Theorem revco 13580

Description: Mapping of words commutes with reversal. (Contributed by Stefan O'Rear, 27-Aug-2015.)

Assertion

Ref	Expression
revco	|- ( ( W e. Word A /\ F : A --> B ) -> ( F o. (reverse ` W ) ) = (reverse ` ( F o. W ) ) )

Proof of Theorem revco

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		wrdfn 13319	. . . . . . 7 |- ( W e. Word A -> W Fn ( 0..^ ( # ` W ) ) )
2	1	ad2antrr 762	. . . . . 6 |- ( ( ( W e. Word A /\ F : A --> B ) /\ x e. ( 0..^ ( # ` W ) ) ) -> W Fn ( 0..^ ( # ` W ) ) )
3		lencl 13324	. . . . . . . . . . . . 13 |- ( W e. Word A -> ( # ` W ) e. NN0 )
4	3	nn0zd 11480	. . . . . . . . . . . 12 |- ( W e. Word A -> ( # ` W ) e. ZZ )
5		fzoval 12471	. . . . . . . . . . . 12 |- ( ( # ` W ) e. ZZ -> ( 0..^ ( # ` W ) ) = ( 0 ... ( ( # ` W ) - 1 ) ) )
6	4, 5	syl 17	. . . . . . . . . . 11 |- ( W e. Word A -> ( 0..^ ( # ` W ) ) = ( 0 ... ( ( # ` W ) - 1 ) ) )
7	6	adantr 481	. . . . . . . . . 10 |- ( ( W e. Word A /\ F : A --> B ) -> ( 0..^ ( # ` W ) ) = ( 0 ... ( ( # ` W ) - 1 ) ) )
8	7	eleq2d 2687	. . . . . . . . 9 |- ( ( W e. Word A /\ F : A --> B ) -> ( x e. ( 0..^ ( # ` W ) ) <-> x e. ( 0 ... ( ( # ` W ) - 1 ) ) ) )
9	8	biimpa 501	. . . . . . . 8 |- ( ( ( W e. Word A /\ F : A --> B ) /\ x e. ( 0..^ ( # ` W ) ) ) -> x e. ( 0 ... ( ( # ` W ) - 1 ) ) )
10		fznn0sub2 12446	. . . . . . . 8 |- ( x e. ( 0 ... ( ( # ` W ) - 1 ) ) -> ( ( ( # ` W ) - 1 ) - x ) e. ( 0 ... ( ( # ` W ) - 1 ) ) )
11	9, 10	syl 17	. . . . . . 7 |- ( ( ( W e. Word A /\ F : A --> B ) /\ x e. ( 0..^ ( # ` W ) ) ) -> ( ( ( # ` W ) - 1 ) - x ) e. ( 0 ... ( ( # ` W ) - 1 ) ) )
12	7	adantr 481	. . . . . . 7 |- ( ( ( W e. Word A /\ F : A --> B ) /\ x e. ( 0..^ ( # ` W ) ) ) -> ( 0..^ ( # ` W ) ) = ( 0 ... ( ( # ` W ) - 1 ) ) )
13	11, 12	eleqtrrd 2704	. . . . . 6 |- ( ( ( W e. Word A /\ F : A --> B ) /\ x e. ( 0..^ ( # ` W ) ) ) -> ( ( ( # ` W ) - 1 ) - x ) e. ( 0..^ ( # ` W ) ) )
14		fvco2 6273	. . . . . 6 |- ( ( W Fn ( 0..^ ( # ` W ) ) /\ ( ( ( # ` W ) - 1 ) - x ) e. ( 0..^ ( # ` W ) ) ) -> ( ( F o. W ) ` ( ( ( # ` W ) - 1 ) - x ) ) = ( F ` ( W ` ( ( ( # ` W ) - 1 ) - x ) ) ) )
15	2, 13, 14	syl2anc 693	. . . . 5 |- ( ( ( W e. Word A /\ F : A --> B ) /\ x e. ( 0..^ ( # ` W ) ) ) -> ( ( F o. W ) ` ( ( ( # ` W ) - 1 ) - x ) ) = ( F ` ( W ` ( ( ( # ` W ) - 1 ) - x ) ) ) )
16		lenco 13578	. . . . . . . . 9 |- ( ( W e. Word A /\ F : A --> B ) -> ( # ` ( F o. W ) ) = ( # ` W ) )
17	16	oveq1d 6665	. . . . . . . 8 |- ( ( W e. Word A /\ F : A --> B ) -> ( ( # ` ( F o. W ) ) - 1 ) = ( ( # ` W ) - 1 ) )
18	17	oveq1d 6665	. . . . . . 7 |- ( ( W e. Word A /\ F : A --> B ) -> ( ( ( # ` ( F o. W ) ) - 1 ) - x ) = ( ( ( # ` W ) - 1 ) - x ) )
19	18	adantr 481	. . . . . 6 |- ( ( ( W e. Word A /\ F : A --> B ) /\ x e. ( 0..^ ( # ` W ) ) ) -> ( ( ( # ` ( F o. W ) ) - 1 ) - x ) = ( ( ( # ` W ) - 1 ) - x ) )
20	19	fveq2d 6195	. . . . 5 |- ( ( ( W e. Word A /\ F : A --> B ) /\ x e. ( 0..^ ( # ` W ) ) ) -> ( ( F o. W ) ` ( ( ( # ` ( F o. W ) ) - 1 ) - x ) ) = ( ( F o. W ) ` ( ( ( # ` W ) - 1 ) - x ) ) )
21		revfv 13512	. . . . . . 7 |- ( ( W e. Word A /\ x e. ( 0..^ ( # ` W ) ) ) -> ( (reverse ` W ) ` x ) = ( W ` ( ( ( # ` W ) - 1 ) - x ) ) )
22	21	adantlr 751	. . . . . 6 |- ( ( ( W e. Word A /\ F : A --> B ) /\ x e. ( 0..^ ( # ` W ) ) ) -> ( (reverse ` W ) ` x ) = ( W ` ( ( ( # ` W ) - 1 ) - x ) ) )
23	22	fveq2d 6195	. . . . 5 |- ( ( ( W e. Word A /\ F : A --> B ) /\ x e. ( 0..^ ( # ` W ) ) ) -> ( F ` ( (reverse ` W ) ` x ) ) = ( F ` ( W ` ( ( ( # ` W ) - 1 ) - x ) ) ) )
24	15, 20, 23	3eqtr4d 2666	. . . 4 |- ( ( ( W e. Word A /\ F : A --> B ) /\ x e. ( 0..^ ( # ` W ) ) ) -> ( ( F o. W ) ` ( ( ( # ` ( F o. W ) ) - 1 ) - x ) ) = ( F ` ( (reverse ` W ) ` x ) ) )
25	24	mpteq2dva 4744	. . 3 |- ( ( W e. Word A /\ F : A --> B ) -> ( x e. ( 0..^ ( # ` W ) ) |-> ( ( F o. W ) ` ( ( ( # ` ( F o. W ) ) - 1 ) - x ) ) ) = ( x e. ( 0..^ ( # ` W ) ) |-> ( F ` ( (reverse ` W ) ` x ) ) ) )
26	16	oveq2d 6666	. . . 4 |- ( ( W e. Word A /\ F : A --> B ) -> ( 0..^ ( # ` ( F o. W ) ) ) = ( 0..^ ( # ` W ) ) )
27	26	mpteq1d 4738	. . 3 |- ( ( W e. Word A /\ F : A --> B ) -> ( x e. ( 0..^ ( # ` ( F o. W ) ) ) |-> ( ( F o. W ) ` ( ( ( # ` ( F o. W ) ) - 1 ) - x ) ) ) = ( x e. ( 0..^ ( # ` W ) ) |-> ( ( F o. W ) ` ( ( ( # ` ( F o. W ) ) - 1 ) - x ) ) ) )
28		revlen 13511	. . . . . 6 |- ( W e. Word A -> ( # ` (reverse ` W ) ) = ( # ` W ) )
29	28	adantr 481	. . . . 5 |- ( ( W e. Word A /\ F : A --> B ) -> ( # ` (reverse ` W ) ) = ( # ` W ) )
30	29	oveq2d 6666	. . . 4 |- ( ( W e. Word A /\ F : A --> B ) -> ( 0..^ ( # ` (reverse ` W ) ) ) = ( 0..^ ( # ` W ) ) )
31	30	mpteq1d 4738	. . 3 |- ( ( W e. Word A /\ F : A --> B ) -> ( x e. ( 0..^ ( # ` (reverse ` W ) ) ) |-> ( F ` ( (reverse ` W ) ` x ) ) ) = ( x e. ( 0..^ ( # ` W ) ) |-> ( F ` ( (reverse ` W ) ` x ) ) ) )
32	25, 27, 31	3eqtr4rd 2667	. 2 |- ( ( W e. Word A /\ F : A --> B ) -> ( x e. ( 0..^ ( # ` (reverse ` W ) ) ) |-> ( F ` ( (reverse ` W ) ` x ) ) ) = ( x e. ( 0..^ ( # ` ( F o. W ) ) ) |-> ( ( F o. W ) ` ( ( ( # ` ( F o. W ) ) - 1 ) - x ) ) ) )
33		simpr 477	. . 3 |- ( ( W e. Word A /\ F : A --> B ) -> F : A --> B )
34		revcl 13510	. . . . 5 |- ( W e. Word A -> (reverse ` W ) e. Word A )
35		wrdf 13310	. . . . 5 |- ( (reverse ` W ) e. Word A -> (reverse ` W ) : ( 0..^ ( # ` (reverse ` W ) ) ) --> A )
36	34, 35	syl 17	. . . 4 |- ( W e. Word A -> (reverse ` W ) : ( 0..^ ( # ` (reverse ` W ) ) ) --> A )
37	36	adantr 481	. . 3 |- ( ( W e. Word A /\ F : A --> B ) -> (reverse ` W ) : ( 0..^ ( # ` (reverse ` W ) ) ) --> A )
38		fcompt 6400	. . 3 |- ( ( F : A --> B /\ (reverse ` W ) : ( 0..^ ( # ` (reverse ` W ) ) ) --> A ) -> ( F o. (reverse ` W ) ) = ( x e. ( 0..^ ( # ` (reverse ` W ) ) ) |-> ( F ` ( (reverse ` W ) ` x ) ) ) )
39	33, 37, 38	syl2anc 693	. 2 |- ( ( W e. Word A /\ F : A --> B ) -> ( F o. (reverse ` W ) ) = ( x e. ( 0..^ ( # ` (reverse ` W ) ) ) |-> ( F ` ( (reverse ` W ) ` x ) ) ) )
40		ffun 6048	. . . . 5 |- ( F : A --> B -> Fun F )
41	40	adantl 482	. . . 4 |- ( ( W e. Word A /\ F : A --> B ) -> Fun F )
42		simpl 473	. . . 4 |- ( ( W e. Word A /\ F : A --> B ) -> W e. Word A )
43		cofunexg 7130	. . . 4 |- ( ( Fun F /\ W e. Word A ) -> ( F o. W ) e. _V )
44	41, 42, 43	syl2anc 693	. . 3 |- ( ( W e. Word A /\ F : A --> B ) -> ( F o. W ) e. _V )
45		revval 13509	. . 3 |- ( ( F o. W ) e. _V -> (reverse ` ( F o. W ) ) = ( x e. ( 0..^ ( # ` ( F o. W ) ) ) |-> ( ( F o. W ) ` ( ( ( # ` ( F o. W ) ) - 1 ) - x ) ) ) )
46	44, 45	syl 17	. 2 |- ( ( W e. Word A /\ F : A --> B ) -> (reverse ` ( F o. W ) ) = ( x e. ( 0..^ ( # ` ( F o. W ) ) ) |-> ( ( F o. W ) ` ( ( ( # ` ( F o. W ) ) - 1 ) - x ) ) ) )
47	32, 39, 46	3eqtr4d 2666	1 |- ( ( W e. Word A /\ F : A --> B ) -> ( F o. (reverse ` W ) ) = (reverse ` ( F o. W ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   |-> cmpt 4729   o. ccom 5118   Fun wfun 5882   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   - cmin 10266   ZZcz 11377   ...cfz 12326  ..^cfzo 12465   #chash 13117  Word cword 13291  reversecreverse 13297

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-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  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-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-reverse 13305

This theorem is referenced by:  efginvrel1  18141