Theorem lswccatn0lsw 11324

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 11308	. . . . . . 7 |- ( ( A e. Word V /\ B e. Word V ) -> (♯ ` ( A ++ B ) ) = ( (♯ ` A ) + (♯ ` B ) ) )
2	1	oveq1d 6073	. . . . . 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 11253	. . . . . . . . . 10 |- ( A e. Word V -> (♯ ` A ) e. NN0 )
5	4	nn0zd 9716	. . . . . . . . 9 |- ( A e. Word V -> (♯ ` A ) e. ZZ )
6		lennncl 11269	. . . . . . . . 9 |- ( ( B e. Word V /\ B =/= (/) ) -> (♯ ` B ) e. NN )
7		simpl 109	. . . . . . . . . 10 |- ( ( (♯ ` A ) e. ZZ /\ (♯ ` B ) e. NN ) -> (♯ ` A ) e. ZZ )
8		zaddcllempos 9631	. . . . . . . . . 10 |- ( ( (♯ ` A ) e. ZZ /\ (♯ ` B ) e. NN ) -> ( (♯ ` A ) + (♯ ` B ) ) e. ZZ )
9		zre 9598	. . . . . . . . . . 11 |- ( (♯ ` A ) e. ZZ -> (♯ ` A ) e. RR )
10		nnrp 10014	. . . . . . . . . . 11 |- ( (♯ ` B ) e. NN -> (♯ ` B ) e. RR+ )
11		ltaddrp 10042	. . . . . . . . . . 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 10510	. . . . . . 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 10589	. . . . . 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 2311	. . . 4 |- ( ( A e. Word V /\ B e. Word V /\ B =/= (/) ) -> ( (♯ ` ( A ++ B ) ) - 1 ) e. ( (♯ ` A )..^ ( (♯ ` A ) + (♯ ` B ) ) ) )
21		ccatval2 11311	. . . 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 6073	. . . . . 6 |- ( ( A e. Word V /\ B e. Word V ) -> ( ( (♯ ` ( A ++ B ) ) - 1 ) - (♯ ` A ) ) = ( ( ( (♯ ` A ) + (♯ ` B ) ) - 1 ) - (♯ ` A ) ) )
24	4	nn0cnd 9572	. . . . . . 7 |- ( A e. Word V -> (♯ ` A ) e. CC )
25		lencl 11253	. . . . . . . 8 |- ( B e. Word V -> (♯ ` B ) e. NN0 )
26	25	nn0cnd 9572	. . . . . . 7 |- ( B e. Word V -> (♯ ` B ) e. CC )
27		addcl 8268	. . . . . . . . 9 |- ( ( (♯ ` A ) e. CC /\ (♯ ` B ) e. CC ) -> ( (♯ ` A ) + (♯ ` B ) ) e. CC )
28		1cnd 8306	. . . . . . . . 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 8632	. . . . . . . 8 |- ( ( (♯ ` A ) e. CC /\ (♯ ` B ) e. CC ) -> ( ( ( (♯ ` A ) + (♯ ` B ) ) - 1 ) - (♯ ` A ) ) = ( ( ( (♯ ` A ) + (♯ ` B ) ) - (♯ ` A ) ) - 1 ) )
31		pncan2 8496	. . . . . . . . 9 |- ( ( (♯ ` A ) e. CC /\ (♯ ` B ) e. CC ) -> ( ( (♯ ` A ) + (♯ ` B ) ) - (♯ ` A ) ) = (♯ ` B ) )
32	31	oveq1d 6073	. . . . . . . 8 |- ( ( (♯ ` A ) e. CC /\ (♯ ` B ) e. CC ) -> ( ( ( (♯ ` A ) + (♯ ` B ) ) - (♯ ` A ) ) - 1 ) = ( (♯ ` B ) - 1 ) )
33	30, 32	eqtrd 2267	. . . . . . 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 2267	. . . . 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 5679	. . 3 |- ( ( A e. Word V /\ B e. Word V /\ B =/= (/) ) -> ( B ` ( ( (♯ ` ( A ++ B ) ) - 1 ) - (♯ ` A ) ) ) = ( B ` ( (♯ ` B ) - 1 ) ) )
38	22, 37	eqtrd 2267	. 2 |- ( ( A e. Word V /\ B e. Word V /\ B =/= (/) ) -> ( ( A ++ B ) ` ( (♯ ` ( A ++ B ) ) - 1 ) ) = ( B ` ( (♯ ` B ) - 1 ) ) )
39		ccatcl 11306	. . . 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 11296	. . 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 11296	. . 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 2277	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 2205   =/= wne 2414   (/)c0 3512   class class class wbr 4114   ` cfv 5357  (class class class)co 6058   CCcc 8141   RRcr 8142   1c1 8144   + caddc 8146   < clt 8324   - cmin 8460   NNcn 9254   ZZcz 9594   RR+crp 10004  ..^cfzo 10498  ♯chash 11163  Word cword 11249  lastSclsw 11294   ++ cconcat 11303

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259

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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-1o 6660  df-er 6780  df-en 6989  df-dom 6990  df-fin 6991  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-inn 9255  df-n0 9514  df-z 9595  df-uz 9872  df-rp 10005  df-fz 10362  df-fzo 10499  df-ihash 11164  df-word 11250  df-lsw 11295  df-concat 11304

This theorem is referenced by:  clwwlkccat  16522