Theorem swrdccatin2d 11436

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 6057	. . . . . . . . . 10 |- ( (♯ ` A ) = L -> ( (♯ ` A ) ... N ) = ( L ... N ) )
9	8	eleq2d 2302	. . . . . . . . 9 |- ( (♯ ` A ) = L -> ( M e. ( (♯ ` A ) ... N ) <-> M e. ( L ... N ) ) )
10		id 19	. . . . . . . . . . 11 |- ( (♯ ` A ) = L -> (♯ ` A ) = L )
11		oveq1 6057	. . . . . . . . . . 11 |- ( (♯ ` A ) = L -> ( (♯ ` A ) + (♯ ` B ) ) = ( L + (♯ ` B ) ) )
12	10, 11	oveq12d 6068	. . . . . . . . . 10 |- ( (♯ ` A ) = L -> ( (♯ ` A ) ... ( (♯ ` A ) + (♯ ` B ) ) ) = ( L ... ( L + (♯ ` B ) ) ) )
13	12	eleq2d 2302	. . . . . . . . 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 2232	. . . . . 6 |- (♯ ` A ) = (♯ ` A )
20	19	swrdccatin2 11421	. . . . 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 6058	. . . . . 6 |- ( (♯ ` A ) = L -> ( M - (♯ ` A ) ) = ( M - L ) )
24		oveq2 6058	. . . . . 6 |- ( (♯ ` A ) = L -> ( N - (♯ ` A ) ) = ( N - L ) )
25	23, 24	opeq12d 3891	. . . . 5 |- ( (♯ ` A ) = L -> <. ( M - (♯ ` A ) ) , ( N - (♯ ` A ) ) >. = <. ( M - L ) , ( N - L ) >. )
26	25	oveq2d 6066	. . . 4 |- ( (♯ ` A ) = L -> ( B substr <. ( M - (♯ ` A ) ) , ( N - (♯ ` A ) ) >. ) = ( B substr <. ( M - L ) , ( N - L ) >. ) )
27	26	eqeq2d 2244	. . 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 1398   e. wcel 2203   <.cop 3692   ` cfv 5352  (class class class)co 6050   + caddc 8130   - cmin 8444   ...cfz 10342  ♯chash 11138  Word cword 11224   ++ cconcat 11278   substr csubstr 11337

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-0id 8235  ax-rnegex 8236  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-iord 4487  df-on 4489  df-ilim 4490  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-frec 6622  df-1o 6647  df-er 6767  df-en 6976  df-dom 6977  df-fin 6978  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-inn 9238  df-n0 9497  df-z 9578  df-uz 9854  df-fz 10343  df-fzo 10477  df-ihash 11139  df-word 11225  df-concat 11279  df-substr 11338

This theorem is referenced by: (None)