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Theorem catass 16347
Description: Associativity of composition in a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
catcocl.b |- B = ( Base ` C )
catcocl.h |- H = ( Hom ` C )
catcocl.o |- .x. = (comp ` C )
catcocl.c |- ( ph -> C e. Cat )
catcocl.x |- ( ph -> X e. B )
catcocl.y |- ( ph -> Y e. B )
catcocl.z |- ( ph -> Z e. B )
catcocl.f |- ( ph -> F e. ( X H Y ) )
catcocl.g |- ( ph -> G e. ( Y H Z ) )
catass.w |- ( ph -> W e. B )
catass.g |- ( ph -> K e. ( Z H W ) )
Assertion
Ref Expression
catass |- ( ph -> ( ( K ( <. Y , Z >. .x. W ) G ) ( <. X , Y >. .x. W ) F ) = ( K ( <. X , Z >. .x. W ) ( G ( <. X , Y >. .x. Z ) F ) ) )
Proof of Theorem catass
Dummy variables f g k w x y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 catcocl.c . . 3 |- ( ph -> C e. Cat )
2 catcocl.b . . . . 5 |- B = ( Base ` C )
3 catcocl.h . . . . 5 |- H = ( Hom ` C )
4 catcocl.o . . . . 5 |- .x. = (comp ` C )
52, 3, 4iscat 16333 . . . 4 |- ( C e. Cat -> ( C e. Cat <-> A. x e. B ( E. g e. ( x H x ) A. y e. B ( A. f e. ( y H x ) ( g ( <. y , x >. .x. x ) f ) = f /\ A. f e. ( x H y ) ( f ( <. x , x >. .x. y ) g ) = f ) /\ A. y e. B A. z e. B A. f e. ( x H y ) A. g e. ( y H z ) ( ( g ( <. x , y >. .x. z ) f ) e. ( x H z ) /\ A. w e. B A. k e. ( z H w ) ( ( k ( <. y , z >. .x. w ) g ) ( <. x , y >. .x. w ) f ) = ( k ( <. x , z >. .x. w ) ( g ( <. x , y >. .x. z ) f ) ) ) ) ) )
65ibi 256 . . 3 |- ( C e. Cat -> A. x e. B ( E. g e. ( x H x ) A. y e. B ( A. f e. ( y H x ) ( g ( <. y , x >. .x. x ) f ) = f /\ A. f e. ( x H y ) ( f ( <. x , x >. .x. y ) g ) = f ) /\ A. y e. B A. z e. B A. f e. ( x H y ) A. g e. ( y H z ) ( ( g ( <. x , y >. .x. z ) f ) e. ( x H z ) /\ A. w e. B A. k e. ( z H w ) ( ( k ( <. y , z >. .x. w ) g ) ( <. x , y >. .x. w ) f ) = ( k ( <. x , z >. .x. w ) ( g ( <. x , y >. .x. z ) f ) ) ) ) )
71, 6syl 17 . 2 |- ( ph -> A. x e. B ( E. g e. ( x H x ) A. y e. B ( A. f e. ( y H x ) ( g ( <. y , x >. .x. x ) f ) = f /\ A. f e. ( x H y ) ( f ( <. x , x >. .x. y ) g ) = f ) /\ A. y e. B A. z e. B A. f e. ( x H y ) A. g e. ( y H z ) ( ( g ( <. x , y >. .x. z ) f ) e. ( x H z ) /\ A. w e. B A. k e. ( z H w ) ( ( k ( <. y , z >. .x. w ) g ) ( <. x , y >. .x. w ) f ) = ( k ( <. x , z >. .x. w ) ( g ( <. x , y >. .x. z ) f ) ) ) ) )
8 catcocl.x . . 3 |- ( ph -> X e. B )
9 catcocl.y . . . . . 6 |- ( ph -> Y e. B )
109adantr 481 . . . . 5 |- ( ( ph /\ x = X ) -> Y e. B )
11 catcocl.z . . . . . . 7 |- ( ph -> Z e. B )
1211ad2antrr 762 . . . . . 6 |- ( ( ( ph /\ x = X ) /\ y = Y ) -> Z e. B )
13 catcocl.f . . . . . . . . 9 |- ( ph -> F e. ( X H Y ) )
1413ad3antrrr 766 . . . . . . . 8 |- ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) -> F e. ( X H Y ) )
15 simpllr 799 . . . . . . . . 9 |- ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) -> x = X )
16 simplr 792 . . . . . . . . 9 |- ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) -> y = Y )
1715, 16oveq12d 6668 . . . . . . . 8 |- ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) -> ( x H y ) = ( X H Y ) )
1814, 17eleqtrrd 2704 . . . . . . 7 |- ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) -> F e. ( x H y ) )
19 catcocl.g . . . . . . . . . 10 |- ( ph -> G e. ( Y H Z ) )
2019ad4antr 768 . . . . . . . . 9 |- ( ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) /\ f = F ) -> G e. ( Y H Z ) )
21 simpllr 799 . . . . . . . . . 10 |- ( ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) /\ f = F ) -> y = Y )
22 simplr 792 . . . . . . . . . 10 |- ( ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) /\ f = F ) -> z = Z )
2321, 22oveq12d 6668 . . . . . . . . 9 |- ( ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) /\ f = F ) -> ( y H z ) = ( Y H Z ) )
2420, 23eleqtrrd 2704 . . . . . . . 8 |- ( ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) /\ f = F ) -> G e. ( y H z ) )
25 catass.w . . . . . . . . . . 11 |- ( ph -> W e. B )
2625ad5antr 770 . . . . . . . . . 10 |- ( ( ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) /\ f = F ) /\ g = G ) -> W e. B )
27 catass.g . . . . . . . . . . . . 13 |- ( ph -> K e. ( Z H W ) )
2827ad6antr 772 . . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) /\ f = F ) /\ g = G ) /\ w = W ) -> K e. ( Z H W ) )
29 simp-4r 807 . . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) /\ f = F ) /\ g = G ) /\ w = W ) -> z = Z )
30 simpr 477 . . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) /\ f = F ) /\ g = G ) /\ w = W ) -> w = W )
3129, 30oveq12d 6668 . . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) /\ f = F ) /\ g = G ) /\ w = W ) -> ( z H w ) = ( Z H W ) )
3228, 31eleqtrrd 2704 . . . . . . . . . . 11 |- ( ( ( ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) /\ f = F ) /\ g = G ) /\ w = W ) -> K e. ( z H w ) )
33 simp-7r 813 . . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) /\ f = F ) /\ g = G ) /\ w = W ) /\ k = K ) -> x = X )
34 simp-6r 811 . . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) /\ f = F ) /\ g = G ) /\ w = W ) /\ k = K ) -> y = Y )
3533, 34opeq12d 4410 . . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) /\ f = F ) /\ g = G ) /\ w = W ) /\ k = K ) -> <. x , y >. = <. X , Y >. )
36 simplr 792 . . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) /\ f = F ) /\ g = G ) /\ w = W ) /\ k = K ) -> w = W )
3735, 36oveq12d 6668 . . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) /\ f = F ) /\ g = G ) /\ w = W ) /\ k = K ) -> ( <. x , y >. .x. w ) = ( <. X , Y >. .x. W ) )
38 simp-5r 809 . . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) /\ f = F ) /\ g = G ) /\ w = W ) /\ k = K ) -> z = Z )
3934, 38opeq12d 4410 . . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) /\ f = F ) /\ g = G ) /\ w = W ) /\ k = K ) -> <. y , z >. = <. Y , Z >. )
4039, 36oveq12d 6668 . . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) /\ f = F ) /\ g = G ) /\ w = W ) /\ k = K ) -> ( <. y , z >. .x. w ) = ( <. Y , Z >. .x. W ) )
41 simpr 477 . . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) /\ f = F ) /\ g = G ) /\ w = W ) /\ k = K ) -> k = K )
42 simpllr 799 . . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) /\ f = F ) /\ g = G ) /\ w = W ) /\ k = K ) -> g = G )
4340, 41, 42oveq123d 6671 . . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) /\ f = F ) /\ g = G ) /\ w = W ) /\ k = K ) -> ( k ( <. y , z >. .x. w ) g ) = ( K ( <. Y , Z >. .x. W ) G ) )
44 simp-4r 807 . . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) /\ f = F ) /\ g = G ) /\ w = W ) /\ k = K ) -> f = F )
4537, 43, 44oveq123d 6671 . . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) /\ f = F ) /\ g = G ) /\ w = W ) /\ k = K ) -> ( ( k ( <. y , z >. .x. w ) g ) ( <. x , y >. .x. w ) f ) = ( ( K ( <. Y , Z >. .x. W ) G ) ( <. X , Y >. .x. W ) F ) )
4633, 38opeq12d 4410 . . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) /\ f = F ) /\ g = G ) /\ w = W ) /\ k = K ) -> <. x , z >. = <. X , Z >. )
4746, 36oveq12d 6668 . . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) /\ f = F ) /\ g = G ) /\ w = W ) /\ k = K ) -> ( <. x , z >. .x. w ) = ( <. X , Z >. .x. W ) )
4835, 38oveq12d 6668 . . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) /\ f = F ) /\ g = G ) /\ w = W ) /\ k = K ) -> ( <. x , y >. .x. z ) = ( <. X , Y >. .x. Z ) )
4948, 42, 44oveq123d 6671 . . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) /\ f = F ) /\ g = G ) /\ w = W ) /\ k = K ) -> ( g ( <. x , y >. .x. z ) f ) = ( G ( <. X , Y >. .x. Z ) F ) )
5047, 41, 49oveq123d 6671 . . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) /\ f = F ) /\ g = G ) /\ w = W ) /\ k = K ) -> ( k ( <. x , z >. .x. w ) ( g ( <. x , y >. .x. z ) f ) ) = ( K ( <. X , Z >. .x. W ) ( G ( <. X , Y >. .x. Z ) F ) ) )
5145, 50eqeq12d 2637 . . . . . . . . . . 11 |- ( ( ( ( ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) /\ f = F ) /\ g = G ) /\ w = W ) /\ k = K ) -> ( ( ( k ( <. y , z >. .x. w ) g ) ( <. x , y >. .x. w ) f ) = ( k ( <. x , z >. .x. w ) ( g ( <. x , y >. .x. z ) f ) ) <-> ( ( K ( <. Y , Z >. .x. W ) G ) ( <. X , Y >. .x. W ) F ) = ( K ( <. X , Z >. .x. W ) ( G ( <. X , Y >. .x. Z ) F ) ) ) )
5232, 51rspcdv 3312 . . . . . . . . . 10 |- ( ( ( ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) /\ f = F ) /\ g = G ) /\ w = W ) -> ( A. k e. ( z H w ) ( ( k ( <. y , z >. .x. w ) g ) ( <. x , y >. .x. w ) f ) = ( k ( <. x , z >. .x. w ) ( g ( <. x , y >. .x. z ) f ) ) -> ( ( K ( <. Y , Z >. .x. W ) G ) ( <. X , Y >. .x. W ) F ) = ( K ( <. X , Z >. .x. W ) ( G ( <. X , Y >. .x. Z ) F ) ) ) )
5326, 52rspcimdv 3310 . . . . . . . . 9 |- ( ( ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) /\ f = F ) /\ g = G ) -> ( A. w e. B A. k e. ( z H w ) ( ( k ( <. y , z >. .x. w ) g ) ( <. x , y >. .x. w ) f ) = ( k ( <. x , z >. .x. w ) ( g ( <. x , y >. .x. z ) f ) ) -> ( ( K ( <. Y , Z >. .x. W ) G ) ( <. X , Y >. .x. W ) F ) = ( K ( <. X , Z >. .x. W ) ( G ( <. X , Y >. .x. Z ) F ) ) ) )
5453adantld 483 . . . . . . . 8 |- ( ( ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) /\ f = F ) /\ g = G ) -> ( ( ( g ( <. x , y >. .x. z ) f ) e. ( x H z ) /\ A. w e. B A. k e. ( z H w ) ( ( k ( <. y , z >. .x. w ) g ) ( <. x , y >. .x. w ) f ) = ( k ( <. x , z >. .x. w ) ( g ( <. x , y >. .x. z ) f ) ) ) -> ( ( K ( <. Y , Z >. .x. W ) G ) ( <. X , Y >. .x. W ) F ) = ( K ( <. X , Z >. .x. W ) ( G ( <. X , Y >. .x. Z ) F ) ) ) )
5524, 54rspcimdv 3310 . . . . . . 7 |- ( ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) /\ f = F ) -> ( A. g e. ( y H z ) ( ( g ( <. x , y >. .x. z ) f ) e. ( x H z ) /\ A. w e. B A. k e. ( z H w ) ( ( k ( <. y , z >. .x. w ) g ) ( <. x , y >. .x. w ) f ) = ( k ( <. x , z >. .x. w ) ( g ( <. x , y >. .x. z ) f ) ) ) -> ( ( K ( <. Y , Z >. .x. W ) G ) ( <. X , Y >. .x. W ) F ) = ( K ( <. X , Z >. .x. W ) ( G ( <. X , Y >. .x. Z ) F ) ) ) )
5618, 55rspcimdv 3310 . . . . . 6 |- ( ( ( ( ph /\ x = X ) /\ y = Y ) /\ z = Z ) -> ( A. f e. ( x H y ) A. g e. ( y H z ) ( ( g ( <. x , y >. .x. z ) f ) e. ( x H z ) /\ A. w e. B A. k e. ( z H w ) ( ( k ( <. y , z >. .x. w ) g ) ( <. x , y >. .x. w ) f ) = ( k ( <. x , z >. .x. w ) ( g ( <. x , y >. .x. z ) f ) ) ) -> ( ( K ( <. Y , Z >. .x. W ) G ) ( <. X , Y >. .x. W ) F ) = ( K ( <. X , Z >. .x. W ) ( G ( <. X , Y >. .x. Z ) F ) ) ) )
5712, 56rspcimdv 3310 . . . . 5 |- ( ( ( ph /\ x = X ) /\ y = Y ) -> ( A. z e. B A. f e. ( x H y ) A. g e. ( y H z ) ( ( g ( <. x , y >. .x. z ) f ) e. ( x H z ) /\ A. w e. B A. k e. ( z H w ) ( ( k ( <. y , z >. .x. w ) g ) ( <. x , y >. .x. w ) f ) = ( k ( <. x , z >. .x. w ) ( g ( <. x , y >. .x. z ) f ) ) ) -> ( ( K ( <. Y , Z >. .x. W ) G ) ( <. X , Y >. .x. W ) F ) = ( K ( <. X , Z >. .x. W ) ( G ( <. X , Y >. .x. Z ) F ) ) ) )
5810, 57rspcimdv 3310 . . . 4 |- ( ( ph /\ x = X ) -> ( A. y e. B A. z e. B A. f e. ( x H y ) A. g e. ( y H z ) ( ( g ( <. x , y >. .x. z ) f ) e. ( x H z ) /\ A. w e. B A. k e. ( z H w ) ( ( k ( <. y , z >. .x. w ) g ) ( <. x , y >. .x. w ) f ) = ( k ( <. x , z >. .x. w ) ( g ( <. x , y >. .x. z ) f ) ) ) -> ( ( K ( <. Y , Z >. .x. W ) G ) ( <. X , Y >. .x. W ) F ) = ( K ( <. X , Z >. .x. W ) ( G ( <. X , Y >. .x. Z ) F ) ) ) )
5958adantld 483 . . 3 |- ( ( ph /\ x = X ) -> ( ( E. g e. ( x H x ) A. y e. B ( A. f e. ( y H x ) ( g ( <. y , x >. .x. x ) f ) = f /\ A. f e. ( x H y ) ( f ( <. x , x >. .x. y ) g ) = f ) /\ A. y e. B A. z e. B A. f e. ( x H y ) A. g e. ( y H z ) ( ( g ( <. x , y >. .x. z ) f ) e. ( x H z ) /\ A. w e. B A. k e. ( z H w ) ( ( k ( <. y , z >. .x. w ) g ) ( <. x , y >. .x. w ) f ) = ( k ( <. x , z >. .x. w ) ( g ( <. x , y >. .x. z ) f ) ) ) ) -> ( ( K ( <. Y , Z >. .x. W ) G ) ( <. X , Y >. .x. W ) F ) = ( K ( <. X , Z >. .x. W ) ( G ( <. X , Y >. .x. Z ) F ) ) ) )
608, 59rspcimdv 3310 . 2 |- ( ph -> ( A. x e. B ( E. g e. ( x H x ) A. y e. B ( A. f e. ( y H x ) ( g ( <. y , x >. .x. x ) f ) = f /\ A. f e. ( x H y ) ( f ( <. x , x >. .x. y ) g ) = f ) /\ A. y e. B A. z e. B A. f e. ( x H y ) A. g e. ( y H z ) ( ( g ( <. x , y >. .x. z ) f ) e. ( x H z ) /\ A. w e. B A. k e. ( z H w ) ( ( k ( <. y , z >. .x. w ) g ) ( <. x , y >. .x. w ) f ) = ( k ( <. x , z >. .x. w ) ( g ( <. x , y >. .x. z ) f ) ) ) ) -> ( ( K ( <. Y , Z >. .x. W ) G ) ( <. X , Y >. .x. W ) F ) = ( K ( <. X , Z >. .x. W ) ( G ( <. X , Y >. .x. Z ) F ) ) ) )
617, 60mpd 15 1 |- ( ph -> ( ( K ( <. Y , Z >. .x. W ) G ) ( <. X , Y >. .x. W ) F ) = ( K ( <. X , Z >. .x. W ) ( G ( <. X , Y >. .x. Z ) F ) ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   <.cop 4183   ` cfv 5888  (class class class)co 6650   Basecbs 15857   Hom chom 15952  compcco 15953   Catccat 16325
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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653  df-cat 16329
This theorem is referenced by:  oppccatid  16379  sectcan  16415  sectco  16416  sectmon  16442  monsect  16443  rcaninv  16454  subccatid  16506  fuccocl  16624  fucass  16628  invfuc  16634  arwass  16724  xpccatid  16828  evlfcllem  16861  hofcllem  16898
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