Theorem lswccatn0lsw 11159

Description: The last symbol of a word concatenated with a nonempty word is the last symbol of the nonempty word. (Contributed by AV, 22-Oct-2018.) (Proof shortened by AV, 1-May-2020.)

Assertion

Ref	Expression
lswccatn0lsw	|- ( ( A e. Word V /\ B e. Word V /\ B =/= (/) ) -> (lastS ` ( A ++ B ) ) = (lastS ` B ) )

Proof of Theorem lswccatn0lsw

Step	Hyp	Ref	Expression
1		ccatlen 11143	. . . . . . 7 |- ( ( A e. Word V /\ B e. Word V ) -> (♯ ` ( A ++ B ) ) = ( (♯ ` A ) + (♯ ` B ) ) )
2	1	oveq1d 6022	. . . . . 6 |- ( ( A e. Word V /\ B e. Word V ) -> ( (♯ ` ( A ++ B ) ) - 1 ) = ( ( (♯ ` A ) + (♯ ` B ) ) - 1 ) )
3	2	3adant3 1041	. . . . 5 |- ( ( A e. Word V /\ B e. Word V /\ B =/= (/) ) -> ( (♯ ` ( A ++ B ) ) - 1 ) = ( ( (♯ ` A ) + (♯ ` B ) ) - 1 ) )
4		lencl 11088	. . . . . . . . . 10 |- ( A e. Word V -> (♯ ` A ) e. NN0 )
5	4	nn0zd 9578	. . . . . . . . 9 |- ( A e. Word V -> (♯ ` A ) e. ZZ )
6		lennncl 11104	. . . . . . . . 9 |- ( ( B e. Word V /\ B =/= (/) ) -> (♯ ` B ) e. NN )
7		simpl 109	. . . . . . . . . 10 |- ( ( (♯ ` A ) e. ZZ /\ (♯ ` B ) e. NN ) -> (♯ ` A ) e. ZZ )
8		zaddcllempos 9494	. . . . . . . . . 10 |- ( ( (♯ ` A ) e. ZZ /\ (♯ ` B ) e. NN ) -> ( (♯ ` A ) + (♯ ` B ) ) e. ZZ )
9		zre 9461	. . . . . . . . . . 11 |- ( (♯ ` A ) e. ZZ -> (♯ ` A ) e. RR )
10		nnrp 9871	. . . . . . . . . . 11 |- ( (♯ ` B ) e. NN -> (♯ ` B ) e. RR+ )
11		ltaddrp 9899	. . . . . . . . . . 11 |- ( ( (♯ ` A ) e. RR /\ (♯ ` B ) e. RR+ ) -> (♯ ` A ) < ( (♯ ` A ) + (♯ ` B ) ) )
12	9, 10, 11	syl2an 289	. . . . . . . . . 10 |- ( ( (♯ ` A ) e. ZZ /\ (♯ ` B ) e. NN ) -> (♯ ` A ) < ( (♯ ` A ) + (♯ ` B ) ) )
13	7, 8, 12	3jca 1201	. . . . . . . . 9 |- ( ( (♯ ` A ) e. ZZ /\ (♯ ` B ) e. NN ) -> ( (♯ ` A ) e. ZZ /\ ( (♯ ` A ) + (♯ ` B ) ) e. ZZ /\ (♯ ` A ) < ( (♯ ` A ) + (♯ ` B ) ) ) )
14	5, 6, 13	syl2an 289	. . . . . . . 8 |- ( ( A e. Word V /\ ( B e. Word V /\ B =/= (/) ) ) -> ( (♯ ` A ) e. ZZ /\ ( (♯ ` A ) + (♯ ` B ) ) e. ZZ /\ (♯ ` A ) < ( (♯ ` A ) + (♯ ` B ) ) ) )
15	14	3impb 1223	. . . . . . 7 |- ( ( A e. Word V /\ B e. Word V /\ B =/= (/) ) -> ( (♯ ` A ) e. ZZ /\ ( (♯ ` A ) + (♯ ` B ) ) e. ZZ /\ (♯ ` A ) < ( (♯ ` A ) + (♯ ` B ) ) ) )
16		fzolb 10362	. . . . . . 7 |- ( (♯ ` A ) e. ( (♯ ` A )..^ ( (♯ ` A ) + (♯ ` B ) ) ) <-> ( (♯ ` A ) e. ZZ /\ ( (♯ ` A ) + (♯ ` B ) ) e. ZZ /\ (♯ ` A ) < ( (♯ ` A ) + (♯ ` B ) ) ) )
17	15, 16	sylibr 134	. . . . . 6 |- ( ( A e. Word V /\ B e. Word V /\ B =/= (/) ) -> (♯ ` A ) e. ( (♯ ` A )..^ ( (♯ ` A ) + (♯ ` B ) ) ) )
18		fzoend 10440	. . . . . 6 |- ( (♯ ` A ) e. ( (♯ ` A )..^ ( (♯ ` A ) + (♯ ` B ) ) ) -> ( ( (♯ ` A ) + (♯ ` B ) ) - 1 ) e. ( (♯ ` A )..^ ( (♯ ` A ) + (♯ ` B ) ) ) )
19	17, 18	syl 14	. . . . 5 |- ( ( A e. Word V /\ B e. Word V /\ B =/= (/) ) -> ( ( (♯ ` A ) + (♯ ` B ) ) - 1 ) e. ( (♯ ` A )..^ ( (♯ ` A ) + (♯ ` B ) ) ) )
20	3, 19	eqeltrd 2306	. . . 4 |- ( ( A e. Word V /\ B e. Word V /\ B =/= (/) ) -> ( (♯ ` ( A ++ B ) ) - 1 ) e. ( (♯ ` A )..^ ( (♯ ` A ) + (♯ ` B ) ) ) )
21		ccatval2 11146	. . . 4 |- ( ( A e. Word V /\ B e. Word V /\ ( (♯ ` ( A ++ B ) ) - 1 ) e. ( (♯ ` A )..^ ( (♯ ` A ) + (♯ ` B ) ) ) ) -> ( ( A ++ B ) ` ( (♯ ` ( A ++ B ) ) - 1 ) ) = ( B ` ( ( (♯ ` ( A ++ B ) ) - 1 ) - (♯ ` A ) ) ) )
22	20, 21	syld3an3 1316	. . 3 |- ( ( A e. Word V /\ B e. Word V /\ B =/= (/) ) -> ( ( A ++ B ) ` ( (♯ ` ( A ++ B ) ) - 1 ) ) = ( B ` ( ( (♯ ` ( A ++ B ) ) - 1 ) - (♯ ` A ) ) ) )
23	2	oveq1d 6022	. . . . . 6 |- ( ( A e. Word V /\ B e. Word V ) -> ( ( (♯ ` ( A ++ B ) ) - 1 ) - (♯ ` A ) ) = ( ( ( (♯ ` A ) + (♯ ` B ) ) - 1 ) - (♯ ` A ) ) )
24	4	nn0cnd 9435	. . . . . . 7 |- ( A e. Word V -> (♯ ` A ) e. CC )
25		lencl 11088	. . . . . . . 8 |- ( B e. Word V -> (♯ ` B ) e. NN0 )
26	25	nn0cnd 9435	. . . . . . 7 |- ( B e. Word V -> (♯ ` B ) e. CC )
27		addcl 8135	. . . . . . . . 9 |- ( ( (♯ ` A ) e. CC /\ (♯ ` B ) e. CC ) -> ( (♯ ` A ) + (♯ ` B ) ) e. CC )
28		1cnd 8173	. . . . . . . . 9 |- ( ( (♯ ` A ) e. CC /\ (♯ ` B ) e. CC ) -> 1 e. CC )
29		simpl 109	. . . . . . . . 9 |- ( ( (♯ ` A ) e. CC /\ (♯ ` B ) e. CC ) -> (♯ ` A ) e. CC )
30	27, 28, 29	sub32d 8500	. . . . . . . 8 |- ( ( (♯ ` A ) e. CC /\ (♯ ` B ) e. CC ) -> ( ( ( (♯ ` A ) + (♯ ` B ) ) - 1 ) - (♯ ` A ) ) = ( ( ( (♯ ` A ) + (♯ ` B ) ) - (♯ ` A ) ) - 1 ) )
31		pncan2 8364	. . . . . . . . 9 |- ( ( (♯ ` A ) e. CC /\ (♯ ` B ) e. CC ) -> ( ( (♯ ` A ) + (♯ ` B ) ) - (♯ ` A ) ) = (♯ ` B ) )
32	31	oveq1d 6022	. . . . . . . 8 |- ( ( (♯ ` A ) e. CC /\ (♯ ` B ) e. CC ) -> ( ( ( (♯ ` A ) + (♯ ` B ) ) - (♯ ` A ) ) - 1 ) = ( (♯ ` B ) - 1 ) )
33	30, 32	eqtrd 2262	. . . . . . 7 |- ( ( (♯ ` A ) e. CC /\ (♯ ` B ) e. CC ) -> ( ( ( (♯ ` A ) + (♯ ` B ) ) - 1 ) - (♯ ` A ) ) = ( (♯ ` B ) - 1 ) )
34	24, 26, 33	syl2an 289	. . . . . 6 |- ( ( A e. Word V /\ B e. Word V ) -> ( ( ( (♯ ` A ) + (♯ ` B ) ) - 1 ) - (♯ ` A ) ) = ( (♯ ` B ) - 1 ) )
35	23, 34	eqtrd 2262	. . . . 5 |- ( ( A e. Word V /\ B e. Word V ) -> ( ( (♯ ` ( A ++ B ) ) - 1 ) - (♯ ` A ) ) = ( (♯ ` B ) - 1 ) )
36	35	3adant3 1041	. . . 4 |- ( ( A e. Word V /\ B e. Word V /\ B =/= (/) ) -> ( ( (♯ ` ( A ++ B ) ) - 1 ) - (♯ ` A ) ) = ( (♯ ` B ) - 1 ) )
37	36	fveq2d 5633	. . 3 |- ( ( A e. Word V /\ B e. Word V /\ B =/= (/) ) -> ( B ` ( ( (♯ ` ( A ++ B ) ) - 1 ) - (♯ ` A ) ) ) = ( B ` ( (♯ ` B ) - 1 ) ) )
38	22, 37	eqtrd 2262	. 2 |- ( ( A e. Word V /\ B e. Word V /\ B =/= (/) ) -> ( ( A ++ B ) ` ( (♯ ` ( A ++ B ) ) - 1 ) ) = ( B ` ( (♯ ` B ) - 1 ) ) )
39		ccatcl 11141	. . . 4 |- ( ( A e. Word V /\ B e. Word V ) -> ( A ++ B ) e. Word V )
40	39	3adant3 1041	. . 3 |- ( ( A e. Word V /\ B e. Word V /\ B =/= (/) ) -> ( A ++ B ) e. Word V )
41		lswwrd 11131	. . 3 |- ( ( A ++ B ) e. Word V -> (lastS ` ( A ++ B ) ) = ( ( A ++ B ) ` ( (♯ ` ( A ++ B ) ) - 1 ) ) )
42	40, 41	syl 14	. 2 |- ( ( A e. Word V /\ B e. Word V /\ B =/= (/) ) -> (lastS ` ( A ++ B ) ) = ( ( A ++ B ) ` ( (♯ ` ( A ++ B ) ) - 1 ) ) )
43		lswwrd 11131	. . 3 |- ( B e. Word V -> (lastS ` B ) = ( B ` ( (♯ ` B ) - 1 ) ) )
44	43	3ad2ant2 1043	. 2 |- ( ( A e. Word V /\ B e. Word V /\ B =/= (/) ) -> (lastS ` B ) = ( B ` ( (♯ ` B ) - 1 ) ) )
45	38, 42, 44	3eqtr4d 2272	1 |- ( ( A e. Word V /\ B e. Word V /\ B =/= (/) ) -> (lastS ` ( A ++ B ) ) = (lastS ` B ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1002   = wceq 1395   e. wcel 2200   =/= wne 2400   (/)c0 3491   class class class wbr 4083   ` cfv 5318  (class class class)co 6007   CCcc 8008   RRcr 8009   1c1 8011   + caddc 8013   < clt 8192   - cmin 8328   NNcn 9121   ZZcz 9457   RR+crp 9861  ..^cfzo 10350  ♯chash 11009  Word cword 11084  lastSclsw 11129   ++ cconcat 11138

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-addcom 8110  ax-addass 8112  ax-distr 8114  ax-i2m1 8115  ax-0lt1 8116  ax-0id 8118  ax-rnegex 8119  ax-cnre 8121  ax-pre-ltirr 8122  ax-pre-ltwlin 8123  ax-pre-lttrn 8124  ax-pre-apti 8125  ax-pre-ltadd 8126

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-frec 6543  df-1o 6568  df-er 6688  df-en 6896  df-dom 6897  df-fin 6898  df-pnf 8194  df-mnf 8195  df-xr 8196  df-ltxr 8197  df-le 8198  df-sub 8330  df-neg 8331  df-inn 9122  df-n0 9381  df-z 9458  df-uz 9734  df-rp 9862  df-fz 10217  df-fzo 10351  df-ihash 11010  df-word 11085  df-lsw 11130  df-concat 11139

This theorem is referenced by:  clwwlkccat  16138