Theorem sectco 16416

Description: Composition of two sections. (Contributed by Mario Carneiro, 2-Jan-2017.)

Hypotheses

Ref	Expression
sectco.b	|- B = ( Base ` C )
sectco.o	|- .x. = (comp ` C )
sectco.s	|- S = (Sect ` C )
sectco.c	|- ( ph -> C e. Cat )
sectco.x	|- ( ph -> X e. B )
sectco.y	|- ( ph -> Y e. B )
sectco.z	|- ( ph -> Z e. B )
sectco.1	|- ( ph -> F ( X S Y ) G )
sectco.2	|- ( ph -> H ( Y S Z ) K )

Assertion

Ref	Expression
sectco	|- ( ph -> ( H ( <. X , Y >. .x. Z ) F ) ( X S Z ) ( G ( <. Z , Y >. .x. X ) K ) )

Proof of Theorem sectco

Step	Hyp	Ref	Expression
1		sectco.b	. . . 4 |- B = ( Base ` C )
2		eqid 2622	. . . 4 |- ( Hom ` C ) = ( Hom ` C )
3		sectco.o	. . . 4 |- .x. = (comp ` C )
4		sectco.c	. . . 4 |- ( ph -> C e. Cat )
5		sectco.x	. . . 4 |- ( ph -> X e. B )
6		sectco.z	. . . 4 |- ( ph -> Z e. B )
7		sectco.y	. . . 4 |- ( ph -> Y e. B )
8		sectco.1	. . . . . . 7 |- ( ph -> F ( X S Y ) G )
9		eqid 2622	. . . . . . . 8 |- ( Id ` C ) = ( Id ` C )
10		sectco.s	. . . . . . . 8 |- S = (Sect ` C )
11	1, 2, 3, 9, 10, 4, 5, 7	issect 16413	. . . . . . 7 |- ( ph -> ( F ( X S Y ) G <-> ( F e. ( X ( Hom ` C ) Y ) /\ G e. ( Y ( Hom ` C ) X ) /\ ( G ( <. X , Y >. .x. X ) F ) = ( ( Id ` C ) ` X ) ) ) )
12	8, 11	mpbid 222	. . . . . 6 |- ( ph -> ( F e. ( X ( Hom ` C ) Y ) /\ G e. ( Y ( Hom ` C ) X ) /\ ( G ( <. X , Y >. .x. X ) F ) = ( ( Id ` C ) ` X ) ) )
13	12	simp1d 1073	. . . . 5 |- ( ph -> F e. ( X ( Hom ` C ) Y ) )
14		sectco.2	. . . . . . 7 |- ( ph -> H ( Y S Z ) K )
15	1, 2, 3, 9, 10, 4, 7, 6	issect 16413	. . . . . . 7 |- ( ph -> ( H ( Y S Z ) K <-> ( H e. ( Y ( Hom ` C ) Z ) /\ K e. ( Z ( Hom ` C ) Y ) /\ ( K ( <. Y , Z >. .x. Y ) H ) = ( ( Id ` C ) ` Y ) ) ) )
16	14, 15	mpbid 222	. . . . . 6 |- ( ph -> ( H e. ( Y ( Hom ` C ) Z ) /\ K e. ( Z ( Hom ` C ) Y ) /\ ( K ( <. Y , Z >. .x. Y ) H ) = ( ( Id ` C ) ` Y ) ) )
17	16	simp1d 1073	. . . . 5 |- ( ph -> H e. ( Y ( Hom ` C ) Z ) )
18	1, 2, 3, 4, 5, 7, 6, 13, 17	catcocl 16346	. . . 4 |- ( ph -> ( H ( <. X , Y >. .x. Z ) F ) e. ( X ( Hom ` C ) Z ) )
19	16	simp2d 1074	. . . 4 |- ( ph -> K e. ( Z ( Hom ` C ) Y ) )
20	12	simp2d 1074	. . . 4 |- ( ph -> G e. ( Y ( Hom ` C ) X ) )
21	1, 2, 3, 4, 5, 6, 7, 18, 19, 5, 20	catass 16347	. . 3 |- ( ph -> ( ( G ( <. Z , Y >. .x. X ) K ) ( <. X , Z >. .x. X ) ( H ( <. X , Y >. .x. Z ) F ) ) = ( G ( <. X , Y >. .x. X ) ( K ( <. X , Z >. .x. Y ) ( H ( <. X , Y >. .x. Z ) F ) ) ) )
22	16	simp3d 1075	. . . . . 6 |- ( ph -> ( K ( <. Y , Z >. .x. Y ) H ) = ( ( Id ` C ) ` Y ) )
23	22	oveq1d 6665	. . . . 5 |- ( ph -> ( ( K ( <. Y , Z >. .x. Y ) H ) ( <. X , Y >. .x. Y ) F ) = ( ( ( Id ` C ) ` Y ) ( <. X , Y >. .x. Y ) F ) )
24	1, 2, 3, 4, 5, 7, 6, 13, 17, 7, 19	catass 16347	. . . . 5 |- ( ph -> ( ( K ( <. Y , Z >. .x. Y ) H ) ( <. X , Y >. .x. Y ) F ) = ( K ( <. X , Z >. .x. Y ) ( H ( <. X , Y >. .x. Z ) F ) ) )
25	1, 2, 9, 4, 5, 3, 7, 13	catlid 16344	. . . . 5 |- ( ph -> ( ( ( Id ` C ) ` Y ) ( <. X , Y >. .x. Y ) F ) = F )
26	23, 24, 25	3eqtr3d 2664	. . . 4 |- ( ph -> ( K ( <. X , Z >. .x. Y ) ( H ( <. X , Y >. .x. Z ) F ) ) = F )
27	26	oveq2d 6666	. . 3 |- ( ph -> ( G ( <. X , Y >. .x. X ) ( K ( <. X , Z >. .x. Y ) ( H ( <. X , Y >. .x. Z ) F ) ) ) = ( G ( <. X , Y >. .x. X ) F ) )
28	12	simp3d 1075	. . 3 |- ( ph -> ( G ( <. X , Y >. .x. X ) F ) = ( ( Id ` C ) ` X ) )
29	21, 27, 28	3eqtrd 2660	. 2 |- ( ph -> ( ( G ( <. Z , Y >. .x. X ) K ) ( <. X , Z >. .x. X ) ( H ( <. X , Y >. .x. Z ) F ) ) = ( ( Id ` C ) ` X ) )
30	1, 2, 3, 4, 6, 7, 5, 19, 20	catcocl 16346	. . 3 |- ( ph -> ( G ( <. Z , Y >. .x. X ) K ) e. ( Z ( Hom ` C ) X ) )
31	1, 2, 3, 9, 10, 4, 5, 6, 18, 30	issect2 16414	. 2 |- ( ph -> ( ( H ( <. X , Y >. .x. Z ) F ) ( X S Z ) ( G ( <. Z , Y >. .x. X ) K ) <-> ( ( G ( <. Z , Y >. .x. X ) K ) ( <. X , Z >. .x. X ) ( H ( <. X , Y >. .x. Z ) F ) ) = ( ( Id ` C ) ` X ) ) )
32	29, 31	mpbird 247	1 |- ( ph -> ( H ( <. X , Y >. .x. Z ) F ) ( X S Z ) ( G ( <. Z , Y >. .x. X ) K ) )

Syntax hints:   -> wi 4   /\ w3a 1037   = wceq 1483   e. wcel 1990   <.cop 4183   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   Hom chom 15952  compcco 15953   Catccat 16325   Idccid 16326  Sectcsect 16404

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

This theorem is referenced by:  invco  16431