Theorem ccatopth 11511

Description: An opth 4379-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 6095	. . . . 5 |- ( ( A ++ B ) = ( C ++ D ) -> ( ( A ++ B ) prefix (♯ ` A ) ) = ( ( C ++ D ) prefix (♯ ` A ) ) )
2		pfxccat1 11497	. . . . . 6 |- ( ( A e. Word X /\ B e. Word X ) -> ( ( A ++ B ) prefix (♯ ` A ) ) = A )
3		oveq2 6096	. . . . . . 7 |- ( (♯ ` A ) = (♯ ` C ) -> ( ( C ++ D ) prefix (♯ ` A ) ) = ( ( C ++ D ) prefix (♯ ` C ) ) )
4		pfxccat1 11497	. . . . . . 7 |- ( ( C e. Word X /\ D e. Word X ) -> ( ( C ++ D ) prefix (♯ ` C ) ) = C )
5	3, 4	sylan9eqr 2293	. . . . . 6 |- ( ( ( C e. Word X /\ D e. Word X ) /\ (♯ ` A ) = (♯ ` C ) ) -> ( ( C ++ D ) prefix (♯ ` A ) ) = C )
6	2, 5	eqeqan12d 2254	. . . . 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 1230	. . 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 1033	. . . . . . 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 5702	. . . . . . . 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 1031	. . . . . . . . 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 11386	. . . . . . . . 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 1032	. . . . . . . . 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 11386	. . . . . . . . 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 2279	. . . . . . 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 3912	. . . . . 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 6106	. . . . 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 11466	. . . . . 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 11466	. . . . . 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 2279	. . . 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 6097	. 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 ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   <.cop 3712   ` cfv 5380  (class class class)co 6088   + caddc 8189  ♯chash 11237  Word cword 11327   ++ cconcat 11381   substr csubstr 11440   prefix cpfx 11467

This proof depends on 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 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8277  ax-resscn 8278  ax-1cn 8279  ax-1re 8280  ax-icn 8281  ax-addcl 8282  ax-addrcl 8283  ax-mulcl 8284  ax-addcom 8286  ax-addass 8288  ax-distr 8290  ax-i2m1 8291  ax-0lt1 8292  ax-0id 8294  ax-rnegex 8295  ax-cnre 8297  ax-pre-ltirr 8298  ax-pre-ltwlin 8299  ax-pre-lttrn 8300  ax-pre-apti 8301  ax-pre-ltadd 8302

This proof 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 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-frec 6666  df-1o 6691  df-er 6811  df-en 7028  df-dom 7029  df-fin 7030  df-pnf 8369  df-mnf 8370  df-xr 8371  df-ltxr 8372  df-le 8373  df-sub 8507  df-neg 8508  df-inn 9315  df-n0 9576  df-z 9657  df-uz 9939  df-fz 10430  df-fzo 10568  df-ihash 11238  df-word 11328  df-concat 11382  df-substr 11441  df-pfx 11468

This theorem is used by:  ccatopth2  11512  ccatlcan  11513