Theorem cats1un 13475

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 13396	. . . . 5 |- ( ( A e. Word X /\ B e. X ) -> ( A ++ <" B "> ) e. Word X )
2		wrdf 13310	. . . . 5 |- ( ( A ++ <" B "> ) e. Word X -> ( A ++ <" B "> ) : ( 0..^ ( # ` ( A ++ <" B "> ) ) ) --> X )
3	1, 2	syl 17	. . . 4 |- ( ( A e. Word X /\ B e. X ) -> ( A ++ <" B "> ) : ( 0..^ ( # ` ( A ++ <" B "> ) ) ) --> X )
4		ccatws1len 13398	. . . . . . 7 |- ( ( A e. Word X /\ B e. X ) -> ( # ` ( A ++ <" B "> ) ) = ( ( # ` A ) + 1 ) )
5	4	oveq2d 6666	. . . . . 6 |- ( ( A e. Word X /\ B e. X ) -> ( 0..^ ( # ` ( A ++ <" B "> ) ) ) = ( 0..^ ( ( # ` A ) + 1 ) ) )
6		lencl 13324	. . . . . . . . 9 |- ( A e. Word X -> ( # ` A ) e. NN0 )
7	6	adantr 481	. . . . . . . 8 |- ( ( A e. Word X /\ B e. X ) -> ( # ` A ) e. NN0 )
8		nn0uz 11722	. . . . . . . 8 |- NN0 = ( ZZ>= ` 0 )
9	7, 8	syl6eleq 2711	. . . . . . 7 |- ( ( A e. Word X /\ B e. X ) -> ( # ` A ) e. ( ZZ>= ` 0 ) )
10		fzosplitsn 12576	. . . . . . 7 |- ( ( # ` A ) e. ( ZZ>= ` 0 ) -> ( 0..^ ( ( # ` A ) + 1 ) ) = ( ( 0..^ ( # ` A ) ) u. { ( # ` A ) } ) )
11	9, 10	syl 17	. . . . . 6 |- ( ( A e. Word X /\ B e. X ) -> ( 0..^ ( ( # ` A ) + 1 ) ) = ( ( 0..^ ( # ` A ) ) u. { ( # ` A ) } ) )
12	5, 11	eqtrd 2656	. . . . 5 |- ( ( A e. Word X /\ B e. X ) -> ( 0..^ ( # ` ( A ++ <" B "> ) ) ) = ( ( 0..^ ( # ` A ) ) u. { ( # ` A ) } ) )
13	12	feq2d 6031	. . . 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 222	. . 3 |- ( ( A e. Word X /\ B e. X ) -> ( A ++ <" B "> ) : ( ( 0..^ ( # ` A ) ) u. { ( # ` A ) } ) --> X )
15		ffn 6045	. . 3 |- ( ( A ++ <" B "> ) : ( ( 0..^ ( # ` A ) ) u. { ( # ` A ) } ) --> X -> ( A ++ <" B "> ) Fn ( ( 0..^ ( # ` A ) ) u. { ( # ` A ) } ) )
16	14, 15	syl 17	. 2 |- ( ( A e. Word X /\ B e. X ) -> ( A ++ <" B "> ) Fn ( ( 0..^ ( # ` A ) ) u. { ( # ` A ) } ) )
17		wrdf 13310	. . . . 5 |- ( A e. Word X -> A : ( 0..^ ( # ` A ) ) --> X )
18	17	adantr 481	. . . 4 |- ( ( A e. Word X /\ B e. X ) -> A : ( 0..^ ( # ` A ) ) --> X )
19		eqid 2622	. . . . . 6 |- { <. ( # ` A ) , B >. } = { <. ( # ` A ) , B >. }
20		fsng 6404	. . . . . 6 |- ( ( ( # ` A ) e. NN0 /\ B e. X ) -> ( { <. ( # ` A ) , B >. } : { ( # ` A ) } --> { B } <-> { <. ( # ` A ) , B >. } = { <. ( # ` A ) , B >. } ) )
21	19, 20	mpbiri 248	. . . . 5 |- ( ( ( # ` A ) e. NN0 /\ B e. X ) -> { <. ( # ` A ) , B >. } : { ( # ` A ) } --> { B } )
22	6, 21	sylan 488	. . . 4 |- ( ( A e. Word X /\ B e. X ) -> { <. ( # ` A ) , B >. } : { ( # ` A ) } --> { B } )
23		fzonel 12483	. . . . . 6 |- -. ( # ` A ) e. ( 0..^ ( # ` A ) )
24	23	a1i 11	. . . . 5 |- ( ( A e. Word X /\ B e. X ) -> -. ( # ` A ) e. ( 0..^ ( # ` A ) ) )
25		disjsn 4246	. . . . 5 |- ( ( ( 0..^ ( # ` A ) ) i^i { ( # ` A ) } ) = (/) <-> -. ( # ` A ) e. ( 0..^ ( # ` A ) ) )
26	24, 25	sylibr 224	. . . 4 |- ( ( A e. Word X /\ B e. X ) -> ( ( 0..^ ( # ` A ) ) i^i { ( # ` A ) } ) = (/) )
27		fun 6066	. . . 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 } ) )
28	18, 22, 26, 27	syl21anc 1325	. . 3 |- ( ( A e. Word X /\ B e. X ) -> ( A u. { <. ( # ` A ) , B >. } ) : ( ( 0..^ ( # ` A ) ) u. { ( # ` A ) } ) --> ( X u. { B } ) )
29		ffn 6045	. . 3 |- ( ( A u. { <. ( # ` A ) , B >. } ) : ( ( 0..^ ( # ` A ) ) u. { ( # ` A ) } ) --> ( X u. { B } ) -> ( A u. { <. ( # ` A ) , B >. } ) Fn ( ( 0..^ ( # ` A ) ) u. { ( # ` A ) } ) )
30	28, 29	syl 17	. 2 |- ( ( A e. Word X /\ B e. X ) -> ( A u. { <. ( # ` A ) , B >. } ) Fn ( ( 0..^ ( # ` A ) ) u. { ( # ` A ) } ) )
31		elun 3753	. . 3 |- ( x e. ( ( 0..^ ( # ` A ) ) u. { ( # ` A ) } ) <-> ( x e. ( 0..^ ( # ` A ) ) \/ x e. { ( # ` A ) } ) )
32		ccats1val1 13403	. . . . . 6 |- ( ( A e. Word X /\ B e. X /\ x e. ( 0..^ ( # ` A ) ) ) -> ( ( A ++ <" B "> ) ` x ) = ( A ` x ) )
33	32	3expa 1265	. . . . 5 |- ( ( ( A e. Word X /\ B e. X ) /\ x e. ( 0..^ ( # ` A ) ) ) -> ( ( A ++ <" B "> ) ` x ) = ( A ` x ) )
34		simpr 477	. . . . . . . 8 |- ( ( ( A e. Word X /\ B e. X ) /\ x e. ( 0..^ ( # ` A ) ) ) -> x e. ( 0..^ ( # ` A ) ) )
35		nelne2 2891	. . . . . . . 8 |- ( ( x e. ( 0..^ ( # ` A ) ) /\ -. ( # ` A ) e. ( 0..^ ( # ` A ) ) ) -> x =/= ( # ` A ) )
36	34, 23, 35	sylancl 694	. . . . . . 7 |- ( ( ( A e. Word X /\ B e. X ) /\ x e. ( 0..^ ( # ` A ) ) ) -> x =/= ( # ` A ) )
37	36	necomd 2849	. . . . . 6 |- ( ( ( A e. Word X /\ B e. X ) /\ x e. ( 0..^ ( # ` A ) ) ) -> ( # ` A ) =/= x )
38		fvunsn 6445	. . . . . 6 |- ( ( # ` A ) =/= x -> ( ( A u. { <. ( # ` A ) , B >. } ) ` x ) = ( A ` x ) )
39	37, 38	syl 17	. . . . 5 |- ( ( ( A e. Word X /\ B e. X ) /\ x e. ( 0..^ ( # ` A ) ) ) -> ( ( A u. { <. ( # ` A ) , B >. } ) ` x ) = ( A ` x ) )
40	33, 39	eqtr4d 2659	. . . 4 |- ( ( ( A e. Word X /\ B e. X ) /\ x e. ( 0..^ ( # ` A ) ) ) -> ( ( A ++ <" B "> ) ` x ) = ( ( A u. { <. ( # ` A ) , B >. } ) ` x ) )
41		fvexd 6203	. . . . . . . 8 |- ( ( A e. Word X /\ B e. X ) -> ( # ` A ) e. _V )
42		elex 3212	. . . . . . . . 9 |- ( B e. X -> B e. _V )
43	42	adantl 482	. . . . . . . 8 |- ( ( A e. Word X /\ B e. X ) -> B e. _V )
44		fdm 6051	. . . . . . . . . . 11 |- ( A : ( 0..^ ( # ` A ) ) --> X -> dom A = ( 0..^ ( # ` A ) ) )
45	18, 44	syl 17	. . . . . . . . . 10 |- ( ( A e. Word X /\ B e. X ) -> dom A = ( 0..^ ( # ` A ) ) )
46	45	eleq2d 2687	. . . . . . . . 9 |- ( ( A e. Word X /\ B e. X ) -> ( ( # ` A ) e. dom A <-> ( # ` A ) e. ( 0..^ ( # ` A ) ) ) )
47	23, 46	mtbiri 317	. . . . . . . 8 |- ( ( A e. Word X /\ B e. X ) -> -. ( # ` A ) e. dom A )
48		fsnunfv 6453	. . . . . . . 8 |- ( ( ( # ` A ) e. _V /\ B e. _V /\ -. ( # ` A ) e. dom A ) -> ( ( A u. { <. ( # ` A ) , B >. } ) ` ( # ` A ) ) = B )
49	41, 43, 47, 48	syl3anc 1326	. . . . . . 7 |- ( ( A e. Word X /\ B e. X ) -> ( ( A u. { <. ( # ` A ) , B >. } ) ` ( # ` A ) ) = B )
50		simpl 473	. . . . . . . . 9 |- ( ( A e. Word X /\ B e. X ) -> A e. Word X )
51		s1cl 13382	. . . . . . . . . 10 |- ( B e. X -> <" B "> e. Word X )
52	51	adantl 482	. . . . . . . . 9 |- ( ( A e. Word X /\ B e. X ) -> <" B "> e. Word X )
53		s1len 13385	. . . . . . . . . . . 12 |- ( # ` <" B "> ) = 1
54		1nn 11031	. . . . . . . . . . . 12 |- 1 e. NN
55	53, 54	eqeltri 2697	. . . . . . . . . . 11 |- ( # ` <" B "> ) e. NN
56	55	a1i 11	. . . . . . . . . 10 |- ( ( A e. Word X /\ B e. X ) -> ( # ` <" B "> ) e. NN )
57		lbfzo0 12507	. . . . . . . . . 10 |- ( 0 e. ( 0..^ ( # ` <" B "> ) ) <-> ( # ` <" B "> ) e. NN )
58	56, 57	sylibr 224	. . . . . . . . 9 |- ( ( A e. Word X /\ B e. X ) -> 0 e. ( 0..^ ( # ` <" B "> ) ) )
59		ccatval3 13363	. . . . . . . . 9 |- ( ( A e. Word X /\ <" B "> e. Word X /\ 0 e. ( 0..^ ( # ` <" B "> ) ) ) -> ( ( A ++ <" B "> ) ` ( 0 + ( # ` A ) ) ) = ( <" B "> ` 0 ) )
60	50, 52, 58, 59	syl3anc 1326	. . . . . . . 8 |- ( ( A e. Word X /\ B e. X ) -> ( ( A ++ <" B "> ) ` ( 0 + ( # ` A ) ) ) = ( <" B "> ` 0 ) )
61		s1fv 13390	. . . . . . . . 9 |- ( B e. X -> ( <" B "> ` 0 ) = B )
62	61	adantl 482	. . . . . . . 8 |- ( ( A e. Word X /\ B e. X ) -> ( <" B "> ` 0 ) = B )
63	60, 62	eqtrd 2656	. . . . . . 7 |- ( ( A e. Word X /\ B e. X ) -> ( ( A ++ <" B "> ) ` ( 0 + ( # ` A ) ) ) = B )
64	7	nn0cnd 11353	. . . . . . . . 9 |- ( ( A e. Word X /\ B e. X ) -> ( # ` A ) e. CC )
65	64	addid2d 10237	. . . . . . . 8 |- ( ( A e. Word X /\ B e. X ) -> ( 0 + ( # ` A ) ) = ( # ` A ) )
66	65	fveq2d 6195	. . . . . . 7 |- ( ( A e. Word X /\ B e. X ) -> ( ( A ++ <" B "> ) ` ( 0 + ( # ` A ) ) ) = ( ( A ++ <" B "> ) ` ( # ` A ) ) )
67	49, 63, 66	3eqtr2rd 2663	. . . . . 6 |- ( ( A e. Word X /\ B e. X ) -> ( ( A ++ <" B "> ) ` ( # ` A ) ) = ( ( A u. { <. ( # ` A ) , B >. } ) ` ( # ` A ) ) )
68		elsni 4194	. . . . . . . 8 |- ( x e. { ( # ` A ) } -> x = ( # ` A ) )
69	68	fveq2d 6195	. . . . . . 7 |- ( x e. { ( # ` A ) } -> ( ( A ++ <" B "> ) ` x ) = ( ( A ++ <" B "> ) ` ( # ` A ) ) )
70	68	fveq2d 6195	. . . . . . 7 |- ( x e. { ( # ` A ) } -> ( ( A u. { <. ( # ` A ) , B >. } ) ` x ) = ( ( A u. { <. ( # ` A ) , B >. } ) ` ( # ` A ) ) )
71	69, 70	eqeq12d 2637	. . . . . 6 |- ( x e. { ( # ` A ) } -> ( ( ( A ++ <" B "> ) ` x ) = ( ( A u. { <. ( # ` A ) , B >. } ) ` x ) <-> ( ( A ++ <" B "> ) ` ( # ` A ) ) = ( ( A u. { <. ( # ` A ) , B >. } ) ` ( # ` A ) ) ) )
72	67, 71	syl5ibrcom 237	. . . . 5 |- ( ( A e. Word X /\ B e. X ) -> ( x e. { ( # ` A ) } -> ( ( A ++ <" B "> ) ` x ) = ( ( A u. { <. ( # ` A ) , B >. } ) ` x ) ) )
73	72	imp 445	. . . 4 |- ( ( ( A e. Word X /\ B e. X ) /\ x e. { ( # ` A ) } ) -> ( ( A ++ <" B "> ) ` x ) = ( ( A u. { <. ( # ` A ) , B >. } ) ` x ) )
74	40, 73	jaodan 826	. . 3 |- ( ( ( A e. Word X /\ B e. X ) /\ ( x e. ( 0..^ ( # ` A ) ) \/ x e. { ( # ` A ) } ) ) -> ( ( A ++ <" B "> ) ` x ) = ( ( A u. { <. ( # ` A ) , B >. } ) ` x ) )
75	31, 74	sylan2b 492	. 2 |- ( ( ( A e. Word X /\ B e. X ) /\ x e. ( ( 0..^ ( # ` A ) ) u. { ( # ` A ) } ) ) -> ( ( A ++ <" B "> ) ` x ) = ( ( A u. { <. ( # ` A ) , B >. } ) ` x ) )
76	16, 30, 75	eqfnfvd 6314	1 |- ( ( A e. Word X /\ B e. X ) -> ( A ++ <" B "> ) = ( A u. { <. ( # ` A ) , B >. } ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   u. cun 3572   i^i cin 3573   (/)c0 3915   {csn 4177   <.cop 4183   dom cdm 5114   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   + caddc 9939   NNcn 11020   NN0cn0 11292   ZZ>=cuz 11687  ..^cfzo 12465   #chash 13117  Word cword 13291   ++ cconcat 13293   <"cs1 13294

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-concat 13301  df-s1 13302

This theorem is referenced by:  s2prop  13652  s3tpop  13654  s4prop  13655  pgpfaclem1  18480  vdegp1ai  26432  vdegp1bi  26433  wwlksnext  26788