Theorem swrdccatin2d 11324

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 6024	. . . . . . . . . 10 |- ( (♯ ` A ) = L -> ( (♯ ` A ) ... N ) = ( L ... N ) )
9	8	eleq2d 2301	. . . . . . . . 9 |- ( (♯ ` A ) = L -> ( M e. ( (♯ ` A ) ... N ) <-> M e. ( L ... N ) ) )
10		id 19	. . . . . . . . . . 11 |- ( (♯ ` A ) = L -> (♯ ` A ) = L )
11		oveq1 6024	. . . . . . . . . . 11 |- ( (♯ ` A ) = L -> ( (♯ ` A ) + (♯ ` B ) ) = ( L + (♯ ` B ) ) )
12	10, 11	oveq12d 6035	. . . . . . . . . 10 |- ( (♯ ` A ) = L -> ( (♯ ` A ) ... ( (♯ ` A ) + (♯ ` B ) ) ) = ( L ... ( L + (♯ ` B ) ) ) )
13	12	eleq2d 2301	. . . . . . . . 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 2231	. . . . . 6 |- (♯ ` A ) = (♯ ` A )
20	19	swrdccatin2 11309	. . . . 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 6025	. . . . . 6 |- ( (♯ ` A ) = L -> ( M - (♯ ` A ) ) = ( M - L ) )
24		oveq2 6025	. . . . . 6 |- ( (♯ ` A ) = L -> ( N - (♯ ` A ) ) = ( N - L ) )
25	23, 24	opeq12d 3870	. . . . 5 |- ( (♯ ` A ) = L -> <. ( M - (♯ ` A ) ) , ( N - (♯ ` A ) ) >. = <. ( M - L ) , ( N - L ) >. )
26	25	oveq2d 6033	. . . 4 |- ( (♯ ` A ) = L -> ( B substr <. ( M - (♯ ` A ) ) , ( N - (♯ ` A ) ) >. ) = ( B substr <. ( M - L ) , ( N - L ) >. ) )
27	26	eqeq2d 2243	. . 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 1397   e. wcel 2202   <.cop 3672   ` cfv 5326  (class class class)co 6017   + caddc 8034   - cmin 8349   ...cfz 10242  ♯chash 11036  Word cword 11112   ++ cconcat 11166   substr csubstr 11225

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-0id 8139  ax-rnegex 8140  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-frec 6556  df-1o 6581  df-er 6701  df-en 6909  df-dom 6910  df-fin 6911  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-inn 9143  df-n0 9402  df-z 9479  df-uz 9755  df-fz 10243  df-fzo 10377  df-ihash 11037  df-word 11113  df-concat 11167  df-substr 11226

This theorem is referenced by: (None)