Theorem ccatopth2 11405

Description: An opth 4352-like theorem for recovering the two halves of a concatenated word. (Contributed by Mario Carneiro, 1-Oct-2015.)

Assertion

Ref	Expression
ccatopth2	|- ( ( ( A e. Word X /\ B e. Word X ) /\ ( C e. Word X /\ D e. Word X ) /\ (♯ ` B ) = (♯ ` D ) ) -> ( ( A ++ B ) = ( C ++ D ) <-> ( A = C /\ B = D ) ) )

Proof of Theorem ccatopth2

Step	Hyp	Ref	Expression
1		fveq2 5669	. . . 4 |- ( ( A ++ B ) = ( C ++ D ) -> (♯ ` ( A ++ B ) ) = (♯ ` ( C ++ D ) ) )
2		ccatlen 11279	. . . . . . . 8 |- ( ( A e. Word X /\ B e. Word X ) -> (♯ ` ( A ++ B ) ) = ( (♯ ` A ) + (♯ ` B ) ) )
3	2	3ad2ant1 1045	. . . . . . 7 |- ( ( ( A e. Word X /\ B e. Word X ) /\ ( C e. Word X /\ D e. Word X ) /\ (♯ ` B ) = (♯ ` D ) ) -> (♯ ` ( A ++ B ) ) = ( (♯ ` A ) + (♯ ` B ) ) )
4		simp3 1026	. . . . . . . 8 |- ( ( ( A e. Word X /\ B e. Word X ) /\ ( C e. Word X /\ D e. Word X ) /\ (♯ ` B ) = (♯ ` D ) ) -> (♯ ` B ) = (♯ ` D ) )
5	4	oveq2d 6065	. . . . . . 7 |- ( ( ( A e. Word X /\ B e. Word X ) /\ ( C e. Word X /\ D e. Word X ) /\ (♯ ` B ) = (♯ ` D ) ) -> ( (♯ ` A ) + (♯ ` B ) ) = ( (♯ ` A ) + (♯ ` D ) ) )
6	3, 5	eqtrd 2265	. . . . . 6 |- ( ( ( A e. Word X /\ B e. Word X ) /\ ( C e. Word X /\ D e. Word X ) /\ (♯ ` B ) = (♯ ` D ) ) -> (♯ ` ( A ++ B ) ) = ( (♯ ` A ) + (♯ ` D ) ) )
7		ccatlen 11279	. . . . . . 7 |- ( ( C e. Word X /\ D e. Word X ) -> (♯ ` ( C ++ D ) ) = ( (♯ ` C ) + (♯ ` D ) ) )
8	7	3ad2ant2 1046	. . . . . 6 |- ( ( ( A e. Word X /\ B e. Word X ) /\ ( C e. Word X /\ D e. Word X ) /\ (♯ ` B ) = (♯ ` D ) ) -> (♯ ` ( C ++ D ) ) = ( (♯ ` C ) + (♯ ` D ) ) )
9	6, 8	eqeq12d 2247	. . . . 5 |- ( ( ( A e. Word X /\ B e. Word X ) /\ ( C e. Word X /\ D e. Word X ) /\ (♯ ` B ) = (♯ ` D ) ) -> ( (♯ ` ( A ++ B ) ) = (♯ ` ( C ++ D ) ) <-> ( (♯ ` A ) + (♯ ` D ) ) = ( (♯ ` C ) + (♯ ` D ) ) ) )
10		simp1l 1048	. . . . . . . 8 |- ( ( ( A e. Word X /\ B e. Word X ) /\ ( C e. Word X /\ D e. Word X ) /\ (♯ ` B ) = (♯ ` D ) ) -> A e. Word X )
11		lencl 11224	. . . . . . . 8 |- ( A e. Word X -> (♯ ` A ) e. NN0 )
12	10, 11	syl 14	. . . . . . 7 |- ( ( ( A e. Word X /\ B e. Word X ) /\ ( C e. Word X /\ D e. Word X ) /\ (♯ ` B ) = (♯ ` D ) ) -> (♯ ` A ) e. NN0 )
13	12	nn0cnd 9554	. . . . . 6 |- ( ( ( A e. Word X /\ B e. Word X ) /\ ( C e. Word X /\ D e. Word X ) /\ (♯ ` B ) = (♯ ` D ) ) -> (♯ ` A ) e. CC )
14		simp2l 1050	. . . . . . . 8 |- ( ( ( A e. Word X /\ B e. Word X ) /\ ( C e. Word X /\ D e. Word X ) /\ (♯ ` B ) = (♯ ` D ) ) -> C e. Word X )
15		lencl 11224	. . . . . . . 8 |- ( C e. Word X -> (♯ ` C ) e. NN0 )
16	14, 15	syl 14	. . . . . . 7 |- ( ( ( A e. Word X /\ B e. Word X ) /\ ( C e. Word X /\ D e. Word X ) /\ (♯ ` B ) = (♯ ` D ) ) -> (♯ ` C ) e. NN0 )
17	16	nn0cnd 9554	. . . . . 6 |- ( ( ( A e. Word X /\ B e. Word X ) /\ ( C e. Word X /\ D e. Word X ) /\ (♯ ` B ) = (♯ ` D ) ) -> (♯ ` C ) e. CC )
18		simp2r 1051	. . . . . . . 8 |- ( ( ( A e. Word X /\ B e. Word X ) /\ ( C e. Word X /\ D e. Word X ) /\ (♯ ` B ) = (♯ ` D ) ) -> D e. Word X )
19		lencl 11224	. . . . . . . 8 |- ( D e. Word X -> (♯ ` D ) e. NN0 )
20	18, 19	syl 14	. . . . . . 7 |- ( ( ( A e. Word X /\ B e. Word X ) /\ ( C e. Word X /\ D e. Word X ) /\ (♯ ` B ) = (♯ ` D ) ) -> (♯ ` D ) e. NN0 )
21	20	nn0cnd 9554	. . . . . 6 |- ( ( ( A e. Word X /\ B e. Word X ) /\ ( C e. Word X /\ D e. Word X ) /\ (♯ ` B ) = (♯ ` D ) ) -> (♯ ` D ) e. CC )
22	13, 17, 21	addcan2d 8457	. . . . 5 |- ( ( ( A e. Word X /\ B e. Word X ) /\ ( C e. Word X /\ D e. Word X ) /\ (♯ ` B ) = (♯ ` D ) ) -> ( ( (♯ ` A ) + (♯ ` D ) ) = ( (♯ ` C ) + (♯ ` D ) ) <-> (♯ ` A ) = (♯ ` C ) ) )
23	9, 22	bitrd 188	. . . 4 |- ( ( ( A e. Word X /\ B e. Word X ) /\ ( C e. Word X /\ D e. Word X ) /\ (♯ ` B ) = (♯ ` D ) ) -> ( (♯ ` ( A ++ B ) ) = (♯ ` ( C ++ D ) ) <-> (♯ ` A ) = (♯ ` C ) ) )
24	1, 23	imbitrid 154	. . 3 |- ( ( ( A e. Word X /\ B e. Word X ) /\ ( C e. Word X /\ D e. Word X ) /\ (♯ ` B ) = (♯ ` D ) ) -> ( ( A ++ B ) = ( C ++ D ) -> (♯ ` A ) = (♯ ` C ) ) )
25		ccatopth 11404	. . . . . . 7 |- ( ( ( A e. Word X /\ B e. Word X ) /\ ( C e. Word X /\ D e. Word X ) /\ (♯ ` A ) = (♯ ` C ) ) -> ( ( A ++ B ) = ( C ++ D ) <-> ( A = C /\ B = D ) ) )
26	25	biimpd 144	. . . . . 6 |- ( ( ( A e. Word X /\ B e. Word X ) /\ ( C e. Word X /\ D e. Word X ) /\ (♯ ` A ) = (♯ ` C ) ) -> ( ( A ++ B ) = ( C ++ D ) -> ( A = C /\ B = D ) ) )
27	26	3expia 1232	. . . . 5 |- ( ( ( A e. Word X /\ B e. Word X ) /\ ( C e. Word X /\ D e. Word X ) ) -> ( (♯ ` A ) = (♯ ` C ) -> ( ( A ++ B ) = ( C ++ D ) -> ( A = C /\ B = D ) ) ) )
28	27	com23 78	. . . 4 |- ( ( ( A e. Word X /\ B e. Word X ) /\ ( C e. Word X /\ D e. Word X ) ) -> ( ( A ++ B ) = ( C ++ D ) -> ( (♯ ` A ) = (♯ ` C ) -> ( A = C /\ B = D ) ) ) )
29	28	3adant3 1044	. . 3 |- ( ( ( A e. Word X /\ B e. Word X ) /\ ( C e. Word X /\ D e. Word X ) /\ (♯ ` B ) = (♯ ` D ) ) -> ( ( A ++ B ) = ( C ++ D ) -> ( (♯ ` A ) = (♯ ` C ) -> ( A = C /\ B = D ) ) ) )
30	24, 29	mpdd 41	. 2 |- ( ( ( A e. Word X /\ B e. Word X ) /\ ( C e. Word X /\ D e. Word X ) /\ (♯ ` B ) = (♯ ` D ) ) -> ( ( A ++ B ) = ( C ++ D ) -> ( A = C /\ B = D ) ) )
31		oveq12 6058	. 2 |- ( ( A = C /\ B = D ) -> ( A ++ B ) = ( C ++ D ) )
32	30, 31	impbid1 142	1 |- ( ( ( A e. Word X /\ B e. Word X ) /\ ( C e. Word X /\ D e. Word X ) /\ (♯ ` B ) = (♯ ` D ) ) -> ( ( A ++ B ) = ( C ++ D ) <-> ( A = C /\ B = D ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2203   ` cfv 5351  (class class class)co 6049   + caddc 8129   NN0cn0 9495  ♯chash 11136  Word cword 11220   ++ cconcat 11274

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 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709  ax-cnex 8217  ax-resscn 8218  ax-1cn 8219  ax-1re 8220  ax-icn 8221  ax-addcl 8222  ax-addrcl 8223  ax-mulcl 8224  ax-addcom 8226  ax-addass 8228  ax-distr 8230  ax-i2m1 8231  ax-0lt1 8232  ax-0id 8234  ax-rnegex 8235  ax-cnre 8237  ax-pre-ltirr 8238  ax-pre-ltwlin 8239  ax-pre-lttrn 8240  ax-pre-apti 8241  ax-pre-ltadd 8242

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 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-if 3620  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-id 4413  df-iord 4486  df-on 4488  df-ilim 4489  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-recs 6535  df-frec 6621  df-1o 6646  df-er 6766  df-en 6975  df-dom 6976  df-fin 6977  df-pnf 8309  df-mnf 8310  df-xr 8311  df-ltxr 8312  df-le 8313  df-sub 8445  df-neg 8446  df-inn 9237  df-n0 9496  df-z 9577  df-uz 9853  df-fz 10342  df-fzo 10476  df-ihash 11137  df-word 11221  df-concat 11275  df-substr 11334  df-pfx 11361

This theorem is referenced by:  ccatrcan  11407