Theorem cats1un 11306

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 11213	. . . . 5 |- ( ( A e. Word X /\ B e. X ) -> ( A ++ <" B "> ) e. Word X )
2		wrdf 11123	. . . . 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 11215	. . . . . . 7 |- ( ( A e. Word X /\ B e. X ) -> (♯ ` ( A ++ <" B "> ) ) = ( (♯ ` A ) + 1 ) )
5	4	oveq2d 6034	. . . . . 6 |- ( ( A e. Word X /\ B e. X ) -> ( 0..^ (♯ ` ( A ++ <" B "> ) ) ) = ( 0..^ ( (♯ ` A ) + 1 ) ) )
6		lencl 11121	. . . . . . . . 9 |- ( A e. Word X -> (♯ ` A ) e. NN0 )
7		nn0uz 9791	. . . . . . . . 9 |- NN0 = ( ZZ>= ` 0 )
8	6, 7	eleqtrdi 2324	. . . . . . . 8 |- ( A e. Word X -> (♯ ` A ) e. ( ZZ>= ` 0 ) )
9		fzosplitsn 10479	. . . . . . . 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 2264	. . . . 5 |- ( ( A e. Word X /\ B e. X ) -> ( 0..^ (♯ ` ( A ++ <" B "> ) ) ) = ( ( 0..^ (♯ ` A ) ) u. { (♯ ` A ) } ) )
13	12	feq2d 5470	. . . 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 5483	. 2 |- ( ( A e. Word X /\ B e. X ) -> ( A ++ <" B "> ) Fn ( ( 0..^ (♯ ` A ) ) u. { (♯ ` A ) } ) )
16		wrdf 11123	. . . . 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 2231	. . . . . 6 |- { <. (♯ ` A ) , B >. } = { <. (♯ ` A ) , B >. }
19		fsng 5820	. . . . . 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 10419	. . . . 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 5508	. . . 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 1272	. . 3 |- ( ( A e. Word X /\ B e. X ) -> ( A u. { <. (♯ ` A ) , B >. } ) : ( ( 0..^ (♯ ` A ) ) u. { (♯ ` A ) } ) --> ( X u. { B } ) )
26	25	ffnd 5483	. 2 |- ( ( A e. Word X /\ B e. X ) -> ( A u. { <. (♯ ` A ) , B >. } ) Fn ( ( 0..^ (♯ ` A ) ) u. { (♯ ` A ) } ) )
27		elun 3348	. . 3 |- ( x e. ( ( 0..^ (♯ ` A ) ) u. { (♯ ` A ) } ) <-> ( x e. ( 0..^ (♯ ` A ) ) \/ x e. { (♯ ` A ) } ) )
28		ccats1val1g 11220	. . . . . 6 |- ( ( A e. Word X /\ B e. X /\ x e. ( 0..^ (♯ ` A ) ) ) -> ( ( A ++ <" B "> ) ` x ) = ( A ` x ) )
29	28	3expa 1229	. . . . 5 |- ( ( ( A e. Word X /\ B e. X ) /\ x e. ( 0..^ (♯ ` A ) ) ) -> ( ( A ++ <" B "> ) ` x ) = ( A ` x ) )
30		vex 2805	. . . . . 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 10396	. . . . . . . 8 |- -. (♯ ` A ) e. ( 0..^ (♯ ` A ) )
33		nelne2 2493	. . . . . . . 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 2488	. . . . . 6 |- ( ( ( A e. Word X /\ B e. X ) /\ x e. ( 0..^ (♯ ` A ) ) ) -> (♯ ` A ) =/= x )
36		fvunsng 5848	. . . . . 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 2267	. . . 4 |- ( ( ( A e. Word X /\ B e. X ) /\ x e. ( 0..^ (♯ ` A ) ) ) -> ( ( A ++ <" B "> ) ` x ) = ( ( A u. { <. (♯ ` A ) , B >. } ) ` x ) )
39	6	elexd 2816	. . . . . . . . 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 5489	. . . . . . . . . 10 |- ( ( A e. Word X /\ B e. X ) -> dom A = ( 0..^ (♯ ` A ) ) )
43	42	eleq2d 2301	. . . . . . . . 9 |- ( ( A e. Word X /\ B e. X ) -> ( (♯ ` A ) e. dom A <-> (♯ ` A ) e. ( 0..^ (♯ ` A ) ) ) )
44	32, 43	mtbiri 681	. . . . . . . 8 |- ( ( A e. Word X /\ B e. X ) -> -. (♯ ` A ) e. dom A )
45		fsnunfv 5855	. . . . . . . 8 |- ( ( (♯ ` A ) e. _V /\ B e. X /\ -. (♯ ` A ) e. dom A ) -> ( ( A u. { <. (♯ ` A ) , B >. } ) ` (♯ ` A ) ) = B )
46	40, 41, 44, 45	syl3anc 1273	. . . . . . 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 11202	. . . . . . . . . 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 11205	. . . . . . . . . . . 12 |- ( B e. X -> (♯ ` <" B "> ) = 1 )
51		1nn 9154	. . . . . . . . . . . 12 |- 1 e. NN
52	50, 51	eqeltrdi 2322	. . . . . . . . . . 11 |- ( B e. X -> (♯ ` <" B "> ) e. NN )
53		lbfzo0 10420	. . . . . . . . . . 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 11180	. . . . . . . . 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 1273	. . . . . . . 8 |- ( ( A e. Word X /\ B e. X ) -> ( ( A ++ <" B "> ) ` ( 0 + (♯ ` A ) ) ) = ( <" B "> ` 0 ) )
58		s1fv 11207	. . . . . . . . 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 2264	. . . . . . 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 9457	. . . . . . . . 9 |- ( ( A e. Word X /\ B e. X ) -> (♯ ` A ) e. CC )
63	62	addlidd 8329	. . . . . . . 8 |- ( ( A e. Word X /\ B e. X ) -> ( 0 + (♯ ` A ) ) = (♯ ` A ) )
64	63	fveq2d 5643	. . . . . . 7 |- ( ( A e. Word X /\ B e. X ) -> ( ( A ++ <" B "> ) ` ( 0 + (♯ ` A ) ) ) = ( ( A ++ <" B "> ) ` (♯ ` A ) ) )
65	46, 60, 64	3eqtr2rd 2271	. . . . . 6 |- ( ( A e. Word X /\ B e. X ) -> ( ( A ++ <" B "> ) ` (♯ ` A ) ) = ( ( A u. { <. (♯ ` A ) , B >. } ) ` (♯ ` A ) ) )
66		elsni 3687	. . . . . . . 8 |- ( x e. { (♯ ` A ) } -> x = (♯ ` A ) )
67	66	fveq2d 5643	. . . . . . 7 |- ( x e. { (♯ ` A ) } -> ( ( A ++ <" B "> ) ` x ) = ( ( A ++ <" B "> ) ` (♯ ` A ) ) )
68	66	fveq2d 5643	. . . . . . 7 |- ( x e. { (♯ ` A ) } -> ( ( A u. { <. (♯ ` A ) , B >. } ) ` x ) = ( ( A u. { <. (♯ ` A ) , B >. } ) ` (♯ ` A ) ) )
69	67, 68	eqeq12d 2246	. . . . . 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 804	. . 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 5747	1 |- ( ( A e. Word X /\ B e. X ) -> ( A ++ <" B "> ) = ( A u. { <. (♯ ` A ) , B >. } ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 715   = wceq 1397   e. wcel 2202   =/= wne 2402   _Vcvv 2802   u. cun 3198   i^i cin 3199   (/)c0 3494   {csn 3669   <.cop 3672   dom cdm 4725   -->wf 5322   ` cfv 5326  (class class class)co 6018   0cc0 8032   1c1 8033   + caddc 8035   NNcn 9143   NN0cn0 9402   ZZ>=cuz 9755  ..^cfzo 10377  ♯chash 11038  Word cword 11117   ++ cconcat 11171   <"cs1 11196

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-addass 8134  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-0id 8140  ax-rnegex 8141  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-frec 6557  df-1o 6582  df-er 6702  df-en 6910  df-dom 6911  df-fin 6912  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-inn 9144  df-n0 9403  df-z 9480  df-uz 9756  df-fz 10244  df-fzo 10378  df-ihash 11039  df-word 11118  df-concat 11172  df-s1 11197

This theorem is referenced by:  vdegp1aid  16171  vdegp1bid  16172