Theorem lswco 13584

Description: Mapping of (nonempty) words commutes with the "last symbol" operation. This theorem would not hold if W = (/), ( F ` (/) ) =/= (/) and (/) e. A, because then ( lastS ` ( F o. W ) ) = ( lastS ` (/) ) = (/) =/= ( F ` (/) ) = ( F ( lastS ` W ) ). (Contributed by AV, 11-Nov-2018.)

Assertion

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

Proof of Theorem lswco

Step	Hyp	Ref	Expression
1		ffun 6048	. . . . . 6 |- ( F : A --> B -> Fun F )
2	1	anim1i 592	. . . . 5 |- ( ( F : A --> B /\ W e. Word A ) -> ( Fun F /\ W e. Word A ) )
3	2	ancoms 469	. . . 4 |- ( ( W e. Word A /\ F : A --> B ) -> ( Fun F /\ W e. Word A ) )
4	3	3adant2 1080	. . 3 |- ( ( W e. Word A /\ W =/= (/) /\ F : A --> B ) -> ( Fun F /\ W e. Word A ) )
5		cofunexg 7130	. . 3 |- ( ( Fun F /\ W e. Word A ) -> ( F o. W ) e. _V )
6		lsw 13351	. . 3 |- ( ( F o. W ) e. _V -> ( lastS ` ( F o. W ) ) = ( ( F o. W ) ` ( ( # ` ( F o. W ) ) - 1 ) ) )
7	4, 5, 6	3syl 18	. 2 |- ( ( W e. Word A /\ W =/= (/) /\ F : A --> B ) -> ( lastS ` ( F o. W ) ) = ( ( F o. W ) ` ( ( # ` ( F o. W ) ) - 1 ) ) )
8		lenco 13578	. . . . 5 |- ( ( W e. Word A /\ F : A --> B ) -> ( # ` ( F o. W ) ) = ( # ` W ) )
9	8	3adant2 1080	. . . 4 |- ( ( W e. Word A /\ W =/= (/) /\ F : A --> B ) -> ( # ` ( F o. W ) ) = ( # ` W ) )
10	9	oveq1d 6665	. . 3 |- ( ( W e. Word A /\ W =/= (/) /\ F : A --> B ) -> ( ( # ` ( F o. W ) ) - 1 ) = ( ( # ` W ) - 1 ) )
11	10	fveq2d 6195	. 2 |- ( ( W e. Word A /\ W =/= (/) /\ F : A --> B ) -> ( ( F o. W ) ` ( ( # ` ( F o. W ) ) - 1 ) ) = ( ( F o. W ) ` ( ( # ` W ) - 1 ) ) )
12		wrdf 13310	. . . . . . 7 |- ( W e. Word A -> W : ( 0..^ ( # ` W ) ) --> A )
13	12	adantr 481	. . . . . 6 |- ( ( W e. Word A /\ W =/= (/) ) -> W : ( 0..^ ( # ` W ) ) --> A )
14		lennncl 13325	. . . . . . 7 |- ( ( W e. Word A /\ W =/= (/) ) -> ( # ` W ) e. NN )
15		fzo0end 12560	. . . . . . 7 |- ( ( # ` W ) e. NN -> ( ( # ` W ) - 1 ) e. ( 0..^ ( # ` W ) ) )
16	14, 15	syl 17	. . . . . 6 |- ( ( W e. Word A /\ W =/= (/) ) -> ( ( # ` W ) - 1 ) e. ( 0..^ ( # ` W ) ) )
17	13, 16	jca 554	. . . . 5 |- ( ( W e. Word A /\ W =/= (/) ) -> ( W : ( 0..^ ( # ` W ) ) --> A /\ ( ( # ` W ) - 1 ) e. ( 0..^ ( # ` W ) ) ) )
18	17	3adant3 1081	. . . 4 |- ( ( W e. Word A /\ W =/= (/) /\ F : A --> B ) -> ( W : ( 0..^ ( # ` W ) ) --> A /\ ( ( # ` W ) - 1 ) e. ( 0..^ ( # ` W ) ) ) )
19		fvco3 6275	. . . 4 |- ( ( W : ( 0..^ ( # ` W ) ) --> A /\ ( ( # ` W ) - 1 ) e. ( 0..^ ( # ` W ) ) ) -> ( ( F o. W ) ` ( ( # ` W ) - 1 ) ) = ( F ` ( W ` ( ( # ` W ) - 1 ) ) ) )
20	18, 19	syl 17	. . 3 |- ( ( W e. Word A /\ W =/= (/) /\ F : A --> B ) -> ( ( F o. W ) ` ( ( # ` W ) - 1 ) ) = ( F ` ( W ` ( ( # ` W ) - 1 ) ) ) )
21		lsw 13351	. . . . . 6 |- ( W e. Word A -> ( lastS ` W ) = ( W ` ( ( # ` W ) - 1 ) ) )
22	21	3ad2ant1 1082	. . . . 5 |- ( ( W e. Word A /\ W =/= (/) /\ F : A --> B ) -> ( lastS ` W ) = ( W ` ( ( # ` W ) - 1 ) ) )
23	22	eqcomd 2628	. . . 4 |- ( ( W e. Word A /\ W =/= (/) /\ F : A --> B ) -> ( W ` ( ( # ` W ) - 1 ) ) = ( lastS ` W ) )
24	23	fveq2d 6195	. . 3 |- ( ( W e. Word A /\ W =/= (/) /\ F : A --> B ) -> ( F ` ( W ` ( ( # ` W ) - 1 ) ) ) = ( F ` ( lastS ` W ) ) )
25	20, 24	eqtrd 2656	. 2 |- ( ( W e. Word A /\ W =/= (/) /\ F : A --> B ) -> ( ( F o. W ) ` ( ( # ` W ) - 1 ) ) = ( F ` ( lastS ` W ) ) )
26	7, 11, 25	3eqtrd 2660	1 |- ( ( W e. Word A /\ W =/= (/) /\ F : A --> B ) -> ( lastS ` ( F o. W ) ) = ( F ` ( lastS ` W ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   (/)c0 3915   o. ccom 5118   Fun wfun 5882   -->wf 5884   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   - cmin 10266   NNcn 11020  ..^cfzo 12465   #chash 13117  Word cword 13291   lastS clsw 13292

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-lsw 13300

This theorem is referenced by: (None)