Theorem fucco 16622

Description: Value of the composition of natural transformations. (Contributed by Mario Carneiro, 6-Jan-2017.)

Hypotheses

Ref	Expression
fucco.q	|- Q = ( C FuncCat D )
fucco.n	|- N = ( C Nat D )
fucco.a	|- A = ( Base ` C )
fucco.o	|- .x. = (comp ` D )
fucco.x	|- .xb = (comp ` Q )
fucco.f	|- ( ph -> R e. ( F N G ) )
fucco.g	|- ( ph -> S e. ( G N H ) )

Assertion

Ref	Expression
fucco	|- ( ph -> ( S ( <. F , G >. .xb H ) R ) = ( x e. A |-> ( ( S ` x ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. .x. ( ( 1st ` H ) ` x ) ) ( R ` x ) ) ) )

Distinct variable groups:   x, A   ph, x   x, R   x, S   x, C   x, D   x, .x.   x, F   x, G   x, H

Allowed substitution hints:   Q( x)   .xb ( x)   N( x)

Proof of Theorem fucco

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

Step	Hyp	Ref	Expression
1		fucco.q	. . . 4 |- Q = ( C FuncCat D )
2		eqid 2622	. . . 4 |- ( C Func D ) = ( C Func D )
3		fucco.n	. . . 4 |- N = ( C Nat D )
4		fucco.a	. . . 4 |- A = ( Base ` C )
5		fucco.o	. . . 4 |- .x. = (comp ` D )
6		fucco.f	. . . . . . . 8 |- ( ph -> R e. ( F N G ) )
7	3	natrcl 16610	. . . . . . . 8 |- ( R e. ( F N G ) -> ( F e. ( C Func D ) /\ G e. ( C Func D ) ) )
8	6, 7	syl 17	. . . . . . 7 |- ( ph -> ( F e. ( C Func D ) /\ G e. ( C Func D ) ) )
9	8	simpld 475	. . . . . 6 |- ( ph -> F e. ( C Func D ) )
10		funcrcl 16523	. . . . . 6 |- ( F e. ( C Func D ) -> ( C e. Cat /\ D e. Cat ) )
11	9, 10	syl 17	. . . . 5 |- ( ph -> ( C e. Cat /\ D e. Cat ) )
12	11	simpld 475	. . . 4 |- ( ph -> C e. Cat )
13	11	simprd 479	. . . 4 |- ( ph -> D e. Cat )
14		fucco.x	. . . 4 |- .xb = (comp ` Q )
15	1, 2, 3, 4, 5, 12, 13, 14	fuccofval 16619	. . 3 |- ( ph -> .xb = ( v e. ( ( C Func D ) X. ( C Func D ) ) , h e. ( C Func D ) |-> [_ ( 1st ` v ) / f ]_ [_ ( 2nd ` v ) / g ]_ ( b e. ( g N h ) , a e. ( f N g ) |-> ( x e. A |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. .x. ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) ) )
16		fvexd 6203	. . . 4 |- ( ( ph /\ ( v = <. F , G >. /\ h = H ) ) -> ( 1st ` v ) e. _V )
17		simprl 794	. . . . . 6 |- ( ( ph /\ ( v = <. F , G >. /\ h = H ) ) -> v = <. F , G >. )
18	17	fveq2d 6195	. . . . 5 |- ( ( ph /\ ( v = <. F , G >. /\ h = H ) ) -> ( 1st ` v ) = ( 1st ` <. F , G >. ) )
19		op1stg 7180	. . . . . . 7 |- ( ( F e. ( C Func D ) /\ G e. ( C Func D ) ) -> ( 1st ` <. F , G >. ) = F )
20	8, 19	syl 17	. . . . . 6 |- ( ph -> ( 1st ` <. F , G >. ) = F )
21	20	adantr 481	. . . . 5 |- ( ( ph /\ ( v = <. F , G >. /\ h = H ) ) -> ( 1st ` <. F , G >. ) = F )
22	18, 21	eqtrd 2656	. . . 4 |- ( ( ph /\ ( v = <. F , G >. /\ h = H ) ) -> ( 1st ` v ) = F )
23		fvexd 6203	. . . . 5 |- ( ( ( ph /\ ( v = <. F , G >. /\ h = H ) ) /\ f = F ) -> ( 2nd ` v ) e. _V )
24	17	adantr 481	. . . . . . 7 |- ( ( ( ph /\ ( v = <. F , G >. /\ h = H ) ) /\ f = F ) -> v = <. F , G >. )
25	24	fveq2d 6195	. . . . . 6 |- ( ( ( ph /\ ( v = <. F , G >. /\ h = H ) ) /\ f = F ) -> ( 2nd ` v ) = ( 2nd ` <. F , G >. ) )
26		op2ndg 7181	. . . . . . . 8 |- ( ( F e. ( C Func D ) /\ G e. ( C Func D ) ) -> ( 2nd ` <. F , G >. ) = G )
27	8, 26	syl 17	. . . . . . 7 |- ( ph -> ( 2nd ` <. F , G >. ) = G )
28	27	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ ( v = <. F , G >. /\ h = H ) ) /\ f = F ) -> ( 2nd ` <. F , G >. ) = G )
29	25, 28	eqtrd 2656	. . . . 5 |- ( ( ( ph /\ ( v = <. F , G >. /\ h = H ) ) /\ f = F ) -> ( 2nd ` v ) = G )
30		simpr 477	. . . . . . 7 |- ( ( ( ( ph /\ ( v = <. F , G >. /\ h = H ) ) /\ f = F ) /\ g = G ) -> g = G )
31		simprr 796	. . . . . . . 8 |- ( ( ph /\ ( v = <. F , G >. /\ h = H ) ) -> h = H )
32	31	ad2antrr 762	. . . . . . 7 |- ( ( ( ( ph /\ ( v = <. F , G >. /\ h = H ) ) /\ f = F ) /\ g = G ) -> h = H )
33	30, 32	oveq12d 6668	. . . . . 6 |- ( ( ( ( ph /\ ( v = <. F , G >. /\ h = H ) ) /\ f = F ) /\ g = G ) -> ( g N h ) = ( G N H ) )
34		simplr 792	. . . . . . 7 |- ( ( ( ( ph /\ ( v = <. F , G >. /\ h = H ) ) /\ f = F ) /\ g = G ) -> f = F )
35	34, 30	oveq12d 6668	. . . . . 6 |- ( ( ( ( ph /\ ( v = <. F , G >. /\ h = H ) ) /\ f = F ) /\ g = G ) -> ( f N g ) = ( F N G ) )
36	34	fveq2d 6195	. . . . . . . . . . 11 |- ( ( ( ( ph /\ ( v = <. F , G >. /\ h = H ) ) /\ f = F ) /\ g = G ) -> ( 1st ` f ) = ( 1st ` F ) )
37	36	fveq1d 6193	. . . . . . . . . 10 |- ( ( ( ( ph /\ ( v = <. F , G >. /\ h = H ) ) /\ f = F ) /\ g = G ) -> ( ( 1st ` f ) ` x ) = ( ( 1st ` F ) ` x ) )
38	30	fveq2d 6195	. . . . . . . . . . 11 |- ( ( ( ( ph /\ ( v = <. F , G >. /\ h = H ) ) /\ f = F ) /\ g = G ) -> ( 1st ` g ) = ( 1st ` G ) )
39	38	fveq1d 6193	. . . . . . . . . 10 |- ( ( ( ( ph /\ ( v = <. F , G >. /\ h = H ) ) /\ f = F ) /\ g = G ) -> ( ( 1st ` g ) ` x ) = ( ( 1st ` G ) ` x ) )
40	37, 39	opeq12d 4410	. . . . . . . . 9 |- ( ( ( ( ph /\ ( v = <. F , G >. /\ h = H ) ) /\ f = F ) /\ g = G ) -> <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. = <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. )
41	32	fveq2d 6195	. . . . . . . . . 10 |- ( ( ( ( ph /\ ( v = <. F , G >. /\ h = H ) ) /\ f = F ) /\ g = G ) -> ( 1st ` h ) = ( 1st ` H ) )
42	41	fveq1d 6193	. . . . . . . . 9 |- ( ( ( ( ph /\ ( v = <. F , G >. /\ h = H ) ) /\ f = F ) /\ g = G ) -> ( ( 1st ` h ) ` x ) = ( ( 1st ` H ) ` x ) )
43	40, 42	oveq12d 6668	. . . . . . . 8 |- ( ( ( ( ph /\ ( v = <. F , G >. /\ h = H ) ) /\ f = F ) /\ g = G ) -> ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. .x. ( ( 1st ` h ) ` x ) ) = ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. .x. ( ( 1st ` H ) ` x ) ) )
44	43	oveqd 6667	. . . . . . 7 |- ( ( ( ( ph /\ ( v = <. F , G >. /\ h = H ) ) /\ f = F ) /\ g = G ) -> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. .x. ( ( 1st ` h ) ` x ) ) ( a ` x ) ) = ( ( b ` x ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. .x. ( ( 1st ` H ) ` x ) ) ( a ` x ) ) )
45	44	mpteq2dv 4745	. . . . . 6 |- ( ( ( ( ph /\ ( v = <. F , G >. /\ h = H ) ) /\ f = F ) /\ g = G ) -> ( x e. A |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. .x. ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) = ( x e. A |-> ( ( b ` x ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. .x. ( ( 1st ` H ) ` x ) ) ( a ` x ) ) ) )
46	33, 35, 45	mpt2eq123dv 6717	. . . . 5 |- ( ( ( ( ph /\ ( v = <. F , G >. /\ h = H ) ) /\ f = F ) /\ g = G ) -> ( b e. ( g N h ) , a e. ( f N g ) |-> ( x e. A |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. .x. ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) = ( b e. ( G N H ) , a e. ( F N G ) |-> ( x e. A |-> ( ( b ` x ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. .x. ( ( 1st ` H ) ` x ) ) ( a ` x ) ) ) ) )
47	23, 29, 46	csbied2 3561	. . . 4 |- ( ( ( ph /\ ( v = <. F , G >. /\ h = H ) ) /\ f = F ) -> [_ ( 2nd ` v ) / g ]_ ( b e. ( g N h ) , a e. ( f N g ) |-> ( x e. A |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. .x. ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) = ( b e. ( G N H ) , a e. ( F N G ) |-> ( x e. A |-> ( ( b ` x ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. .x. ( ( 1st ` H ) ` x ) ) ( a ` x ) ) ) ) )
48	16, 22, 47	csbied2 3561	. . 3 |- ( ( ph /\ ( v = <. F , G >. /\ h = H ) ) -> [_ ( 1st ` v ) / f ]_ [_ ( 2nd ` v ) / g ]_ ( b e. ( g N h ) , a e. ( f N g ) |-> ( x e. A |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. .x. ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) = ( b e. ( G N H ) , a e. ( F N G ) |-> ( x e. A |-> ( ( b ` x ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. .x. ( ( 1st ` H ) ` x ) ) ( a ` x ) ) ) ) )
49		opelxpi 5148	. . . 4 |- ( ( F e. ( C Func D ) /\ G e. ( C Func D ) ) -> <. F , G >. e. ( ( C Func D ) X. ( C Func D ) ) )
50	8, 49	syl 17	. . 3 |- ( ph -> <. F , G >. e. ( ( C Func D ) X. ( C Func D ) ) )
51		fucco.g	. . . . 5 |- ( ph -> S e. ( G N H ) )
52	3	natrcl 16610	. . . . 5 |- ( S e. ( G N H ) -> ( G e. ( C Func D ) /\ H e. ( C Func D ) ) )
53	51, 52	syl 17	. . . 4 |- ( ph -> ( G e. ( C Func D ) /\ H e. ( C Func D ) ) )
54	53	simprd 479	. . 3 |- ( ph -> H e. ( C Func D ) )
55		ovex 6678	. . . . 5 |- ( G N H ) e. _V
56		ovex 6678	. . . . 5 |- ( F N G ) e. _V
57	55, 56	mpt2ex 7247	. . . 4 |- ( b e. ( G N H ) , a e. ( F N G ) |-> ( x e. A |-> ( ( b ` x ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. .x. ( ( 1st ` H ) ` x ) ) ( a ` x ) ) ) ) e. _V
58	57	a1i 11	. . 3 |- ( ph -> ( b e. ( G N H ) , a e. ( F N G ) |-> ( x e. A |-> ( ( b ` x ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. .x. ( ( 1st ` H ) ` x ) ) ( a ` x ) ) ) ) e. _V )
59	15, 48, 50, 54, 58	ovmpt2d 6788	. 2 |- ( ph -> ( <. F , G >. .xb H ) = ( b e. ( G N H ) , a e. ( F N G ) |-> ( x e. A |-> ( ( b ` x ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. .x. ( ( 1st ` H ) ` x ) ) ( a ` x ) ) ) ) )
60		simprl 794	. . . . 5 |- ( ( ph /\ ( b = S /\ a = R ) ) -> b = S )
61	60	fveq1d 6193	. . . 4 |- ( ( ph /\ ( b = S /\ a = R ) ) -> ( b ` x ) = ( S ` x ) )
62		simprr 796	. . . . 5 |- ( ( ph /\ ( b = S /\ a = R ) ) -> a = R )
63	62	fveq1d 6193	. . . 4 |- ( ( ph /\ ( b = S /\ a = R ) ) -> ( a ` x ) = ( R ` x ) )
64	61, 63	oveq12d 6668	. . 3 |- ( ( ph /\ ( b = S /\ a = R ) ) -> ( ( b ` x ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. .x. ( ( 1st ` H ) ` x ) ) ( a ` x ) ) = ( ( S ` x ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. .x. ( ( 1st ` H ) ` x ) ) ( R ` x ) ) )
65	64	mpteq2dv 4745	. 2 |- ( ( ph /\ ( b = S /\ a = R ) ) -> ( x e. A |-> ( ( b ` x ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. .x. ( ( 1st ` H ) ` x ) ) ( a ` x ) ) ) = ( x e. A |-> ( ( S ` x ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. .x. ( ( 1st ` H ) ` x ) ) ( R ` x ) ) ) )
66		fvex 6201	. . . . 5 |- ( Base ` C ) e. _V
67	4, 66	eqeltri 2697	. . . 4 |- A e. _V
68	67	mptex 6486	. . 3 |- ( x e. A |-> ( ( S ` x ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. .x. ( ( 1st ` H ) ` x ) ) ( R ` x ) ) ) e. _V
69	68	a1i 11	. 2 |- ( ph -> ( x e. A |-> ( ( S ` x ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. .x. ( ( 1st ` H ) ` x ) ) ( R ` x ) ) ) e. _V )
70	59, 65, 51, 6, 69	ovmpt2d 6788	1 |- ( ph -> ( S ( <. F , G >. .xb H ) R ) = ( x e. A |-> ( ( S ` x ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. .x. ( ( 1st ` H ) ` x ) ) ( R ` x ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   [_csb 3533   <.cop 4183   |-> cmpt 4729   X. cxp 5112   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1stc1st 7166   2ndc2nd 7167   Basecbs 15857  compcco 15953   Catccat 16325   Func cfunc 16514   Nat cnat 16601   FuncCat cfuc 16602

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-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-hom 15966  df-cco 15967  df-func 16518  df-nat 16603  df-fuc 16604

This theorem is referenced by:  fuccoval  16623  fuccocl  16624  fuclid  16626  fucrid  16627  fucass  16628  fucsect  16632  curfcl  16872