Theorem swrdccatin2d 13500

Description: The subword of a concatenation of two words within the second of the concatenated words. (Contributed by AV, 31-May-2018.) (Revised by Mario Carneiro/AV, 21-Oct-2018.)

Hypotheses

Ref	Expression
swrdccatind.l	|- ( ph -> ( # ` A ) = L )
swrdccatind.w	|- ( ph -> ( A e. Word V /\ B e. Word V ) )
swrdccatin2d.1	|- ( ph -> M e. ( L ... N ) )
swrdccatin2d.2	|- ( ph -> N e. ( L ... ( L + ( # ` B ) ) ) )

Assertion

Ref	Expression
swrdccatin2d	|- ( ph -> ( ( A ++ B ) substr <. M , N >. ) = ( B substr <. ( M - L ) , ( N - L ) >. ) )

Proof of Theorem swrdccatin2d

Step	Hyp	Ref	Expression
1		swrdccatind.l	. 2 |- ( ph -> ( # ` A ) = L )
2		swrdccatind.w	. . . . . . 7 |- ( ph -> ( A e. Word V /\ B e. Word V ) )
3	2	adantl 482	. . . . . 6 |- ( ( ( # ` A ) = L /\ ph ) -> ( A e. Word V /\ B e. Word V ) )
4		swrdccatin2d.1	. . . . . . . . 9 |- ( ph -> M e. ( L ... N ) )
5		swrdccatin2d.2	. . . . . . . . 9 |- ( ph -> N e. ( L ... ( L + ( # ` B ) ) ) )
6	4, 5	jca 554	. . . . . . . 8 |- ( ph -> ( M e. ( L ... N ) /\ N e. ( L ... ( L + ( # ` B ) ) ) ) )
7	6	adantl 482	. . . . . . 7 |- ( ( ( # ` A ) = L /\ ph ) -> ( M e. ( L ... N ) /\ N e. ( L ... ( L + ( # ` B ) ) ) ) )
8		oveq1 6657	. . . . . . . . . 10 |- ( ( # ` A ) = L -> ( ( # ` A ) ... N ) = ( L ... N ) )
9	8	eleq2d 2687	. . . . . . . . 9 |- ( ( # ` A ) = L -> ( M e. ( ( # ` A ) ... N ) <-> M e. ( L ... N ) ) )
10		id 22	. . . . . . . . . . 11 |- ( ( # ` A ) = L -> ( # ` A ) = L )
11		oveq1 6657	. . . . . . . . . . 11 |- ( ( # ` A ) = L -> ( ( # ` A ) + ( # ` B ) ) = ( L + ( # ` B ) ) )
12	10, 11	oveq12d 6668	. . . . . . . . . 10 |- ( ( # ` A ) = L -> ( ( # ` A ) ... ( ( # ` A ) + ( # ` B ) ) ) = ( L ... ( L + ( # ` B ) ) ) )
13	12	eleq2d 2687	. . . . . . . . 9 |- ( ( # ` A ) = L -> ( N e. ( ( # ` A ) ... ( ( # ` A ) + ( # ` B ) ) ) <-> N e. ( L ... ( L + ( # ` B ) ) ) ) )
14	9, 13	anbi12d 747	. . . . . . . 8 |- ( ( # ` A ) = L -> ( ( M e. ( ( # ` A ) ... N ) /\ N e. ( ( # ` A ) ... ( ( # ` A ) + ( # ` B ) ) ) ) <-> ( M e. ( L ... N ) /\ N e. ( L ... ( L + ( # ` B ) ) ) ) ) )
15	14	adantr 481	. . . . . . 7 |- ( ( ( # ` A ) = L /\ ph ) -> ( ( M e. ( ( # ` A ) ... N ) /\ N e. ( ( # ` A ) ... ( ( # ` A ) + ( # ` B ) ) ) ) <-> ( M e. ( L ... N ) /\ N e. ( L ... ( L + ( # ` B ) ) ) ) ) )
16	7, 15	mpbird 247	. . . . . 6 |- ( ( ( # ` A ) = L /\ ph ) -> ( M e. ( ( # ` A ) ... N ) /\ N e. ( ( # ` A ) ... ( ( # ` A ) + ( # ` B ) ) ) ) )
17	3, 16	jca 554	. . . . 5 |- ( ( ( # ` A ) = L /\ ph ) -> ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( ( # ` A ) ... N ) /\ N e. ( ( # ` A ) ... ( ( # ` A ) + ( # ` B ) ) ) ) ) )
18	17	ex 450	. . . 4 |- ( ( # ` A ) = L -> ( ph -> ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( ( # ` A ) ... N ) /\ N e. ( ( # ` A ) ... ( ( # ` A ) + ( # ` B ) ) ) ) ) ) )
19		eqid 2622	. . . . . 6 |- ( # ` A ) = ( # ` A )
20	19	swrdccatin2 13487	. . . . 5 |- ( ( A e. Word V /\ B e. Word V ) -> ( ( M e. ( ( # ` A ) ... N ) /\ N e. ( ( # ` A ) ... ( ( # ` A ) + ( # ` B ) ) ) ) -> ( ( A ++ B ) substr <. M , N >. ) = ( B substr <. ( M - ( # ` A ) ) , ( N - ( # ` A ) ) >. ) ) )
21	20	imp 445	. . . 4 |- ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( ( # ` A ) ... N ) /\ N e. ( ( # ` A ) ... ( ( # ` A ) + ( # ` B ) ) ) ) ) -> ( ( A ++ B ) substr <. M , N >. ) = ( B substr <. ( M - ( # ` A ) ) , ( N - ( # ` A ) ) >. ) )
22	18, 21	syl6 35	. . 3 |- ( ( # ` A ) = L -> ( ph -> ( ( A ++ B ) substr <. M , N >. ) = ( B substr <. ( M - ( # ` A ) ) , ( N - ( # ` A ) ) >. ) ) )
23		oveq2 6658	. . . . . 6 |- ( ( # ` A ) = L -> ( M - ( # ` A ) ) = ( M - L ) )
24		oveq2 6658	. . . . . 6 |- ( ( # ` A ) = L -> ( N - ( # ` A ) ) = ( N - L ) )
25	23, 24	opeq12d 4410	. . . . 5 |- ( ( # ` A ) = L -> <. ( M - ( # ` A ) ) , ( N - ( # ` A ) ) >. = <. ( M - L ) , ( N - L ) >. )
26	25	oveq2d 6666	. . . 4 |- ( ( # ` A ) = L -> ( B substr <. ( M - ( # ` A ) ) , ( N - ( # ` A ) ) >. ) = ( B substr <. ( M - L ) , ( N - L ) >. ) )
27	26	eqeq2d 2632	. . 3 |- ( ( # ` A ) = L -> ( ( ( A ++ B ) substr <. M , N >. ) = ( B substr <. ( M - ( # ` A ) ) , ( N - ( # ` A ) ) >. ) <-> ( ( A ++ B ) substr <. M , N >. ) = ( B substr <. ( M - L ) , ( N - L ) >. ) ) )
28	22, 27	sylibd 229	. 2 |- ( ( # ` A ) = L -> ( ph -> ( ( A ++ B ) substr <. M , N >. ) = ( B substr <. ( M - L ) , ( N - L ) >. ) ) )
29	1, 28	mpcom 38	1 |- ( ph -> ( ( A ++ B ) substr <. M , N >. ) = ( B substr <. ( M - L ) , ( N - L ) >. ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   <.cop 4183   ` cfv 5888  (class class class)co 6650   + caddc 9939   - cmin 10266   ...cfz 12326   #chash 13117  Word cword 13291   ++ cconcat 13293   substr csubstr 13295

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: (None)