Theorem estrcco 16770

Description: Composition in the category of extensible structures. (Contributed by AV, 7-Mar-2020.)

Hypotheses

Ref	Expression
estrcbas.c	|- C = (ExtStrCat ` U )
estrcbas.u	|- ( ph -> U e. V )
estrcco.o	|- .x. = (comp ` C )
estrcco.x	|- ( ph -> X e. U )
estrcco.y	|- ( ph -> Y e. U )
estrcco.z	|- ( ph -> Z e. U )
estrcco.a	|- A = ( Base ` X )
estrcco.b	|- B = ( Base ` Y )
estrcco.d	|- D = ( Base ` Z )
estrcco.f	|- ( ph -> F : A --> B )
estrcco.g	|- ( ph -> G : B --> D )

Assertion

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

Proof of Theorem estrcco

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

Step	Hyp	Ref	Expression
1		estrcbas.c	. . . 4 |- C = (ExtStrCat ` U )
2		estrcbas.u	. . . 4 |- ( ph -> U e. V )
3		estrcco.o	. . . 4 |- .x. = (comp ` C )
4	1, 2, 3	estrccofval 16769	. . 3 |- ( ph -> .x. = ( v e. ( U X. U ) , z e. U |-> ( g e. ( ( Base ` z ) ^m ( Base ` ( 2nd ` v ) ) ) , f e. ( ( Base ` ( 2nd ` v ) ) ^m ( Base ` ( 1st ` v ) ) ) |-> ( g o. f ) ) ) )
5		fveq2 6191	. . . . . . 7 |- ( z = Z -> ( Base ` z ) = ( Base ` Z ) )
6	5	adantl 482	. . . . . 6 |- ( ( v = <. X , Y >. /\ z = Z ) -> ( Base ` z ) = ( Base ` Z ) )
7	6	adantl 482	. . . . 5 |- ( ( ph /\ ( v = <. X , Y >. /\ z = Z ) ) -> ( Base ` z ) = ( Base ` Z ) )
8		simprl 794	. . . . . . . 8 |- ( ( ph /\ ( v = <. X , Y >. /\ z = Z ) ) -> v = <. X , Y >. )
9	8	fveq2d 6195	. . . . . . 7 |- ( ( ph /\ ( v = <. X , Y >. /\ z = Z ) ) -> ( 2nd ` v ) = ( 2nd ` <. X , Y >. ) )
10		estrcco.x	. . . . . . . . 9 |- ( ph -> X e. U )
11		estrcco.y	. . . . . . . . 9 |- ( ph -> Y e. U )
12		op2ndg 7181	. . . . . . . . 9 |- ( ( X e. U /\ Y e. U ) -> ( 2nd ` <. X , Y >. ) = Y )
13	10, 11, 12	syl2anc 693	. . . . . . . 8 |- ( ph -> ( 2nd ` <. X , Y >. ) = Y )
14	13	adantr 481	. . . . . . 7 |- ( ( ph /\ ( v = <. X , Y >. /\ z = Z ) ) -> ( 2nd ` <. X , Y >. ) = Y )
15	9, 14	eqtrd 2656	. . . . . 6 |- ( ( ph /\ ( v = <. X , Y >. /\ z = Z ) ) -> ( 2nd ` v ) = Y )
16	15	fveq2d 6195	. . . . 5 |- ( ( ph /\ ( v = <. X , Y >. /\ z = Z ) ) -> ( Base ` ( 2nd ` v ) ) = ( Base ` Y ) )
17	7, 16	oveq12d 6668	. . . 4 |- ( ( ph /\ ( v = <. X , Y >. /\ z = Z ) ) -> ( ( Base ` z ) ^m ( Base ` ( 2nd ` v ) ) ) = ( ( Base ` Z ) ^m ( Base ` Y ) ) )
18	8	fveq2d 6195	. . . . . . 7 |- ( ( ph /\ ( v = <. X , Y >. /\ z = Z ) ) -> ( 1st ` v ) = ( 1st ` <. X , Y >. ) )
19	18	fveq2d 6195	. . . . . 6 |- ( ( ph /\ ( v = <. X , Y >. /\ z = Z ) ) -> ( Base ` ( 1st ` v ) ) = ( Base ` ( 1st ` <. X , Y >. ) ) )
20		op1stg 7180	. . . . . . . . 9 |- ( ( X e. U /\ Y e. U ) -> ( 1st ` <. X , Y >. ) = X )
21	10, 11, 20	syl2anc 693	. . . . . . . 8 |- ( ph -> ( 1st ` <. X , Y >. ) = X )
22	21	fveq2d 6195	. . . . . . 7 |- ( ph -> ( Base ` ( 1st ` <. X , Y >. ) ) = ( Base ` X ) )
23	22	adantr 481	. . . . . 6 |- ( ( ph /\ ( v = <. X , Y >. /\ z = Z ) ) -> ( Base ` ( 1st ` <. X , Y >. ) ) = ( Base ` X ) )
24	19, 23	eqtrd 2656	. . . . 5 |- ( ( ph /\ ( v = <. X , Y >. /\ z = Z ) ) -> ( Base ` ( 1st ` v ) ) = ( Base ` X ) )
25	16, 24	oveq12d 6668	. . . 4 |- ( ( ph /\ ( v = <. X , Y >. /\ z = Z ) ) -> ( ( Base ` ( 2nd ` v ) ) ^m ( Base ` ( 1st ` v ) ) ) = ( ( Base ` Y ) ^m ( Base ` X ) ) )
26		eqidd 2623	. . . 4 |- ( ( ph /\ ( v = <. X , Y >. /\ z = Z ) ) -> ( g o. f ) = ( g o. f ) )
27	17, 25, 26	mpt2eq123dv 6717	. . 3 |- ( ( ph /\ ( v = <. X , Y >. /\ z = Z ) ) -> ( g e. ( ( Base ` z ) ^m ( Base ` ( 2nd ` v ) ) ) , f e. ( ( Base ` ( 2nd ` v ) ) ^m ( Base ` ( 1st ` v ) ) ) |-> ( g o. f ) ) = ( g e. ( ( Base ` Z ) ^m ( Base ` Y ) ) , f e. ( ( Base ` Y ) ^m ( Base ` X ) ) |-> ( g o. f ) ) )
28		opelxpi 5148	. . . 4 |- ( ( X e. U /\ Y e. U ) -> <. X , Y >. e. ( U X. U ) )
29	10, 11, 28	syl2anc 693	. . 3 |- ( ph -> <. X , Y >. e. ( U X. U ) )
30		estrcco.z	. . 3 |- ( ph -> Z e. U )
31		ovex 6678	. . . . 5 |- ( ( Base ` Z ) ^m ( Base ` Y ) ) e. _V
32		ovex 6678	. . . . 5 |- ( ( Base ` Y ) ^m ( Base ` X ) ) e. _V
33	31, 32	mpt2ex 7247	. . . 4 |- ( g e. ( ( Base ` Z ) ^m ( Base ` Y ) ) , f e. ( ( Base ` Y ) ^m ( Base ` X ) ) |-> ( g o. f ) ) e. _V
34	33	a1i 11	. . 3 |- ( ph -> ( g e. ( ( Base ` Z ) ^m ( Base ` Y ) ) , f e. ( ( Base ` Y ) ^m ( Base ` X ) ) |-> ( g o. f ) ) e. _V )
35	4, 27, 29, 30, 34	ovmpt2d 6788	. 2 |- ( ph -> ( <. X , Y >. .x. Z ) = ( g e. ( ( Base ` Z ) ^m ( Base ` Y ) ) , f e. ( ( Base ` Y ) ^m ( Base ` X ) ) |-> ( g o. f ) ) )
36		simpl 473	. . . 4 |- ( ( g = G /\ f = F ) -> g = G )
37		simpr 477	. . . 4 |- ( ( g = G /\ f = F ) -> f = F )
38	36, 37	coeq12d 5286	. . 3 |- ( ( g = G /\ f = F ) -> ( g o. f ) = ( G o. F ) )
39	38	adantl 482	. 2 |- ( ( ph /\ ( g = G /\ f = F ) ) -> ( g o. f ) = ( G o. F ) )
40		estrcco.g	. . . 4 |- ( ph -> G : B --> D )
41		estrcco.b	. . . . . . 7 |- B = ( Base ` Y )
42	41	a1i 11	. . . . . 6 |- ( ph -> B = ( Base ` Y ) )
43	42	eqcomd 2628	. . . . 5 |- ( ph -> ( Base ` Y ) = B )
44		estrcco.d	. . . . . . 7 |- D = ( Base ` Z )
45	44	a1i 11	. . . . . 6 |- ( ph -> D = ( Base ` Z ) )
46	45	eqcomd 2628	. . . . 5 |- ( ph -> ( Base ` Z ) = D )
47	43, 46	feq23d 6040	. . . 4 |- ( ph -> ( G : ( Base ` Y ) --> ( Base ` Z ) <-> G : B --> D ) )
48	40, 47	mpbird 247	. . 3 |- ( ph -> G : ( Base ` Y ) --> ( Base ` Z ) )
49		fvexd 6203	. . . 4 |- ( ph -> ( Base ` Z ) e. _V )
50		fvexd 6203	. . . 4 |- ( ph -> ( Base ` Y ) e. _V )
51		elmapg 7870	. . . 4 |- ( ( ( Base ` Z ) e. _V /\ ( Base ` Y ) e. _V ) -> ( G e. ( ( Base ` Z ) ^m ( Base ` Y ) ) <-> G : ( Base ` Y ) --> ( Base ` Z ) ) )
52	49, 50, 51	syl2anc 693	. . 3 |- ( ph -> ( G e. ( ( Base ` Z ) ^m ( Base ` Y ) ) <-> G : ( Base ` Y ) --> ( Base ` Z ) ) )
53	48, 52	mpbird 247	. 2 |- ( ph -> G e. ( ( Base ` Z ) ^m ( Base ` Y ) ) )
54		estrcco.f	. . . 4 |- ( ph -> F : A --> B )
55		estrcco.a	. . . . . . 7 |- A = ( Base ` X )
56	55	a1i 11	. . . . . 6 |- ( ph -> A = ( Base ` X ) )
57	56	eqcomd 2628	. . . . 5 |- ( ph -> ( Base ` X ) = A )
58	57, 43	feq23d 6040	. . . 4 |- ( ph -> ( F : ( Base ` X ) --> ( Base ` Y ) <-> F : A --> B ) )
59	54, 58	mpbird 247	. . 3 |- ( ph -> F : ( Base ` X ) --> ( Base ` Y ) )
60		fvexd 6203	. . . 4 |- ( ph -> ( Base ` X ) e. _V )
61		elmapg 7870	. . . 4 |- ( ( ( Base ` Y ) e. _V /\ ( Base ` X ) e. _V ) -> ( F e. ( ( Base ` Y ) ^m ( Base ` X ) ) <-> F : ( Base ` X ) --> ( Base ` Y ) ) )
62	50, 60, 61	syl2anc 693	. . 3 |- ( ph -> ( F e. ( ( Base ` Y ) ^m ( Base ` X ) ) <-> F : ( Base ` X ) --> ( Base ` Y ) ) )
63	59, 62	mpbird 247	. 2 |- ( ph -> F e. ( ( Base ` Y ) ^m ( Base ` X ) ) )
64		coexg 7117	. . 3 |- ( ( G e. ( ( Base ` Z ) ^m ( Base ` Y ) ) /\ F e. ( ( Base ` Y ) ^m ( Base ` X ) ) ) -> ( G o. F ) e. _V )
65	53, 63, 64	syl2anc 693	. 2 |- ( ph -> ( G o. F ) e. _V )
66	35, 39, 53, 63, 65	ovmpt2d 6788	1 |- ( ph -> ( G ( <. X , Y >. .x. Z ) F ) = ( G o. F ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   <.cop 4183   X. cxp 5112   o. ccom 5118   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1stc1st 7166   2ndc2nd 7167   ^m cmap 7857   Basecbs 15857  compcco 15953  ExtStrCatcestrc 16762

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-map 7859  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-estrc 16763

This theorem is referenced by:  estrccatid  16772  funcestrcsetclem9  16788  funcsetcestrclem9  16803  rngcco  41971  rnghmsubcsetclem2  41976  ringcco  42017  rhmsubcsetclem2  42022