Theorem cats1un 11476

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 11383	. . . . 5 |- ( ( A e. Word X /\ B e. X ) -> ( A ++ <" B "> ) e. Word X )
2		wrdf 11293	. . . . 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 11385	. . . . . . 7 |- ( ( A e. Word X /\ B e. X ) -> (♯ ` ( A ++ <" B "> ) ) = ( (♯ ` A ) + 1 ) )
5	4	oveq2d 6095	. . . . . 6 |- ( ( A e. Word X /\ B e. X ) -> ( 0..^ (♯ ` ( A ++ <" B "> ) ) ) = ( 0..^ ( (♯ ` A ) + 1 ) ) )
6		lencl 11291	. . . . . . . . 9 |- ( A e. Word X -> (♯ ` A ) e. NN0 )
7		nn0uz 9940	. . . . . . . . 9 |- NN0 = ( ZZ>= ` 0 )
8	6, 7	eleqtrdi 2331	. . . . . . . 8 |- ( A e. Word X -> (♯ ` A ) e. ( ZZ>= ` 0 ) )
9		fzosplitsn 10634	. . . . . . . 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 2271	. . . . 5 |- ( ( A e. Word X /\ B e. X ) -> ( 0..^ (♯ ` ( A ++ <" B "> ) ) ) = ( ( 0..^ (♯ ` A ) ) u. { (♯ ` A ) } ) )
13	12	feq2d 5519	. . . 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 5532	. 2 |- ( ( A e. Word X /\ B e. X ) -> ( A ++ <" B "> ) Fn ( ( 0..^ (♯ ` A ) ) u. { (♯ ` A ) } ) )
16		wrdf 11293	. . . . 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 2238	. . . . . 6 |- { <. (♯ ` A ) , B >. } = { <. (♯ ` A ) , B >. }
19		fsng 5875	. . . . . 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 10574	. . . . 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 5559	. . . 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 1277	. . 3 |- ( ( A e. Word X /\ B e. X ) -> ( A u. { <. (♯ ` A ) , B >. } ) : ( ( 0..^ (♯ ` A ) ) u. { (♯ ` A ) } ) --> ( X u. { B } ) )
26	25	ffnd 5532	. 2 |- ( ( A e. Word X /\ B e. X ) -> ( A u. { <. (♯ ` A ) , B >. } ) Fn ( ( 0..^ (♯ ` A ) ) u. { (♯ ` A ) } ) )
27		elun 3370	. . 3 |- ( x e. ( ( 0..^ (♯ ` A ) ) u. { (♯ ` A ) } ) <-> ( x e. ( 0..^ (♯ ` A ) ) \/ x e. { (♯ ` A ) } ) )
28		ccats1val1g 11390	. . . . . 6 |- ( ( A e. Word X /\ B e. X /\ x e. ( 0..^ (♯ ` A ) ) ) -> ( ( A ++ <" B "> ) ` x ) = ( A ` x ) )
29	28	3expa 1234	. . . . 5 |- ( ( ( A e. Word X /\ B e. X ) /\ x e. ( 0..^ (♯ ` A ) ) ) -> ( ( A ++ <" B "> ) ` x ) = ( A ` x ) )
30		vex 2824	. . . . . 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 10551	. . . . . . . 8 |- -. (♯ ` A ) e. ( 0..^ (♯ ` A ) )
33		nelne2 2511	. . . . . . . 8 |- ( ( x e. ( 0..^ (♯ ` A ) ) /\ -. (♯ ` A ) e. ( 0..^ (♯ ` A ) ) ) -> x =/= (♯ ` A ) )
34	31, 32, 33	sylancl 417	. . . . . . 7 |- ( ( ( A e. Word X /\ B e. X ) /\ x e. ( 0..^ (♯ ` A ) ) ) -> x =/= (♯ ` A ) )
35	34	necomd 2506	. . . . . 6 |- ( ( ( A e. Word X /\ B e. X ) /\ x e. ( 0..^ (♯ ` A ) ) ) -> (♯ ` A ) =/= x )
36		fvunsng 5903	. . . . . 6 |- ( ( x e. _V /\ (♯ ` A ) =/= x ) -> ( ( A u. { <. (♯ ` A ) , B >. } ) ` x ) = ( A ` x ) )
37	30, 35, 36	sylancr 418	. . . . 5 |- ( ( ( A e. Word X /\ B e. X ) /\ x e. ( 0..^ (♯ ` A ) ) ) -> ( ( A u. { <. (♯ ` A ) , B >. } ) ` x ) = ( A ` x ) )
38	29, 37	eqtr4d 2274	. . . 4 |- ( ( ( A e. Word X /\ B e. X ) /\ x e. ( 0..^ (♯ ` A ) ) ) -> ( ( A ++ <" B "> ) ` x ) = ( ( A u. { <. (♯ ` A ) , B >. } ) ` x ) )
39	6	elexd 2835	. . . . . . . . 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 5538	. . . . . . . . . 10 |- ( ( A e. Word X /\ B e. X ) -> dom A = ( 0..^ (♯ ` A ) ) )
43	42	eleq2d 2308	. . . . . . . . 9 |- ( ( A e. Word X /\ B e. X ) -> ( (♯ ` A ) e. dom A <-> (♯ ` A ) e. ( 0..^ (♯ ` A ) ) ) )
44	32, 43	mtbiri 686	. . . . . . . 8 |- ( ( A e. Word X /\ B e. X ) -> -. (♯ ` A ) e. dom A )
45		fsnunfv 5910	. . . . . . . 8 |- ( ( (♯ ` A ) e. _V /\ B e. X /\ -. (♯ ` A ) e. dom A ) -> ( ( A u. { <. (♯ ` A ) , B >. } ) ` (♯ ` A ) ) = B )
46	40, 41, 44, 45	syl3anc 1278	. . . . . . 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 11372	. . . . . . . . . 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 11375	. . . . . . . . . . . 12 |- ( B e. X -> (♯ ` <" B "> ) = 1 )
51		1nn 9298	. . . . . . . . . . . 12 |- 1 e. NN
52	50, 51	eqeltrdi 2329	. . . . . . . . . . 11 |- ( B e. X -> (♯ ` <" B "> ) e. NN )
53		lbfzo0 10575	. . . . . . . . . . 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 11350	. . . . . . . . 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 1278	. . . . . . . 8 |- ( ( A e. Word X /\ B e. X ) -> ( ( A ++ <" B "> ) ` ( 0 + (♯ ` A ) ) ) = ( <" B "> ` 0 ) )
58		s1fv 11377	. . . . . . . . 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 2271	. . . . . . 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 9605	. . . . . . . . 9 |- ( ( A e. Word X /\ B e. X ) -> (♯ ` A ) e. CC )
63	62	addlidd 8470	. . . . . . . 8 |- ( ( A e. Word X /\ B e. X ) -> ( 0 + (♯ ` A ) ) = (♯ ` A ) )
64	63	fveq2d 5697	. . . . . . 7 |- ( ( A e. Word X /\ B e. X ) -> ( ( A ++ <" B "> ) ` ( 0 + (♯ ` A ) ) ) = ( ( A ++ <" B "> ) ` (♯ ` A ) ) )
65	46, 60, 64	3eqtr2rd 2278	. . . . . 6 |- ( ( A e. Word X /\ B e. X ) -> ( ( A ++ <" B "> ) ` (♯ ` A ) ) = ( ( A u. { <. (♯ ` A ) , B >. } ) ` (♯ ` A ) ) )
66		elsni 3726	. . . . . . . 8 |- ( x e. { (♯ ` A ) } -> x = (♯ ` A ) )
67	66	fveq2d 5697	. . . . . . 7 |- ( x e. { (♯ ` A ) } -> ( ( A ++ <" B "> ) ` x ) = ( ( A ++ <" B "> ) ` (♯ ` A ) ) )
68	66	fveq2d 5697	. . . . . . 7 |- ( x e. { (♯ ` A ) } -> ( ( A u. { <. (♯ ` A ) , B >. } ) ` x ) = ( ( A u. { <. (♯ ` A ) , B >. } ) ` (♯ ` A ) ) )
69	67, 68	eqeq12d 2253	. . . . . 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 809	. . 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 5803	1 |- ( ( A e. Word X /\ B e. X ) -> ( A ++ <" B "> ) = ( A u. { <. (♯ ` A ) , B >. } ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 720   = wceq 1402   e. wcel 2209   =/= wne 2420   _Vcvv 2821   u. cun 3218   i^i cin 3219   (/)c0 3520   {csn 3708   <.cop 3711   dom cdm 4772   -->wf 5371   ` cfv 5375  (class class class)co 6079   0cc0 8173   1c1 8174   + caddc 8176   NNcn 9287   NN0cn0 9546   ZZ>=cuz 9904  ..^cfzo 10532  ♯chash 11197  Word cword 11287   ++ cconcat 11341   <"cs1 11366

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-addcom 8273  ax-addass 8275  ax-distr 8277  ax-i2m1 8278  ax-0lt1 8279  ax-0id 8281  ax-rnegex 8282  ax-cnre 8284  ax-pre-ltirr 8285  ax-pre-ltwlin 8286  ax-pre-lttrn 8287  ax-pre-apti 8288  ax-pre-ltadd 8289

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-iord 4509  df-on 4511  df-ilim 4512  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-recs 6570  df-frec 6656  df-1o 6681  df-er 6801  df-en 7017  df-dom 7018  df-fin 7019  df-pnf 8356  df-mnf 8357  df-xr 8358  df-ltxr 8359  df-le 8360  df-sub 8493  df-neg 8494  df-inn 9288  df-n0 9547  df-z 9628  df-uz 9905  df-fz 10395  df-fzo 10533  df-ihash 11198  df-word 11288  df-concat 11342  df-s1 11367

This theorem is referenced by:  vdegp1aid  16538  vdegp1bid  16539