Theorem cats1un 11239

Description: Express a word with an extra symbol as the union of the word and the new value. (Contributed by Mario Carneiro, 28-Feb-2016.)

Assertion

Ref	Expression
cats1un	|- ( ( A e. Word X /\ B e. X ) -> ( A ++ <" B "> ) = ( A u. { <. (♯ ` A ) , B >. } ) )

Proof of Theorem cats1un

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ccatws1cl 11151	. . . . 5 |- ( ( A e. Word X /\ B e. X ) -> ( A ++ <" B "> ) e. Word X )
2		wrdf 11064	. . . . 5 |- ( ( A ++ <" B "> ) e. Word X -> ( A ++ <" B "> ) : ( 0..^ (♯ ` ( A ++ <" B "> ) ) ) --> X )
3	1, 2	syl 14	. . . 4 |- ( ( A e. Word X /\ B e. X ) -> ( A ++ <" B "> ) : ( 0..^ (♯ ` ( A ++ <" B "> ) ) ) --> X )
4		ccatws1leng 11153	. . . . . . 7 |- ( ( A e. Word X /\ B e. X ) -> (♯ ` ( A ++ <" B "> ) ) = ( (♯ ` A ) + 1 ) )
5	4	oveq2d 6010	. . . . . 6 |- ( ( A e. Word X /\ B e. X ) -> ( 0..^ (♯ ` ( A ++ <" B "> ) ) ) = ( 0..^ ( (♯ ` A ) + 1 ) ) )
6		lencl 11062	. . . . . . . . 9 |- ( A e. Word X -> (♯ ` A ) e. NN0 )
7		nn0uz 9745	. . . . . . . . 9 |- NN0 = ( ZZ>= ` 0 )
8	6, 7	eleqtrdi 2322	. . . . . . . 8 |- ( A e. Word X -> (♯ ` A ) e. ( ZZ>= ` 0 ) )
9		fzosplitsn 10426	. . . . . . . 8 |- ( (♯ ` A ) e. ( ZZ>= ` 0 ) -> ( 0..^ ( (♯ ` A ) + 1 ) ) = ( ( 0..^ (♯ ` A ) ) u. { (♯ ` A ) } ) )
10	8, 9	syl 14	. . . . . . 7 |- ( A e. Word X -> ( 0..^ ( (♯ ` A ) + 1 ) ) = ( ( 0..^ (♯ ` A ) ) u. { (♯ ` A ) } ) )
11	10	adantr 276	. . . . . 6 |- ( ( A e. Word X /\ B e. X ) -> ( 0..^ ( (♯ ` A ) + 1 ) ) = ( ( 0..^ (♯ ` A ) ) u. { (♯ ` A ) } ) )
12	5, 11	eqtrd 2262	. . . . 5 |- ( ( A e. Word X /\ B e. X ) -> ( 0..^ (♯ ` ( A ++ <" B "> ) ) ) = ( ( 0..^ (♯ ` A ) ) u. { (♯ ` A ) } ) )
13	12	feq2d 5457	. . . 4 |- ( ( A e. Word X /\ B e. X ) -> ( ( A ++ <" B "> ) : ( 0..^ (♯ ` ( A ++ <" B "> ) ) ) --> X <-> ( A ++ <" B "> ) : ( ( 0..^ (♯ ` A ) ) u. { (♯ ` A ) } ) --> X ) )
14	3, 13	mpbid 147	. . 3 |- ( ( A e. Word X /\ B e. X ) -> ( A ++ <" B "> ) : ( ( 0..^ (♯ ` A ) ) u. { (♯ ` A ) } ) --> X )
15	14	ffnd 5470	. 2 |- ( ( A e. Word X /\ B e. X ) -> ( A ++ <" B "> ) Fn ( ( 0..^ (♯ ` A ) ) u. { (♯ ` A ) } ) )
16		wrdf 11064	. . . . 5 |- ( A e. Word X -> A : ( 0..^ (♯ ` A ) ) --> X )
17	16	adantr 276	. . . 4 |- ( ( A e. Word X /\ B e. X ) -> A : ( 0..^ (♯ ` A ) ) --> X )
18		eqid 2229	. . . . . 6 |- { <. (♯ ` A ) , B >. } = { <. (♯ ` A ) , B >. }
19		fsng 5801	. . . . . 6 |- ( ( (♯ ` A ) e. NN0 /\ B e. X ) -> ( { <. (♯ ` A ) , B >. } : { (♯ ` A ) } --> { B } <-> { <. (♯ ` A ) , B >. } = { <. (♯ ` A ) , B >. } ) )
20	18, 19	mpbiri 168	. . . . 5 |- ( ( (♯ ` A ) e. NN0 /\ B e. X ) -> { <. (♯ ` A ) , B >. } : { (♯ ` A ) } --> { B } )
21	6, 20	sylan 283	. . . 4 |- ( ( A e. Word X /\ B e. X ) -> { <. (♯ ` A ) , B >. } : { (♯ ` A ) } --> { B } )
22		fzodisjsn 10368	. . . . 5 |- ( ( 0..^ (♯ ` A ) ) i^i { (♯ ` A ) } ) = (/)
23	22	a1i 9	. . . 4 |- ( ( A e. Word X /\ B e. X ) -> ( ( 0..^ (♯ ` A ) ) i^i { (♯ ` A ) } ) = (/) )
24		fun 5493	. . . 4 |- ( ( ( A : ( 0..^ (♯ ` A ) ) --> X /\ { <. (♯ ` A ) , B >. } : { (♯ ` A ) } --> { B } ) /\ ( ( 0..^ (♯ ` A ) ) i^i { (♯ ` A ) } ) = (/) ) -> ( A u. { <. (♯ ` A ) , B >. } ) : ( ( 0..^ (♯ ` A ) ) u. { (♯ ` A ) } ) --> ( X u. { B } ) )
25	17, 21, 23, 24	syl21anc 1270	. . 3 |- ( ( A e. Word X /\ B e. X ) -> ( A u. { <. (♯ ` A ) , B >. } ) : ( ( 0..^ (♯ ` A ) ) u. { (♯ ` A ) } ) --> ( X u. { B } ) )
26	25	ffnd 5470	. 2 |- ( ( A e. Word X /\ B e. X ) -> ( A u. { <. (♯ ` A ) , B >. } ) Fn ( ( 0..^ (♯ ` A ) ) u. { (♯ ` A ) } ) )
27		elun 3345	. . 3 |- ( x e. ( ( 0..^ (♯ ` A ) ) u. { (♯ ` A ) } ) <-> ( x e. ( 0..^ (♯ ` A ) ) \/ x e. { (♯ ` A ) } ) )
28		ccats1val1g 11156	. . . . . 6 |- ( ( A e. Word X /\ B e. X /\ x e. ( 0..^ (♯ ` A ) ) ) -> ( ( A ++ <" B "> ) ` x ) = ( A ` x ) )
29	28	3expa 1227	. . . . 5 |- ( ( ( A e. Word X /\ B e. X ) /\ x e. ( 0..^ (♯ ` A ) ) ) -> ( ( A ++ <" B "> ) ` x ) = ( A ` x ) )
30		vex 2802	. . . . . 6 |- x e. _V
31		simpr 110	. . . . . . . 8 |- ( ( ( A e. Word X /\ B e. X ) /\ x e. ( 0..^ (♯ ` A ) ) ) -> x e. ( 0..^ (♯ ` A ) ) )
32		fzonel 10345	. . . . . . . 8 |- -. (♯ ` A ) e. ( 0..^ (♯ ` A ) )
33		nelne2 2491	. . . . . . . 8 |- ( ( x e. ( 0..^ (♯ ` A ) ) /\ -. (♯ ` A ) e. ( 0..^ (♯ ` A ) ) ) -> x =/= (♯ ` A ) )
34	31, 32, 33	sylancl 413	. . . . . . 7 |- ( ( ( A e. Word X /\ B e. X ) /\ x e. ( 0..^ (♯ ` A ) ) ) -> x =/= (♯ ` A ) )
35	34	necomd 2486	. . . . . 6 |- ( ( ( A e. Word X /\ B e. X ) /\ x e. ( 0..^ (♯ ` A ) ) ) -> (♯ ` A ) =/= x )
36		fvunsng 5826	. . . . . 6 |- ( ( x e. _V /\ (♯ ` A ) =/= x ) -> ( ( A u. { <. (♯ ` A ) , B >. } ) ` x ) = ( A ` x ) )
37	30, 35, 36	sylancr 414	. . . . 5 |- ( ( ( A e. Word X /\ B e. X ) /\ x e. ( 0..^ (♯ ` A ) ) ) -> ( ( A u. { <. (♯ ` A ) , B >. } ) ` x ) = ( A ` x ) )
38	29, 37	eqtr4d 2265	. . . 4 |- ( ( ( A e. Word X /\ B e. X ) /\ x e. ( 0..^ (♯ ` A ) ) ) -> ( ( A ++ <" B "> ) ` x ) = ( ( A u. { <. (♯ ` A ) , B >. } ) ` x ) )
39	6	elexd 2813	. . . . . . . . 9 |- ( A e. Word X -> (♯ ` A ) e. _V )
40	39	adantr 276	. . . . . . . 8 |- ( ( A e. Word X /\ B e. X ) -> (♯ ` A ) e. _V )
41		simpr 110	. . . . . . . 8 |- ( ( A e. Word X /\ B e. X ) -> B e. X )
42	17	fdmd 5476	. . . . . . . . . 10 |- ( ( A e. Word X /\ B e. X ) -> dom A = ( 0..^ (♯ ` A ) ) )
43	42	eleq2d 2299	. . . . . . . . 9 |- ( ( A e. Word X /\ B e. X ) -> ( (♯ ` A ) e. dom A <-> (♯ ` A ) e. ( 0..^ (♯ ` A ) ) ) )
44	32, 43	mtbiri 679	. . . . . . . 8 |- ( ( A e. Word X /\ B e. X ) -> -. (♯ ` A ) e. dom A )
45		fsnunfv 5833	. . . . . . . 8 |- ( ( (♯ ` A ) e. _V /\ B e. X /\ -. (♯ ` A ) e. dom A ) -> ( ( A u. { <. (♯ ` A ) , B >. } ) ` (♯ ` A ) ) = B )
46	40, 41, 44, 45	syl3anc 1271	. . . . . . 7 |- ( ( A e. Word X /\ B e. X ) -> ( ( A u. { <. (♯ ` A ) , B >. } ) ` (♯ ` A ) ) = B )
47		simpl 109	. . . . . . . . 9 |- ( ( A e. Word X /\ B e. X ) -> A e. Word X )
48		s1cl 11140	. . . . . . . . . 10 |- ( B e. X -> <" B "> e. Word X )
49	48	adantl 277	. . . . . . . . 9 |- ( ( A e. Word X /\ B e. X ) -> <" B "> e. Word X )
50		s1leng 11143	. . . . . . . . . . . 12 |- ( B e. X -> (♯ ` <" B "> ) = 1 )
51		1nn 9109	. . . . . . . . . . . 12 |- 1 e. NN
52	50, 51	eqeltrdi 2320	. . . . . . . . . . 11 |- ( B e. X -> (♯ ` <" B "> ) e. NN )
53		lbfzo0 10369	. . . . . . . . . . 11 |- ( 0 e. ( 0..^ (♯ ` <" B "> ) ) <-> (♯ ` <" B "> ) e. NN )
54	52, 53	sylibr 134	. . . . . . . . . 10 |- ( B e. X -> 0 e. ( 0..^ (♯ ` <" B "> ) ) )
55	54	adantl 277	. . . . . . . . 9 |- ( ( A e. Word X /\ B e. X ) -> 0 e. ( 0..^ (♯ ` <" B "> ) ) )
56		ccatval3 11120	. . . . . . . . 9 |- ( ( A e. Word X /\ <" B "> e. Word X /\ 0 e. ( 0..^ (♯ ` <" B "> ) ) ) -> ( ( A ++ <" B "> ) ` ( 0 + (♯ ` A ) ) ) = ( <" B "> ` 0 ) )
57	47, 49, 55, 56	syl3anc 1271	. . . . . . . 8 |- ( ( A e. Word X /\ B e. X ) -> ( ( A ++ <" B "> ) ` ( 0 + (♯ ` A ) ) ) = ( <" B "> ` 0 ) )
58		s1fv 11145	. . . . . . . . 9 |- ( B e. X -> ( <" B "> ` 0 ) = B )
59	58	adantl 277	. . . . . . . 8 |- ( ( A e. Word X /\ B e. X ) -> ( <" B "> ` 0 ) = B )
60	57, 59	eqtrd 2262	. . . . . . 7 |- ( ( A e. Word X /\ B e. X ) -> ( ( A ++ <" B "> ) ` ( 0 + (♯ ` A ) ) ) = B )
61	6	adantr 276	. . . . . . . . . 10 |- ( ( A e. Word X /\ B e. X ) -> (♯ ` A ) e. NN0 )
62	61	nn0cnd 9412	. . . . . . . . 9 |- ( ( A e. Word X /\ B e. X ) -> (♯ ` A ) e. CC )
63	62	addlidd 8284	. . . . . . . 8 |- ( ( A e. Word X /\ B e. X ) -> ( 0 + (♯ ` A ) ) = (♯ ` A ) )
64	63	fveq2d 5627	. . . . . . 7 |- ( ( A e. Word X /\ B e. X ) -> ( ( A ++ <" B "> ) ` ( 0 + (♯ ` A ) ) ) = ( ( A ++ <" B "> ) ` (♯ ` A ) ) )
65	46, 60, 64	3eqtr2rd 2269	. . . . . 6 |- ( ( A e. Word X /\ B e. X ) -> ( ( A ++ <" B "> ) ` (♯ ` A ) ) = ( ( A u. { <. (♯ ` A ) , B >. } ) ` (♯ ` A ) ) )
66		elsni 3684	. . . . . . . 8 |- ( x e. { (♯ ` A ) } -> x = (♯ ` A ) )
67	66	fveq2d 5627	. . . . . . 7 |- ( x e. { (♯ ` A ) } -> ( ( A ++ <" B "> ) ` x ) = ( ( A ++ <" B "> ) ` (♯ ` A ) ) )
68	66	fveq2d 5627	. . . . . . 7 |- ( x e. { (♯ ` A ) } -> ( ( A u. { <. (♯ ` A ) , B >. } ) ` x ) = ( ( A u. { <. (♯ ` A ) , B >. } ) ` (♯ ` A ) ) )
69	67, 68	eqeq12d 2244	. . . . . 6 |- ( x e. { (♯ ` A ) } -> ( ( ( A ++ <" B "> ) ` x ) = ( ( A u. { <. (♯ ` A ) , B >. } ) ` x ) <-> ( ( A ++ <" B "> ) ` (♯ ` A ) ) = ( ( A u. { <. (♯ ` A ) , B >. } ) ` (♯ ` A ) ) ) )
70	65, 69	syl5ibrcom 157	. . . . 5 |- ( ( A e. Word X /\ B e. X ) -> ( x e. { (♯ ` A ) } -> ( ( A ++ <" B "> ) ` x ) = ( ( A u. { <. (♯ ` A ) , B >. } ) ` x ) ) )
71	70	imp 124	. . . 4 |- ( ( ( A e. Word X /\ B e. X ) /\ x e. { (♯ ` A ) } ) -> ( ( A ++ <" B "> ) ` x ) = ( ( A u. { <. (♯ ` A ) , B >. } ) ` x ) )
72	38, 71	jaodan 802	. . 3 |- ( ( ( A e. Word X /\ B e. X ) /\ ( x e. ( 0..^ (♯ ` A ) ) \/ x e. { (♯ ` A ) } ) ) -> ( ( A ++ <" B "> ) ` x ) = ( ( A u. { <. (♯ ` A ) , B >. } ) ` x ) )
73	27, 72	sylan2b 287	. 2 |- ( ( ( A e. Word X /\ B e. X ) /\ x e. ( ( 0..^ (♯ ` A ) ) u. { (♯ ` A ) } ) ) -> ( ( A ++ <" B "> ) ` x ) = ( ( A u. { <. (♯ ` A ) , B >. } ) ` x ) )
74	15, 26, 73	eqfnfvd 5728	1 |- ( ( A e. Word X /\ B e. X ) -> ( A ++ <" B "> ) = ( A u. { <. (♯ ` A ) , B >. } ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 713   = wceq 1395   e. wcel 2200   =/= wne 2400   _Vcvv 2799   u. cun 3195   i^i cin 3196   (/)c0 3491   {csn 3666   <.cop 3669   dom cdm 4716   -->wf 5310   ` cfv 5314  (class class class)co 5994   0cc0 7987   1c1 7988   + caddc 7990   NNcn 9098   NN0cn0 9357   ZZ>=cuz 9710  ..^cfzo 10326  ♯chash 10984  Word cword 11058   ++ cconcat 11111   <"cs1 11134

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 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4521  ax-setind 4626  ax-iinf 4677  ax-cnex 8078  ax-resscn 8079  ax-1cn 8080  ax-1re 8081  ax-icn 8082  ax-addcl 8083  ax-addrcl 8084  ax-mulcl 8085  ax-addcom 8087  ax-addass 8089  ax-distr 8091  ax-i2m1 8092  ax-0lt1 8093  ax-0id 8095  ax-rnegex 8096  ax-cnre 8098  ax-pre-ltirr 8099  ax-pre-ltwlin 8100  ax-pre-lttrn 8101  ax-pre-apti 8102  ax-pre-ltadd 8103

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 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4381  df-iord 4454  df-on 4456  df-ilim 4457  df-suc 4459  df-iom 4680  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729  df-iota 5274  df-fun 5316  df-fn 5317  df-f 5318  df-f1 5319  df-fo 5320  df-f1o 5321  df-fv 5322  df-riota 5947  df-ov 5997  df-oprab 5998  df-mpo 5999  df-1st 6276  df-2nd 6277  df-recs 6441  df-frec 6527  df-1o 6552  df-er 6670  df-en 6878  df-dom 6879  df-fin 6880  df-pnf 8171  df-mnf 8172  df-xr 8173  df-ltxr 8174  df-le 8175  df-sub 8307  df-neg 8308  df-inn 9099  df-n0 9358  df-z 9435  df-uz 9711  df-fz 10193  df-fzo 10327  df-ihash 10985  df-word 11059  df-concat 11112  df-s1 11135

This theorem is referenced by: (None)