Theorem ccatopth 11256

Description: An opth 4323-like theorem for recovering the two halves of a concatenated word. (Contributed by Mario Carneiro, 1-Oct-2015.) (Proof shortened by AV, 12-Oct-2022.)

Assertion

Ref	Expression
ccatopth	|- ( ( ( 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 ) ) )

Proof of Theorem ccatopth

Step	Hyp	Ref	Expression
1		oveq1 6014	. . . . 5 |- ( ( A ++ B ) = ( C ++ D ) -> ( ( A ++ B ) prefix (♯ ` A ) ) = ( ( C ++ D ) prefix (♯ ` A ) ) )
2		pfxccat1 11242	. . . . . 6 |- ( ( A e. Word X /\ B e. Word X ) -> ( ( A ++ B ) prefix (♯ ` A ) ) = A )
3		oveq2 6015	. . . . . . 7 |- ( (♯ ` A ) = (♯ ` C ) -> ( ( C ++ D ) prefix (♯ ` A ) ) = ( ( C ++ D ) prefix (♯ ` C ) ) )
4		pfxccat1 11242	. . . . . . 7 |- ( ( C e. Word X /\ D e. Word X ) -> ( ( C ++ D ) prefix (♯ ` C ) ) = C )
5	3, 4	sylan9eqr 2284	. . . . . 6 |- ( ( ( C e. Word X /\ D e. Word X ) /\ (♯ ` A ) = (♯ ` C ) ) -> ( ( C ++ D ) prefix (♯ ` A ) ) = C )
6	2, 5	eqeqan12d 2245	. . . . 5 |- ( ( ( A e. Word X /\ B e. Word X ) /\ ( ( C e. Word X /\ D e. Word X ) /\ (♯ ` A ) = (♯ ` C ) ) ) -> ( ( ( A ++ B ) prefix (♯ ` A ) ) = ( ( C ++ D ) prefix (♯ ` A ) ) <-> A = C ) )
7	1, 6	imbitrid 154	. . . 4 |- ( ( ( A e. Word X /\ B e. Word X ) /\ ( ( C e. Word X /\ D e. Word X ) /\ (♯ ` A ) = (♯ ` C ) ) ) -> ( ( A ++ B ) = ( C ++ D ) -> A = C ) )
8	7	3impb 1223	. . 3 |- ( ( ( A e. Word X /\ B e. Word X ) /\ ( C e. Word X /\ D e. Word X ) /\ (♯ ` A ) = (♯ ` C ) ) -> ( ( A ++ B ) = ( C ++ D ) -> A = C ) )
9		simpr 110	. . . . . 6 |- ( ( ( ( A e. Word X /\ B e. Word X ) /\ ( C e. Word X /\ D e. Word X ) /\ (♯ ` A ) = (♯ ` C ) ) /\ ( A ++ B ) = ( C ++ D ) ) -> ( A ++ B ) = ( C ++ D ) )
10		simpl3 1026	. . . . . . 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 ) )
11	9	fveq2d 5633	. . . . . . . 8 |- ( ( ( ( A e. Word X /\ B e. Word X ) /\ ( C e. Word X /\ D e. Word X ) /\ (♯ ` A ) = (♯ ` C ) ) /\ ( A ++ B ) = ( C ++ D ) ) -> (♯ ` ( A ++ B ) ) = (♯ ` ( C ++ D ) ) )
12		simpl1 1024	. . . . . . . . 9 |- ( ( ( ( A e. Word X /\ B e. Word X ) /\ ( C e. Word X /\ D e. Word X ) /\ (♯ ` A ) = (♯ ` C ) ) /\ ( A ++ B ) = ( C ++ D ) ) -> ( A e. Word X /\ B e. Word X ) )
13		ccatlen 11138	. . . . . . . . 9 |- ( ( A e. Word X /\ B e. Word X ) -> (♯ ` ( A ++ B ) ) = ( (♯ ` A ) + (♯ ` B ) ) )
14	12, 13	syl 14	. . . . . . . 8 |- ( ( ( ( A e. Word X /\ B e. Word X ) /\ ( C e. Word X /\ D e. Word X ) /\ (♯ ` A ) = (♯ ` C ) ) /\ ( A ++ B ) = ( C ++ D ) ) -> (♯ ` ( A ++ B ) ) = ( (♯ ` A ) + (♯ ` B ) ) )
15		simpl2 1025	. . . . . . . . 9 |- ( ( ( ( A e. Word X /\ B e. Word X ) /\ ( C e. Word X /\ D e. Word X ) /\ (♯ ` A ) = (♯ ` C ) ) /\ ( A ++ B ) = ( C ++ D ) ) -> ( C e. Word X /\ D e. Word X ) )
16		ccatlen 11138	. . . . . . . . 9 |- ( ( C e. Word X /\ D e. Word X ) -> (♯ ` ( C ++ D ) ) = ( (♯ ` C ) + (♯ ` D ) ) )
17	15, 16	syl 14	. . . . . . . 8 |- ( ( ( ( A e. Word X /\ B e. Word X ) /\ ( C e. Word X /\ D e. Word X ) /\ (♯ ` A ) = (♯ ` C ) ) /\ ( A ++ B ) = ( C ++ D ) ) -> (♯ ` ( C ++ D ) ) = ( (♯ ` C ) + (♯ ` D ) ) )
18	11, 14, 17	3eqtr3d 2270	. . . . . . 7 |- ( ( ( ( A e. Word X /\ B e. Word X ) /\ ( C e. Word X /\ D e. Word X ) /\ (♯ ` A ) = (♯ ` C ) ) /\ ( A ++ B ) = ( C ++ D ) ) -> ( (♯ ` A ) + (♯ ` B ) ) = ( (♯ ` C ) + (♯ ` D ) ) )
19	10, 18	opeq12d 3865	. . . . . 6 |- ( ( ( ( A e. Word X /\ B e. Word X ) /\ ( C e. Word X /\ D e. Word X ) /\ (♯ ` A ) = (♯ ` C ) ) /\ ( A ++ B ) = ( C ++ D ) ) -> <. (♯ ` A ) , ( (♯ ` A ) + (♯ ` B ) ) >. = <. (♯ ` C ) , ( (♯ ` C ) + (♯ ` D ) ) >. )
20	9, 19	oveq12d 6025	. . . . 5 |- ( ( ( ( A e. Word X /\ B e. Word X ) /\ ( C e. Word X /\ D e. Word X ) /\ (♯ ` A ) = (♯ ` C ) ) /\ ( A ++ B ) = ( C ++ D ) ) -> ( ( A ++ B ) substr <. (♯ ` A ) , ( (♯ ` A ) + (♯ ` B ) ) >. ) = ( ( C ++ D ) substr <. (♯ ` C ) , ( (♯ ` C ) + (♯ ` D ) ) >. ) )
21		swrdccat2 11211	. . . . . 6 |- ( ( A e. Word X /\ B e. Word X ) -> ( ( A ++ B ) substr <. (♯ ` A ) , ( (♯ ` A ) + (♯ ` B ) ) >. ) = B )
22	12, 21	syl 14	. . . . 5 |- ( ( ( ( A e. Word X /\ B e. Word X ) /\ ( C e. Word X /\ D e. Word X ) /\ (♯ ` A ) = (♯ ` C ) ) /\ ( A ++ B ) = ( C ++ D ) ) -> ( ( A ++ B ) substr <. (♯ ` A ) , ( (♯ ` A ) + (♯ ` B ) ) >. ) = B )
23		swrdccat2 11211	. . . . . 6 |- ( ( C e. Word X /\ D e. Word X ) -> ( ( C ++ D ) substr <. (♯ ` C ) , ( (♯ ` C ) + (♯ ` D ) ) >. ) = D )
24	15, 23	syl 14	. . . . 5 |- ( ( ( ( A e. Word X /\ B e. Word X ) /\ ( C e. Word X /\ D e. Word X ) /\ (♯ ` A ) = (♯ ` C ) ) /\ ( A ++ B ) = ( C ++ D ) ) -> ( ( C ++ D ) substr <. (♯ ` C ) , ( (♯ ` C ) + (♯ ` D ) ) >. ) = D )
25	20, 22, 24	3eqtr3d 2270	. . . 4 |- ( ( ( ( A e. Word X /\ B e. Word X ) /\ ( C e. Word X /\ D e. Word X ) /\ (♯ ` A ) = (♯ ` C ) ) /\ ( A ++ B ) = ( C ++ D ) ) -> B = D )
26	25	ex 115	. . 3 |- ( ( ( A e. Word X /\ B e. Word X ) /\ ( C e. Word X /\ D e. Word X ) /\ (♯ ` A ) = (♯ ` C ) ) -> ( ( A ++ B ) = ( C ++ D ) -> B = D ) )
27	8, 26	jcad 307	. 2 |- ( ( ( 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		oveq12 6016	. 2 |- ( ( A = C /\ B = D ) -> ( A ++ B ) = ( C ++ D ) )
29	27, 28	impbid1 142	1 |- ( ( ( 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 ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   <.cop 3669   ` cfv 5318  (class class class)co 6007   + caddc 8010  ♯chash 11005  Word cword 11079   ++ cconcat 11133   substr csubstr 11185   prefix cpfx 11212

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8098  ax-resscn 8099  ax-1cn 8100  ax-1re 8101  ax-icn 8102  ax-addcl 8103  ax-addrcl 8104  ax-mulcl 8105  ax-addcom 8107  ax-addass 8109  ax-distr 8111  ax-i2m1 8112  ax-0lt1 8113  ax-0id 8115  ax-rnegex 8116  ax-cnre 8118  ax-pre-ltirr 8119  ax-pre-ltwlin 8120  ax-pre-lttrn 8121  ax-pre-apti 8122  ax-pre-ltadd 8123

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-frec 6543  df-1o 6568  df-er 6688  df-en 6896  df-dom 6897  df-fin 6898  df-pnf 8191  df-mnf 8192  df-xr 8193  df-ltxr 8194  df-le 8195  df-sub 8327  df-neg 8328  df-inn 9119  df-n0 9378  df-z 9455  df-uz 9731  df-fz 10213  df-fzo 10347  df-ihash 11006  df-word 11080  df-concat 11134  df-substr 11186  df-pfx 11213

This theorem is referenced by:  ccatopth2  11257  ccatlcan  11258