Theorem ccatopth 11187

Description: An opth 4288-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 5963	. . . . 5 |- ( ( A ++ B ) = ( C ++ D ) -> ( ( A ++ B ) prefix (♯ ` A ) ) = ( ( C ++ D ) prefix (♯ ` A ) ) )
2		pfxccat1 11173	. . . . . 6 |- ( ( A e. Word X /\ B e. Word X ) -> ( ( A ++ B ) prefix (♯ ` A ) ) = A )
3		oveq2 5964	. . . . . . 7 |- ( (♯ ` A ) = (♯ ` C ) -> ( ( C ++ D ) prefix (♯ ` A ) ) = ( ( C ++ D ) prefix (♯ ` C ) ) )
4		pfxccat1 11173	. . . . . . 7 |- ( ( C e. Word X /\ D e. Word X ) -> ( ( C ++ D ) prefix (♯ ` C ) ) = C )
5	3, 4	sylan9eqr 2261	. . . . . 6 |- ( ( ( C e. Word X /\ D e. Word X ) /\ (♯ ` A ) = (♯ ` C ) ) -> ( ( C ++ D ) prefix (♯ ` A ) ) = C )
6	2, 5	eqeqan12d 2222	. . . . 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 1202	. . 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 1005	. . . . . . 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 5592	. . . . . . . 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 1003	. . . . . . . . 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 11069	. . . . . . . . 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 1004	. . . . . . . . 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 11069	. . . . . . . . 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 2247	. . . . . . 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 3832	. . . . . 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 5974	. . . . 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 11142	. . . . . 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 11142	. . . . . 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 2247	. . . 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 5965	. 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 981   = wceq 1373   e. wcel 2177   <.cop 3640   ` cfv 5279  (class class class)co 5956   + caddc 7943  ♯chash 10937  Word cword 11011   ++ cconcat 11064   substr csubstr 11116   prefix cpfx 11143

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4166  ax-sep 4169  ax-nul 4177  ax-pow 4225  ax-pr 4260  ax-un 4487  ax-setind 4592  ax-iinf 4643  ax-cnex 8031  ax-resscn 8032  ax-1cn 8033  ax-1re 8034  ax-icn 8035  ax-addcl 8036  ax-addrcl 8037  ax-mulcl 8038  ax-addcom 8040  ax-addass 8042  ax-distr 8044  ax-i2m1 8045  ax-0lt1 8046  ax-0id 8048  ax-rnegex 8049  ax-cnre 8051  ax-pre-ltirr 8052  ax-pre-ltwlin 8053  ax-pre-lttrn 8054  ax-pre-apti 8055  ax-pre-ltadd 8056

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-if 3576  df-pw 3622  df-sn 3643  df-pr 3644  df-op 3646  df-uni 3856  df-int 3891  df-iun 3934  df-br 4051  df-opab 4113  df-mpt 4114  df-tr 4150  df-id 4347  df-iord 4420  df-on 4422  df-ilim 4423  df-suc 4425  df-iom 4646  df-xp 4688  df-rel 4689  df-cnv 4690  df-co 4691  df-dm 4692  df-rn 4693  df-res 4694  df-ima 4695  df-iota 5240  df-fun 5281  df-fn 5282  df-f 5283  df-f1 5284  df-fo 5285  df-f1o 5286  df-fv 5287  df-riota 5911  df-ov 5959  df-oprab 5960  df-mpo 5961  df-1st 6238  df-2nd 6239  df-recs 6403  df-frec 6489  df-1o 6514  df-er 6632  df-en 6840  df-dom 6841  df-fin 6842  df-pnf 8124  df-mnf 8125  df-xr 8126  df-ltxr 8127  df-le 8128  df-sub 8260  df-neg 8261  df-inn 9052  df-n0 9311  df-z 9388  df-uz 9664  df-fz 10146  df-fzo 10280  df-ihash 10938  df-word 11012  df-concat 11065  df-substr 11117  df-pfx 11144

This theorem is referenced by:  ccatopth2  11188  ccatlcan  11189