Theorem cats1un 11438

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 11345	. . . . 5 |- ( ( A e. Word X /\ B e. X ) -> ( A ++ <" B "> ) e. Word X )
2		wrdf 11255	. . . . 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 11347	. . . . . . 7 |- ( ( A e. Word X /\ B e. X ) -> (♯ ` ( A ++ <" B "> ) ) = ( (♯ ` A ) + 1 ) )
5	4	oveq2d 6074	. . . . . 6 |- ( ( A e. Word X /\ B e. X ) -> ( 0..^ (♯ ` ( A ++ <" B "> ) ) ) = ( 0..^ ( (♯ ` A ) + 1 ) ) )
6		lencl 11253	. . . . . . . . 9 |- ( A e. Word X -> (♯ ` A ) e. NN0 )
7		nn0uz 9907	. . . . . . . . 9 |- NN0 = ( ZZ>= ` 0 )
8	6, 7	eleqtrdi 2327	. . . . . . . 8 |- ( A e. Word X -> (♯ ` A ) e. ( ZZ>= ` 0 ) )
9		fzosplitsn 10600	. . . . . . . 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 2267	. . . . 5 |- ( ( A e. Word X /\ B e. X ) -> ( 0..^ (♯ ` ( A ++ <" B "> ) ) ) = ( ( 0..^ (♯ ` A ) ) u. { (♯ ` A ) } ) )
13	12	feq2d 5501	. . . 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 5514	. 2 |- ( ( A e. Word X /\ B e. X ) -> ( A ++ <" B "> ) Fn ( ( 0..^ (♯ ` A ) ) u. { (♯ ` A ) } ) )
16		wrdf 11255	. . . . 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 2234	. . . . . 6 |- { <. (♯ ` A ) , B >. } = { <. (♯ ` A ) , B >. }
19		fsng 5855	. . . . . 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 10540	. . . . 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 5541	. . . 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 1273	. . 3 |- ( ( A e. Word X /\ B e. X ) -> ( A u. { <. (♯ ` A ) , B >. } ) : ( ( 0..^ (♯ ` A ) ) u. { (♯ ` A ) } ) --> ( X u. { B } ) )
26	25	ffnd 5514	. 2 |- ( ( A e. Word X /\ B e. X ) -> ( A u. { <. (♯ ` A ) , B >. } ) Fn ( ( 0..^ (♯ ` A ) ) u. { (♯ ` A ) } ) )
27		elun 3364	. . 3 |- ( x e. ( ( 0..^ (♯ ` A ) ) u. { (♯ ` A ) } ) <-> ( x e. ( 0..^ (♯ ` A ) ) \/ x e. { (♯ ` A ) } ) )
28		ccats1val1g 11352	. . . . . 6 |- ( ( A e. Word X /\ B e. X /\ x e. ( 0..^ (♯ ` A ) ) ) -> ( ( A ++ <" B "> ) ` x ) = ( A ` x ) )
29	28	3expa 1230	. . . . 5 |- ( ( ( A e. Word X /\ B e. X ) /\ x e. ( 0..^ (♯ ` A ) ) ) -> ( ( A ++ <" B "> ) ` x ) = ( A ` x ) )
30		vex 2818	. . . . . 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 10517	. . . . . . . 8 |- -. (♯ ` A ) e. ( 0..^ (♯ ` A ) )
33		nelne2 2505	. . . . . . . 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 2500	. . . . . 6 |- ( ( ( A e. Word X /\ B e. X ) /\ x e. ( 0..^ (♯ ` A ) ) ) -> (♯ ` A ) =/= x )
36		fvunsng 5883	. . . . . 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 2270	. . . 4 |- ( ( ( A e. Word X /\ B e. X ) /\ x e. ( 0..^ (♯ ` A ) ) ) -> ( ( A ++ <" B "> ) ` x ) = ( ( A u. { <. (♯ ` A ) , B >. } ) ` x ) )
39	6	elexd 2829	. . . . . . . . 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 5520	. . . . . . . . . 10 |- ( ( A e. Word X /\ B e. X ) -> dom A = ( 0..^ (♯ ` A ) ) )
43	42	eleq2d 2304	. . . . . . . . 9 |- ( ( A e. Word X /\ B e. X ) -> ( (♯ ` A ) e. dom A <-> (♯ ` A ) e. ( 0..^ (♯ ` A ) ) ) )
44	32, 43	mtbiri 682	. . . . . . . 8 |- ( ( A e. Word X /\ B e. X ) -> -. (♯ ` A ) e. dom A )
45		fsnunfv 5890	. . . . . . . 8 |- ( ( (♯ ` A ) e. _V /\ B e. X /\ -. (♯ ` A ) e. dom A ) -> ( ( A u. { <. (♯ ` A ) , B >. } ) ` (♯ ` A ) ) = B )
46	40, 41, 44, 45	syl3anc 1274	. . . . . . 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 11334	. . . . . . . . . 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 11337	. . . . . . . . . . . 12 |- ( B e. X -> (♯ ` <" B "> ) = 1 )
51		1nn 9265	. . . . . . . . . . . 12 |- 1 e. NN
52	50, 51	eqeltrdi 2325	. . . . . . . . . . 11 |- ( B e. X -> (♯ ` <" B "> ) e. NN )
53		lbfzo0 10541	. . . . . . . . . . 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 11312	. . . . . . . . 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 1274	. . . . . . . 8 |- ( ( A e. Word X /\ B e. X ) -> ( ( A ++ <" B "> ) ` ( 0 + (♯ ` A ) ) ) = ( <" B "> ` 0 ) )
58		s1fv 11339	. . . . . . . . 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 2267	. . . . . . 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 9572	. . . . . . . . 9 |- ( ( A e. Word X /\ B e. X ) -> (♯ ` A ) e. CC )
63	62	addlidd 8439	. . . . . . . 8 |- ( ( A e. Word X /\ B e. X ) -> ( 0 + (♯ ` A ) ) = (♯ ` A ) )
64	63	fveq2d 5679	. . . . . . 7 |- ( ( A e. Word X /\ B e. X ) -> ( ( A ++ <" B "> ) ` ( 0 + (♯ ` A ) ) ) = ( ( A ++ <" B "> ) ` (♯ ` A ) ) )
65	46, 60, 64	3eqtr2rd 2274	. . . . . 6 |- ( ( A e. Word X /\ B e. X ) -> ( ( A ++ <" B "> ) ` (♯ ` A ) ) = ( ( A u. { <. (♯ ` A ) , B >. } ) ` (♯ ` A ) ) )
66		elsni 3712	. . . . . . . 8 |- ( x e. { (♯ ` A ) } -> x = (♯ ` A ) )
67	66	fveq2d 5679	. . . . . . 7 |- ( x e. { (♯ ` A ) } -> ( ( A ++ <" B "> ) ` x ) = ( ( A ++ <" B "> ) ` (♯ ` A ) ) )
68	66	fveq2d 5679	. . . . . . 7 |- ( x e. { (♯ ` A ) } -> ( ( A u. { <. (♯ ` A ) , B >. } ) ` x ) = ( ( A u. { <. (♯ ` A ) , B >. } ) ` (♯ ` A ) ) )
69	67, 68	eqeq12d 2249	. . . . . 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 805	. . 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 5783	1 |- ( ( A e. Word X /\ B e. X ) -> ( A ++ <" B "> ) = ( A u. { <. (♯ ` A ) , B >. } ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 716   = wceq 1398   e. wcel 2205   =/= wne 2414   _Vcvv 2815   u. cun 3212   i^i cin 3213   (/)c0 3512   {csn 3694   <.cop 3697   dom cdm 4754   -->wf 5353   ` cfv 5357  (class class class)co 6058   0cc0 8143   1c1 8144   + caddc 8146   NNcn 9254   NN0cn0 9513   ZZ>=cuz 9871  ..^cfzo 10498  ♯chash 11163  Word cword 11249   ++ cconcat 11303   <"cs1 11328

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-fz 10362  df-fzo 10499  df-ihash 11164  df-word 11250  df-concat 11304  df-s1 11329

This theorem is referenced by:  vdegp1aid  16421  vdegp1bid  16422