Theorem rcaninv 16454

Description: Right cancellation of an inverse of an isomorphism. (Contributed by AV, 5-Apr-2017.)

Hypotheses

Ref	Expression
rcaninv.b	|- B = ( Base ` C )
rcaninv.n	|- N = (Inv ` C )
rcaninv.c	|- ( ph -> C e. Cat )
rcaninv.x	|- ( ph -> X e. B )
rcaninv.y	|- ( ph -> Y e. B )
rcaninv.z	|- ( ph -> Z e. B )
rcaninv.f	|- ( ph -> F e. ( Y ( Iso ` C ) X ) )
rcaninv.g	|- ( ph -> G e. ( Y ( Hom ` C ) Z ) )
rcaninv.h	|- ( ph -> H e. ( Y ( Hom ` C ) Z ) )
rcaninv.1	|- R = ( ( Y N X ) ` F )
rcaninv.o	|- .o. = ( <. X , Y >. (comp ` C ) Z )

Assertion

Ref	Expression
rcaninv	|- ( ph -> ( ( G .o. R ) = ( H .o. R ) -> G = H ) )

Proof of Theorem rcaninv

Step	Hyp	Ref	Expression
1		rcaninv.b	. . . . . 6 |- B = ( Base ` C )
2		eqid 2622	. . . . . 6 |- ( Hom ` C ) = ( Hom ` C )
3		eqid 2622	. . . . . 6 |- (comp ` C ) = (comp ` C )
4		rcaninv.c	. . . . . 6 |- ( ph -> C e. Cat )
5		rcaninv.y	. . . . . 6 |- ( ph -> Y e. B )
6		rcaninv.x	. . . . . 6 |- ( ph -> X e. B )
7		eqid 2622	. . . . . . . 8 |- ( Iso ` C ) = ( Iso ` C )
8	1, 2, 7, 4, 5, 6	isohom 16436	. . . . . . 7 |- ( ph -> ( Y ( Iso ` C ) X ) C_ ( Y ( Hom ` C ) X ) )
9		rcaninv.f	. . . . . . 7 |- ( ph -> F e. ( Y ( Iso ` C ) X ) )
10	8, 9	sseldd 3604	. . . . . 6 |- ( ph -> F e. ( Y ( Hom ` C ) X ) )
11	1, 2, 7, 4, 6, 5	isohom 16436	. . . . . . 7 |- ( ph -> ( X ( Iso ` C ) Y ) C_ ( X ( Hom ` C ) Y ) )
12		rcaninv.n	. . . . . . . . 9 |- N = (Inv ` C )
13	1, 12, 4, 5, 6, 7	invf 16428	. . . . . . . 8 |- ( ph -> ( Y N X ) : ( Y ( Iso ` C ) X ) --> ( X ( Iso ` C ) Y ) )
14	13, 9	ffvelrnd 6360	. . . . . . 7 |- ( ph -> ( ( Y N X ) ` F ) e. ( X ( Iso ` C ) Y ) )
15	11, 14	sseldd 3604	. . . . . 6 |- ( ph -> ( ( Y N X ) ` F ) e. ( X ( Hom ` C ) Y ) )
16		rcaninv.z	. . . . . 6 |- ( ph -> Z e. B )
17		rcaninv.g	. . . . . 6 |- ( ph -> G e. ( Y ( Hom ` C ) Z ) )
18	1, 2, 3, 4, 5, 6, 5, 10, 15, 16, 17	catass 16347	. . . . 5 |- ( ph -> ( ( G ( <. X , Y >. (comp ` C ) Z ) ( ( Y N X ) ` F ) ) ( <. Y , X >. (comp ` C ) Z ) F ) = ( G ( <. Y , Y >. (comp ` C ) Z ) ( ( ( Y N X ) ` F ) ( <. Y , X >. (comp ` C ) Y ) F ) ) )
19		eqid 2622	. . . . . . . 8 |- ( Id ` C ) = ( Id ` C )
20		eqid 2622	. . . . . . . 8 |- ( <. Y , X >. (comp ` C ) Y ) = ( <. Y , X >. (comp ` C ) Y )
21	1, 7, 12, 4, 5, 6, 9, 19, 20	invcoisoid 16452	. . . . . . 7 |- ( ph -> ( ( ( Y N X ) ` F ) ( <. Y , X >. (comp ` C ) Y ) F ) = ( ( Id ` C ) ` Y ) )
22	21	eqcomd 2628	. . . . . 6 |- ( ph -> ( ( Id ` C ) ` Y ) = ( ( ( Y N X ) ` F ) ( <. Y , X >. (comp ` C ) Y ) F ) )
23	22	oveq2d 6666	. . . . 5 |- ( ph -> ( G ( <. Y , Y >. (comp ` C ) Z ) ( ( Id ` C ) ` Y ) ) = ( G ( <. Y , Y >. (comp ` C ) Z ) ( ( ( Y N X ) ` F ) ( <. Y , X >. (comp ` C ) Y ) F ) ) )
24	1, 2, 19, 4, 5, 3, 16, 17	catrid 16345	. . . . 5 |- ( ph -> ( G ( <. Y , Y >. (comp ` C ) Z ) ( ( Id ` C ) ` Y ) ) = G )
25	18, 23, 24	3eqtr2rd 2663	. . . 4 |- ( ph -> G = ( ( G ( <. X , Y >. (comp ` C ) Z ) ( ( Y N X ) ` F ) ) ( <. Y , X >. (comp ` C ) Z ) F ) )
26	25	adantr 481	. . 3 |- ( ( ph /\ ( G .o. R ) = ( H .o. R ) ) -> G = ( ( G ( <. X , Y >. (comp ` C ) Z ) ( ( Y N X ) ` F ) ) ( <. Y , X >. (comp ` C ) Z ) F ) )
27		rcaninv.o	. . . . . . . . 9 |- .o. = ( <. X , Y >. (comp ` C ) Z )
28	27	eqcomi 2631	. . . . . . . 8 |- ( <. X , Y >. (comp ` C ) Z ) = .o.
29	28	a1i 11	. . . . . . 7 |- ( ph -> ( <. X , Y >. (comp ` C ) Z ) = .o. )
30		eqidd 2623	. . . . . . 7 |- ( ph -> G = G )
31		rcaninv.1	. . . . . . . . 9 |- R = ( ( Y N X ) ` F )
32	31	eqcomi 2631	. . . . . . . 8 |- ( ( Y N X ) ` F ) = R
33	32	a1i 11	. . . . . . 7 |- ( ph -> ( ( Y N X ) ` F ) = R )
34	29, 30, 33	oveq123d 6671	. . . . . 6 |- ( ph -> ( G ( <. X , Y >. (comp ` C ) Z ) ( ( Y N X ) ` F ) ) = ( G .o. R ) )
35	34	adantr 481	. . . . 5 |- ( ( ph /\ ( G .o. R ) = ( H .o. R ) ) -> ( G ( <. X , Y >. (comp ` C ) Z ) ( ( Y N X ) ` F ) ) = ( G .o. R ) )
36		simpr 477	. . . . 5 |- ( ( ph /\ ( G .o. R ) = ( H .o. R ) ) -> ( G .o. R ) = ( H .o. R ) )
37	35, 36	eqtrd 2656	. . . 4 |- ( ( ph /\ ( G .o. R ) = ( H .o. R ) ) -> ( G ( <. X , Y >. (comp ` C ) Z ) ( ( Y N X ) ` F ) ) = ( H .o. R ) )
38	37	oveq1d 6665	. . 3 |- ( ( ph /\ ( G .o. R ) = ( H .o. R ) ) -> ( ( G ( <. X , Y >. (comp ` C ) Z ) ( ( Y N X ) ` F ) ) ( <. Y , X >. (comp ` C ) Z ) F ) = ( ( H .o. R ) ( <. Y , X >. (comp ` C ) Z ) F ) )
39	27	oveqi 6663	. . . . . . 7 |- ( H .o. R ) = ( H ( <. X , Y >. (comp ` C ) Z ) R )
40	39	oveq1i 6660	. . . . . 6 |- ( ( H .o. R ) ( <. Y , X >. (comp ` C ) Z ) F ) = ( ( H ( <. X , Y >. (comp ` C ) Z ) R ) ( <. Y , X >. (comp ` C ) Z ) F )
41	40	a1i 11	. . . . 5 |- ( ph -> ( ( H .o. R ) ( <. Y , X >. (comp ` C ) Z ) F ) = ( ( H ( <. X , Y >. (comp ` C ) Z ) R ) ( <. Y , X >. (comp ` C ) Z ) F ) )
42	31, 15	syl5eqel 2705	. . . . . . 7 |- ( ph -> R e. ( X ( Hom ` C ) Y ) )
43		rcaninv.h	. . . . . . 7 |- ( ph -> H e. ( Y ( Hom ` C ) Z ) )
44	1, 2, 3, 4, 5, 6, 5, 10, 42, 16, 43	catass 16347	. . . . . 6 |- ( ph -> ( ( H ( <. X , Y >. (comp ` C ) Z ) R ) ( <. Y , X >. (comp ` C ) Z ) F ) = ( H ( <. Y , Y >. (comp ` C ) Z ) ( R ( <. Y , X >. (comp ` C ) Y ) F ) ) )
45	31	oveq1i 6660	. . . . . . . 8 |- ( R ( <. Y , X >. (comp ` C ) Y ) F ) = ( ( ( Y N X ) ` F ) ( <. Y , X >. (comp ` C ) Y ) F )
46	45	oveq2i 6661	. . . . . . 7 |- ( H ( <. Y , Y >. (comp ` C ) Z ) ( R ( <. Y , X >. (comp ` C ) Y ) F ) ) = ( H ( <. Y , Y >. (comp ` C ) Z ) ( ( ( Y N X ) ` F ) ( <. Y , X >. (comp ` C ) Y ) F ) )
47	46	a1i 11	. . . . . 6 |- ( ph -> ( H ( <. Y , Y >. (comp ` C ) Z ) ( R ( <. Y , X >. (comp ` C ) Y ) F ) ) = ( H ( <. Y , Y >. (comp ` C ) Z ) ( ( ( Y N X ) ` F ) ( <. Y , X >. (comp ` C ) Y ) F ) ) )
48	21	oveq2d 6666	. . . . . 6 |- ( ph -> ( H ( <. Y , Y >. (comp ` C ) Z ) ( ( ( Y N X ) ` F ) ( <. Y , X >. (comp ` C ) Y ) F ) ) = ( H ( <. Y , Y >. (comp ` C ) Z ) ( ( Id ` C ) ` Y ) ) )
49	44, 47, 48	3eqtrd 2660	. . . . 5 |- ( ph -> ( ( H ( <. X , Y >. (comp ` C ) Z ) R ) ( <. Y , X >. (comp ` C ) Z ) F ) = ( H ( <. Y , Y >. (comp ` C ) Z ) ( ( Id ` C ) ` Y ) ) )
50	1, 2, 19, 4, 5, 3, 16, 43	catrid 16345	. . . . 5 |- ( ph -> ( H ( <. Y , Y >. (comp ` C ) Z ) ( ( Id ` C ) ` Y ) ) = H )
51	41, 49, 50	3eqtrd 2660	. . . 4 |- ( ph -> ( ( H .o. R ) ( <. Y , X >. (comp ` C ) Z ) F ) = H )
52	51	adantr 481	. . 3 |- ( ( ph /\ ( G .o. R ) = ( H .o. R ) ) -> ( ( H .o. R ) ( <. Y , X >. (comp ` C ) Z ) F ) = H )
53	26, 38, 52	3eqtrd 2660	. 2 |- ( ( ph /\ ( G .o. R ) = ( H .o. R ) ) -> G = H )
54	53	ex 450	1 |- ( ph -> ( ( G .o. R ) = ( H .o. R ) -> G = H ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   <.cop 4183   ` cfv 5888  (class class class)co 6650   Basecbs 15857   Hom chom 15952  compcco 15953   Catccat 16325   Idccid 16326  Invcinv 16405   Iso ciso 16406

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

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-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  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-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  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-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-1st 7168  df-2nd 7169  df-cat 16329  df-cid 16330  df-sect 16407  df-inv 16408  df-iso 16409

This theorem is referenced by:  initoeu2lem0  16663