Theorem swrdccatin2d 11291

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 277	. . . . . 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 306	. . . . . . . 8 |- ( ph -> ( M e. ( L ... N ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) )
7	6	adantl 277	. . . . . . 7 |- ( ( (♯ ` A ) = L /\ ph ) -> ( M e. ( L ... N ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) )
8		oveq1 6014	. . . . . . . . . 10 |- ( (♯ ` A ) = L -> ( (♯ ` A ) ... N ) = ( L ... N ) )
9	8	eleq2d 2299	. . . . . . . . 9 |- ( (♯ ` A ) = L -> ( M e. ( (♯ ` A ) ... N ) <-> M e. ( L ... N ) ) )
10		id 19	. . . . . . . . . . 11 |- ( (♯ ` A ) = L -> (♯ ` A ) = L )
11		oveq1 6014	. . . . . . . . . . 11 |- ( (♯ ` A ) = L -> ( (♯ ` A ) + (♯ ` B ) ) = ( L + (♯ ` B ) ) )
12	10, 11	oveq12d 6025	. . . . . . . . . 10 |- ( (♯ ` A ) = L -> ( (♯ ` A ) ... ( (♯ ` A ) + (♯ ` B ) ) ) = ( L ... ( L + (♯ ` B ) ) ) )
13	12	eleq2d 2299	. . . . . . . . 9 |- ( (♯ ` A ) = L -> ( N e. ( (♯ ` A ) ... ( (♯ ` A ) + (♯ ` B ) ) ) <-> N e. ( L ... ( L + (♯ ` B ) ) ) ) )
14	9, 13	anbi12d 473	. . . . . . . 8 |- ( (♯ ` A ) = L -> ( ( M e. ( (♯ ` A ) ... N ) /\ N e. ( (♯ ` A ) ... ( (♯ ` A ) + (♯ ` B ) ) ) ) <-> ( M e. ( L ... N ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) )
15	14	adantr 276	. . . . . . 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 167	. . . . . 6 |- ( ( (♯ ` A ) = L /\ ph ) -> ( M e. ( (♯ ` A ) ... N ) /\ N e. ( (♯ ` A ) ... ( (♯ ` A ) + (♯ ` B ) ) ) ) )
17	3, 16	jca 306	. . . . 5 |- ( ( (♯ ` A ) = L /\ ph ) -> ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( (♯ ` A ) ... N ) /\ N e. ( (♯ ` A ) ... ( (♯ ` A ) + (♯ ` B ) ) ) ) ) )
18	17	ex 115	. . . 4 |- ( (♯ ` A ) = L -> ( ph -> ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( (♯ ` A ) ... N ) /\ N e. ( (♯ ` A ) ... ( (♯ ` A ) + (♯ ` B ) ) ) ) ) ) )
19		eqid 2229	. . . . . 6 |- (♯ ` A ) = (♯ ` A )
20	19	swrdccatin2 11276	. . . . 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 124	. . . 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 33	. . 3 |- ( (♯ ` A ) = L -> ( ph -> ( ( A ++ B ) substr <. M , N >. ) = ( B substr <. ( M - (♯ ` A ) ) , ( N - (♯ ` A ) ) >. ) ) )
23		oveq2 6015	. . . . . 6 |- ( (♯ ` A ) = L -> ( M - (♯ ` A ) ) = ( M - L ) )
24		oveq2 6015	. . . . . 6 |- ( (♯ ` A ) = L -> ( N - (♯ ` A ) ) = ( N - L ) )
25	23, 24	opeq12d 3865	. . . . 5 |- ( (♯ ` A ) = L -> <. ( M - (♯ ` A ) ) , ( N - (♯ ` A ) ) >. = <. ( M - L ) , ( N - L ) >. )
26	25	oveq2d 6023	. . . 4 |- ( (♯ ` A ) = L -> ( B substr <. ( M - (♯ ` A ) ) , ( N - (♯ ` A ) ) >. ) = ( B substr <. ( M - L ) , ( N - L ) >. ) )
27	26	eqeq2d 2241	. . 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 149	. 2 |- ( (♯ ` A ) = L -> ( ph -> ( ( A ++ B ) substr <. M , N >. ) = ( B substr <. ( M - L ) , ( N - L ) >. ) ) )
29	1, 28	mpcom 36	1 |- ( ph -> ( ( A ++ B ) substr <. M , N >. ) = ( B substr <. ( M - L ) , ( N - L ) >. ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   <.cop 3669   ` cfv 5318  (class class class)co 6007   + caddc 8013   - cmin 8328   ...cfz 10216  ♯chash 11009  Word cword 11084   ++ cconcat 11138   substr csubstr 11192

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-addcom 8110  ax-addass 8112  ax-distr 8114  ax-i2m1 8115  ax-0lt1 8116  ax-0id 8118  ax-rnegex 8119  ax-cnre 8121  ax-pre-ltirr 8122  ax-pre-ltwlin 8123  ax-pre-lttrn 8124  ax-pre-apti 8125  ax-pre-ltadd 8126

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-frec 6543  df-1o 6568  df-er 6688  df-en 6896  df-dom 6897  df-fin 6898  df-pnf 8194  df-mnf 8195  df-xr 8196  df-ltxr 8197  df-le 8198  df-sub 8330  df-neg 8331  df-inn 9122  df-n0 9381  df-z 9458  df-uz 9734  df-fz 10217  df-fzo 10351  df-ihash 11010  df-word 11085  df-concat 11139  df-substr 11193

This theorem is referenced by: (None)