7, 12	eqtrd 2656	. . . . 5 |- ( ( ph /\ ( v = <. X , Y >. /\ z = Z ) ) -> ( 2nd ` v ) = Y )
14		simprr 796	. . . . 5 |- ( ( ph /\ ( v = <. X , Y >. /\ z = Z ) ) -> z = Z )
15	13, 14	oveq12d 6668	. . . 4 |- ( ( ph /\ ( v = <. X , Y >. /\ z = Z ) ) -> ( ( 2nd ` v ) RngHomo z ) = ( Y RngHomo Z ) )
16	6	fveq2d 6195	. . . . . 6 |- ( ( ph /\ ( v = <. X , Y >. /\ z = Z ) ) -> ( 1st ` v ) = ( 1st ` <. X , Y >. ) )
17		op1stg 7180	. . . . . . . 8 |- ( ( X e. B /\ Y e. B ) -> ( 1st ` <. X , Y >. ) = X )
18	8, 9, 17	syl2anc 693	. . . . . . 7 |- ( ph -> ( 1st ` <. X , Y >. ) = X )
19	18	adantr 481	. . . . . 6 |- ( ( ph /\ ( v = <. X , Y >. /\ z = Z ) ) -> ( 1st ` <. X , Y >. ) = X )
20	16, 19	eqtrd 2656	. . . . 5 |- ( ( ph /\ ( v = <. X , Y >. /\ z = Z ) ) -> ( 1st ` v ) = X )
21	20, 13	oveq12d 6668	. . . 4 |- ( ( ph /\ ( v = <. X , Y >. /\ z = Z ) ) -> ( ( 1st ` v ) RngHomo ( 2nd ` v ) ) = ( X RngHomo Y ) )
22		eqidd 2623	. . . 4 |- ( ( ph /\ ( v = <. X , Y >. /\ z = Z ) ) -> ( g o. f ) = ( g o. f ) )
23	15, 21, 22	mpt2eq123dv 6717	. . 3 |- ( ( ph /\ ( v = <. X , Y >. /\ z = Z ) ) -> ( g e. ( ( 2nd ` v ) RngHomo z ) , f e. ( ( 1st ` v ) RngHomo ( 2nd ` v ) ) |-> ( g o. f ) ) = ( g e. ( Y RngHomo Z ) , f e. ( X RngHomo Y ) |-> ( g o. f ) ) )
24		opelxpi 5148	. . . 4 |- ( ( X e. B /\ Y e. B ) -> <. X , Y >. e. ( B X. B ) )
25	8, 9, 24	syl2anc 693	. . 3 |- ( ph -> <. X , Y >. e. ( B X. B ) )
26		rngccoALTV.z	. . 3 |- ( ph -> Z e. B )
27		ovex 6678	. . . . 5 |- ( Y RngHomo Z ) e. _V
28		ovex 6678	. . . . 5 |- ( X RngHomo Y ) e. _V
29	27, 28	mpt2ex 7247	. . . 4 |- ( g e. ( Y RngHomo Z ) , f e. ( X RngHomo Y ) |-> ( g o. f ) ) e. _V
30	29	a1i 11	. . 3 |- ( ph -> ( g e. ( Y RngHomo Z ) , f e. ( X RngHomo Y ) |-> ( g o. f ) ) e. _V )
31	5, 23, 25, 26, 30	ovmpt2d 6788	. 2 |- ( ph -> ( <. X , Y >. .x. Z ) = ( g e. ( Y RngHomo Z ) , f e. ( X RngHomo Y ) |-> ( g o. f ) ) )
32		simprl 794	. . 3 |- ( ( ph /\ ( g = G /\ f = F ) ) -> g = G )
33		simprr 796	. . 3 |- ( ( ph /\ ( g = G /\ f = F ) ) -> f = F )
34	32, 33	coeq12d 5286	. 2 |- ( ( ph /\ ( g = G /\ f = F ) ) -> ( g o. f ) = ( G o. F ) )
35		rngccoALTV.g	. 2 |- ( ph -> G e. ( Y RngHomo Z ) )
36		rngccoALTV.f	. 2 |- ( ph -> F e. ( X RngHomo Y ) )
37		coexg 7117	. . 3 |- ( ( G e. ( Y RngHomo Z ) /\ F e. ( X RngHomo Y ) ) -> ( G o. F ) e. _V )
38	35, 36, 37	syl2anc 693	. 2 |- ( ph -> ( G o. F ) e. _V )
39	31, 34, 35, 36, 38	ovmpt2d 6788	1 |- ( ph -> ( G ( <. X , Y >. .x. Z ) F ) = ( G o. F ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   <.cop 4183   X. cxp 5112   o. ccom 5118   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1stc1st 7166   2ndc2nd 7167   Basecbs 15857  compcco 15953   RngHomo crngh 41885  RngCatALTVcrngcALTV 41958

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-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-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-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-hom 15966  df-cco 15967  df-rngcALTV 41960

This theorem is referenced by:  rngccatidALTV  41989  rngcsectALTV  41992  rhmsubcALTVlem4  42107