Theorem ccatopth2 13471

Description: An opth 4945-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 6191	. . . 4 |- ( ( A ++ B ) = ( C ++ D ) -> ( # ` ( A ++ B ) ) = ( # ` ( C ++ D ) ) )
2		ccatlen 13360	. . . . . . . 8 |- ( ( A e. Word X /\ B e. Word X ) -> ( # ` ( A ++ B ) ) = ( ( # ` A ) + ( # ` B ) ) )
3	2	3ad2ant1 1082	. . . . . . 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 1063	. . . . . . . 8 |- ( ( ( A e. Word X /\ B e. Word X ) /\ ( C e. Word X /\ D e. Word X ) /\ ( # ` B ) = ( # ` D ) ) -> ( # ` B ) = ( # ` D ) )
5	4	oveq2d 6666	. . . . . . 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 2656	. . . . . 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 13360	. . . . . . 7 |- ( ( C e. Word X /\ D e. Word X ) -> ( # ` ( C ++ D ) ) = ( ( # ` C ) + ( # ` D ) ) )
8	7	3ad2ant2 1083	. . . . . 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 2637	. . . . 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 1085	. . . . . . . 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 13324	. . . . . . . 8 |- ( A e. Word X -> ( # ` A ) e. NN0 )
12	10, 11	syl 17	. . . . . . 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 11353	. . . . . 6 |- ( ( ( A e. Word X /\ B e. Word X ) /\ ( C e. Word X /\ D e. Word X ) /\ ( # ` B ) = ( # ` D ) ) -> ( # ` A ) e. CC )
14		simp2l 1087	. . . . . . . 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 13324	. . . . . . . 8 |- ( C e. Word X -> ( # ` C ) e. NN0 )
16	14, 15	syl 17	. . . . . . 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 11353	. . . . . 6 |- ( ( ( A e. Word X /\ B e. Word X ) /\ ( C e. Word X /\ D e. Word X ) /\ ( # ` B ) = ( # ` D ) ) -> ( # ` C ) e. CC )
18		simp2r 1088	. . . . . . . 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 13324	. . . . . . . 8 |- ( D e. Word X -> ( # ` D ) e. NN0 )
20	18, 19	syl 17	. . . . . . 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 11353	. . . . . 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 10240	. . . . 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 268	. . . 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	syl5ib 234	. . 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 13470	. . . . . . 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 219	. . . . . 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 1267	. . . . 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 86	. . . 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 1081	. . 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 43	. 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 6659	. 2 |- ( ( A = C /\ B = D ) -> ( A ++ B ) = ( C ++ D ) )
32	30, 31	impbid1 215	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   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   ` cfv 5888  (class class class)co 6650   + caddc 9939   NN0cn0 11292   #chash 13117  Word cword 13291   ++ cconcat 13293

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-concat 13301  df-substr 13303

This theorem is referenced by:  ccatrcan  13473