Theorem swrdccatin2d 11516

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 6092	. . . . . . . . . 10 |- ( (♯ ` A ) = L -> ( (♯ ` A ) ... N ) = ( L ... N ) )
9	8	eleq2d 2308	. . . . . . . . 9 |- ( (♯ ` A ) = L -> ( M e. ( (♯ ` A ) ... N ) <-> M e. ( L ... N ) ) )
10		id 19	. . . . . . . . . . 11 |- ( (♯ ` A ) = L -> (♯ ` A ) = L )
11		oveq1 6092	. . . . . . . . . . 11 |- ( (♯ ` A ) = L -> ( (♯ ` A ) + (♯ ` B ) ) = ( L + (♯ ` B ) ) )
12	10, 11	oveq12d 6103	. . . . . . . . . 10 |- ( (♯ ` A ) = L -> ( (♯ ` A ) ... ( (♯ ` A ) + (♯ ` B ) ) ) = ( L ... ( L + (♯ ` B ) ) ) )
13	12	eleq2d 2308	. . . . . . . . 9 |- ( (♯ ` A ) = L -> ( N e. ( (♯ ` A ) ... ( (♯ ` A ) + (♯ ` B ) ) ) <-> N e. ( L ... ( L + (♯ ` B ) ) ) ) )
14	9, 13	anbi12d 477	. . . . . . . 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 2238	. . . . . 6 |- (♯ ` A ) = (♯ ` A )
20	19	swrdccatin2 11501	. . . . 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 6093	. . . . . 6 |- ( (♯ ` A ) = L -> ( M - (♯ ` A ) ) = ( M - L ) )
24		oveq2 6093	. . . . . 6 |- ( (♯ ` A ) = L -> ( N - (♯ ` A ) ) = ( N - L ) )
25	23, 24	opeq12d 3912	. . . . 5 |- ( (♯ ` A ) = L -> <. ( M - (♯ ` A ) ) , ( N - (♯ ` A ) ) >. = <. ( M - L ) , ( N - L ) >. )
26	25	oveq2d 6101	. . . 4 |- ( (♯ ` A ) = L -> ( B substr <. ( M - (♯ ` A ) ) , ( N - (♯ ` A ) ) >. ) = ( B substr <. ( M - L ) , ( N - L ) >. ) )
27	26	eqeq2d 2250	. . 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 ) >. ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   <.cop 3712   ` cfv 5377  (class class class)co 6085   + caddc 8182   - cmin 8497   ...cfz 10411  ♯chash 11214  Word cword 11304   ++ cconcat 11358   substr csubstr 11417

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-1o 6687  df-er 6807  df-en 7023  df-dom 7024  df-fin 7025  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-inn 9305  df-n0 9564  df-z 9645  df-uz 9922  df-fz 10412  df-fzo 10550  df-ihash 11215  df-word 11305  df-concat 11359  df-substr 11418

This theorem is used by: (None)