Theorem lswccatn0lsw 11299

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 11283	. . . . . . 7 |- ( ( A e. Word V /\ B e. Word V ) -> (♯ ` ( A ++ B ) ) = ( (♯ ` A ) + (♯ ` B ) ) )
2	1	oveq1d 6065	. . . . . 6 |- ( ( A e. Word V /\ B e. Word V ) -> ( (♯ ` ( A ++ B ) ) - 1 ) = ( ( (♯ ` A ) + (♯ ` B ) ) - 1 ) )
3	2	3adant3 1044	. . . . 5 |- ( ( A e. Word V /\ B e. Word V /\ B =/= (/) ) -> ( (♯ ` ( A ++ B ) ) - 1 ) = ( ( (♯ ` A ) + (♯ ` B ) ) - 1 ) )
4		lencl 11228	. . . . . . . . . 10 |- ( A e. Word V -> (♯ ` A ) e. NN0 )
5	4	nn0zd 9698	. . . . . . . . 9 |- ( A e. Word V -> (♯ ` A ) e. ZZ )
6		lennncl 11244	. . . . . . . . 9 |- ( ( B e. Word V /\ B =/= (/) ) -> (♯ ` B ) e. NN )
7		simpl 109	. . . . . . . . . 10 |- ( ( (♯ ` A ) e. ZZ /\ (♯ ` B ) e. NN ) -> (♯ ` A ) e. ZZ )
8		zaddcllempos 9614	. . . . . . . . . 10 |- ( ( (♯ ` A ) e. ZZ /\ (♯ ` B ) e. NN ) -> ( (♯ ` A ) + (♯ ` B ) ) e. ZZ )
9		zre 9581	. . . . . . . . . . 11 |- ( (♯ ` A ) e. ZZ -> (♯ ` A ) e. RR )
10		nnrp 9996	. . . . . . . . . . 11 |- ( (♯ ` B ) e. NN -> (♯ ` B ) e. RR+ )
11		ltaddrp 10024	. . . . . . . . . . 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 1204	. . . . . . . . 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 1226	. . . . . . 7 |- ( ( A e. Word V /\ B e. Word V /\ B =/= (/) ) -> ( (♯ ` A ) e. ZZ /\ ( (♯ ` A ) + (♯ ` B ) ) e. ZZ /\ (♯ ` A ) < ( (♯ ` A ) + (♯ ` B ) ) ) )
16		fzolb 10488	. . . . . . 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 10567	. . . . . 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 2309	. . . 4 |- ( ( A e. Word V /\ B e. Word V /\ B =/= (/) ) -> ( (♯ ` ( A ++ B ) ) - 1 ) e. ( (♯ ` A )..^ ( (♯ ` A ) + (♯ ` B ) ) ) )
21		ccatval2 11286	. . . 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 1319	. . 3 |- ( ( A e. Word V /\ B e. Word V /\ B =/= (/) ) -> ( ( A ++ B ) ` ( (♯ ` ( A ++ B ) ) - 1 ) ) = ( B ` ( ( (♯ ` ( A ++ B ) ) - 1 ) - (♯ ` A ) ) ) )
23	2	oveq1d 6065	. . . . . 6 |- ( ( A e. Word V /\ B e. Word V ) -> ( ( (♯ ` ( A ++ B ) ) - 1 ) - (♯ ` A ) ) = ( ( ( (♯ ` A ) + (♯ ` B ) ) - 1 ) - (♯ ` A ) ) )
24	4	nn0cnd 9555	. . . . . . 7 |- ( A e. Word V -> (♯ ` A ) e. CC )
25		lencl 11228	. . . . . . . 8 |- ( B e. Word V -> (♯ ` B ) e. NN0 )
26	25	nn0cnd 9555	. . . . . . 7 |- ( B e. Word V -> (♯ ` B ) e. CC )
27		addcl 8252	. . . . . . . . 9 |- ( ( (♯ ` A ) e. CC /\ (♯ ` B ) e. CC ) -> ( (♯ ` A ) + (♯ ` B ) ) e. CC )
28		1cnd 8290	. . . . . . . . 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 8616	. . . . . . . 8 |- ( ( (♯ ` A ) e. CC /\ (♯ ` B ) e. CC ) -> ( ( ( (♯ ` A ) + (♯ ` B ) ) - 1 ) - (♯ ` A ) ) = ( ( ( (♯ ` A ) + (♯ ` B ) ) - (♯ ` A ) ) - 1 ) )
31		pncan2 8480	. . . . . . . . 9 |- ( ( (♯ ` A ) e. CC /\ (♯ ` B ) e. CC ) -> ( ( (♯ ` A ) + (♯ ` B ) ) - (♯ ` A ) ) = (♯ ` B ) )
32	31	oveq1d 6065	. . . . . . . 8 |- ( ( (♯ ` A ) e. CC /\ (♯ ` B ) e. CC ) -> ( ( ( (♯ ` A ) + (♯ ` B ) ) - (♯ ` A ) ) - 1 ) = ( (♯ ` B ) - 1 ) )
33	30, 32	eqtrd 2265	. . . . . . 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 2265	. . . . 5 |- ( ( A e. Word V /\ B e. Word V ) -> ( ( (♯ ` ( A ++ B ) ) - 1 ) - (♯ ` A ) ) = ( (♯ ` B ) - 1 ) )
36	35	3adant3 1044	. . . 4 |- ( ( A e. Word V /\ B e. Word V /\ B =/= (/) ) -> ( ( (♯ ` ( A ++ B ) ) - 1 ) - (♯ ` A ) ) = ( (♯ ` B ) - 1 ) )
37	36	fveq2d 5674	. . 3 |- ( ( A e. Word V /\ B e. Word V /\ B =/= (/) ) -> ( B ` ( ( (♯ ` ( A ++ B ) ) - 1 ) - (♯ ` A ) ) ) = ( B ` ( (♯ ` B ) - 1 ) ) )
38	22, 37	eqtrd 2265	. 2 |- ( ( A e. Word V /\ B e. Word V /\ B =/= (/) ) -> ( ( A ++ B ) ` ( (♯ ` ( A ++ B ) ) - 1 ) ) = ( B ` ( (♯ ` B ) - 1 ) ) )
39		ccatcl 11281	. . . 4 |- ( ( A e. Word V /\ B e. Word V ) -> ( A ++ B ) e. Word V )
40	39	3adant3 1044	. . 3 |- ( ( A e. Word V /\ B e. Word V /\ B =/= (/) ) -> ( A ++ B ) e. Word V )
41		lswwrd 11271	. . 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 11271	. . 3 |- ( B e. Word V -> (lastS ` B ) = ( B ` ( (♯ ` B ) - 1 ) ) )
44	43	3ad2ant2 1046	. 2 |- ( ( A e. Word V /\ B e. Word V /\ B =/= (/) ) -> (lastS ` B ) = ( B ` ( (♯ ` B ) - 1 ) ) )
45	38, 42, 44	3eqtr4d 2275	1 |- ( ( A e. Word V /\ B e. Word V /\ B =/= (/) ) -> (lastS ` ( A ++ B ) ) = (lastS ` B ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   = wceq 1398   e. wcel 2203   =/= wne 2412   (/)c0 3508   class class class wbr 4109   ` cfv 5352  (class class class)co 6050   CCcc 8125   RRcr 8126   1c1 8128   + caddc 8130   < clt 8308   - cmin 8444   NNcn 9237   ZZcz 9577   RR+crp 9986  ..^cfzo 10476  ♯chash 11138  Word cword 11224  lastSclsw 11269   ++ cconcat 11278

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-0id 8235  ax-rnegex 8236  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-iord 4487  df-on 4489  df-ilim 4490  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-frec 6622  df-1o 6647  df-er 6767  df-en 6976  df-dom 6977  df-fin 6978  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-inn 9238  df-n0 9497  df-z 9578  df-uz 9854  df-rp 9987  df-fz 10343  df-fzo 10477  df-ihash 11139  df-word 11225  df-lsw 11270  df-concat 11279

This theorem is referenced by:  clwwlkccat  16396