Theorem ccatopth 11301

Description: An opth 4329-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 6025	. . . . 5 |- ( ( A ++ B ) = ( C ++ D ) -> ( ( A ++ B ) prefix (♯ ` A ) ) = ( ( C ++ D ) prefix (♯ ` A ) ) )
2		pfxccat1 11287	. . . . . 6 |- ( ( A e. Word X /\ B e. Word X ) -> ( ( A ++ B ) prefix (♯ ` A ) ) = A )
3		oveq2 6026	. . . . . . 7 |- ( (♯ ` A ) = (♯ ` C ) -> ( ( C ++ D ) prefix (♯ ` A ) ) = ( ( C ++ D ) prefix (♯ ` C ) ) )
4		pfxccat1 11287	. . . . . . 7 |- ( ( C e. Word X /\ D e. Word X ) -> ( ( C ++ D ) prefix (♯ ` C ) ) = C )
5	3, 4	sylan9eqr 2286	. . . . . 6 |- ( ( ( C e. Word X /\ D e. Word X ) /\ (♯ ` A ) = (♯ ` C ) ) -> ( ( C ++ D ) prefix (♯ ` A ) ) = C )
6	2, 5	eqeqan12d 2247	. . . . 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 1225	. . 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 1028	. . . . . . 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 5643	. . . . . . . 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 1026	. . . . . . . . 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 11176	. . . . . . . . 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 1027	. . . . . . . . 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 11176	. . . . . . . . 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 2272	. . . . . . 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 3870	. . . . . 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 6036	. . . . 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 11256	. . . . . 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 11256	. . . . . 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 2272	. . . 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 6027	. 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 1004   = wceq 1397   e. wcel 2202   <.cop 3672   ` cfv 5326  (class class class)co 6018   + caddc 8035  ♯chash 11038  Word cword 11117   ++ cconcat 11171   substr csubstr 11230   prefix cpfx 11257

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-addass 8134  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-0id 8140  ax-rnegex 8141  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-frec 6557  df-1o 6582  df-er 6702  df-en 6910  df-dom 6911  df-fin 6912  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-inn 9144  df-n0 9403  df-z 9480  df-uz 9756  df-fz 10244  df-fzo 10378  df-ihash 11039  df-word 11118  df-concat 11172  df-substr 11231  df-pfx 11258

This theorem is referenced by:  ccatopth2  11302  ccatlcan  11303