Theorem swrdval 11278

Description: Value of a subword. (Contributed by Stefan O'Rear, 15-Aug-2015.)

Assertion

Ref	Expression
swrdval	⊢ ((𝑆 ∈ 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (𝑆 substr ⟨𝐹, 𝐿⟩) = if((𝐹..^𝐿) ⊆ dom 𝑆, (𝑥 ∈ (0..^(𝐿 − 𝐹)) ↦ (𝑆‘(𝑥 + 𝐹))), ∅))

Distinct variable groups:   𝑥,𝑆   𝑥,𝐹   𝑥,𝐿   𝑥,𝑉

Proof of Theorem swrdval

Dummy variables 𝑠 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-substr 11276	. . 3 ⊢ substr = (𝑠 ∈ V, 𝑏 ∈ (ℤ × ℤ) ↦ if(((1st ‘𝑏)..^(2nd ‘𝑏)) ⊆ dom 𝑠, (𝑥 ∈ (0..^((2nd ‘𝑏) − (1st ‘𝑏))) ↦ (𝑠‘(𝑥 + (1st ‘𝑏)))), ∅))
2	1	a1i 9	. 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → substr = (𝑠 ∈ V, 𝑏 ∈ (ℤ × ℤ) ↦ if(((1st ‘𝑏)..^(2nd ‘𝑏)) ⊆ dom 𝑠, (𝑥 ∈ (0..^((2nd ‘𝑏) − (1st ‘𝑏))) ↦ (𝑠‘(𝑥 + (1st ‘𝑏)))), ∅)))
3		simprl 531	. . 3 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∧ (𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝐿⟩)) → 𝑠 = 𝑆)
4		fveq2 5648	. . . . 5 ⊢ (𝑏 = ⟨𝐹, 𝐿⟩ → (1st ‘𝑏) = (1st ‘⟨𝐹, 𝐿⟩))
5	4	adantl 277	. . . 4 ⊢ ((𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝐿⟩) → (1st ‘𝑏) = (1st ‘⟨𝐹, 𝐿⟩))
6		op1stg 6322	. . . . 5 ⊢ ((𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (1st ‘⟨𝐹, 𝐿⟩) = 𝐹)
7	6	3adant1 1042	. . . 4 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (1st ‘⟨𝐹, 𝐿⟩) = 𝐹)
8	5, 7	sylan9eqr 2286	. . 3 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∧ (𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝐿⟩)) → (1st ‘𝑏) = 𝐹)
9		fveq2 5648	. . . . 5 ⊢ (𝑏 = ⟨𝐹, 𝐿⟩ → (2nd ‘𝑏) = (2nd ‘⟨𝐹, 𝐿⟩))
10	9	adantl 277	. . . 4 ⊢ ((𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝐿⟩) → (2nd ‘𝑏) = (2nd ‘⟨𝐹, 𝐿⟩))
11		op2ndg 6323	. . . . 5 ⊢ ((𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (2nd ‘⟨𝐹, 𝐿⟩) = 𝐿)
12	11	3adant1 1042	. . . 4 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (2nd ‘⟨𝐹, 𝐿⟩) = 𝐿)
13	10, 12	sylan9eqr 2286	. . 3 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∧ (𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝐿⟩)) → (2nd ‘𝑏) = 𝐿)
14		simp2 1025	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ (1st ‘𝑏) = 𝐹 ∧ (2nd ‘𝑏) = 𝐿) → (1st ‘𝑏) = 𝐹)
15		simp3 1026	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ (1st ‘𝑏) = 𝐹 ∧ (2nd ‘𝑏) = 𝐿) → (2nd ‘𝑏) = 𝐿)
16	14, 15	oveq12d 6046	. . . . 5 ⊢ ((𝑠 = 𝑆 ∧ (1st ‘𝑏) = 𝐹 ∧ (2nd ‘𝑏) = 𝐿) → ((1st ‘𝑏)..^(2nd ‘𝑏)) = (𝐹..^𝐿))
17		simp1 1024	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ (1st ‘𝑏) = 𝐹 ∧ (2nd ‘𝑏) = 𝐿) → 𝑠 = 𝑆)
18	17	dmeqd 4939	. . . . 5 ⊢ ((𝑠 = 𝑆 ∧ (1st ‘𝑏) = 𝐹 ∧ (2nd ‘𝑏) = 𝐿) → dom 𝑠 = dom 𝑆)
19	16, 18	sseq12d 3259	. . . 4 ⊢ ((𝑠 = 𝑆 ∧ (1st ‘𝑏) = 𝐹 ∧ (2nd ‘𝑏) = 𝐿) → (((1st ‘𝑏)..^(2nd ‘𝑏)) ⊆ dom 𝑠 ↔ (𝐹..^𝐿) ⊆ dom 𝑆))
20	15, 14	oveq12d 6046	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ (1st ‘𝑏) = 𝐹 ∧ (2nd ‘𝑏) = 𝐿) → ((2nd ‘𝑏) − (1st ‘𝑏)) = (𝐿 − 𝐹))
21	20	oveq2d 6044	. . . . 5 ⊢ ((𝑠 = 𝑆 ∧ (1st ‘𝑏) = 𝐹 ∧ (2nd ‘𝑏) = 𝐿) → (0..^((2nd ‘𝑏) − (1st ‘𝑏))) = (0..^(𝐿 − 𝐹)))
22	14	oveq2d 6044	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ (1st ‘𝑏) = 𝐹 ∧ (2nd ‘𝑏) = 𝐿) → (𝑥 + (1st ‘𝑏)) = (𝑥 + 𝐹))
23	17, 22	fveq12d 5655	. . . . 5 ⊢ ((𝑠 = 𝑆 ∧ (1st ‘𝑏) = 𝐹 ∧ (2nd ‘𝑏) = 𝐿) → (𝑠‘(𝑥 + (1st ‘𝑏))) = (𝑆‘(𝑥 + 𝐹)))
24	21, 23	mpteq12dv 4176	. . . 4 ⊢ ((𝑠 = 𝑆 ∧ (1st ‘𝑏) = 𝐹 ∧ (2nd ‘𝑏) = 𝐿) → (𝑥 ∈ (0..^((2nd ‘𝑏) − (1st ‘𝑏))) ↦ (𝑠‘(𝑥 + (1st ‘𝑏)))) = (𝑥 ∈ (0..^(𝐿 − 𝐹)) ↦ (𝑆‘(𝑥 + 𝐹))))
25	19, 24	ifbieq1d 3632	. . 3 ⊢ ((𝑠 = 𝑆 ∧ (1st ‘𝑏) = 𝐹 ∧ (2nd ‘𝑏) = 𝐿) → if(((1st ‘𝑏)..^(2nd ‘𝑏)) ⊆ dom 𝑠, (𝑥 ∈ (0..^((2nd ‘𝑏) − (1st ‘𝑏))) ↦ (𝑠‘(𝑥 + (1st ‘𝑏)))), ∅) = if((𝐹..^𝐿) ⊆ dom 𝑆, (𝑥 ∈ (0..^(𝐿 − 𝐹)) ↦ (𝑆‘(𝑥 + 𝐹))), ∅))
26	3, 8, 13, 25	syl3anc 1274	. 2 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∧ (𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝐿⟩)) → if(((1st ‘𝑏)..^(2nd ‘𝑏)) ⊆ dom 𝑠, (𝑥 ∈ (0..^((2nd ‘𝑏) − (1st ‘𝑏))) ↦ (𝑠‘(𝑥 + (1st ‘𝑏)))), ∅) = if((𝐹..^𝐿) ⊆ dom 𝑆, (𝑥 ∈ (0..^(𝐿 − 𝐹)) ↦ (𝑆‘(𝑥 + 𝐹))), ∅))
27		elex 2815	. . 3 ⊢ (𝑆 ∈ 𝑉 → 𝑆 ∈ V)
28	27	3ad2ant1 1045	. 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → 𝑆 ∈ V)
29		opelxpi 4763	. . 3 ⊢ ((𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → ⟨𝐹, 𝐿⟩ ∈ (ℤ × ℤ))
30	29	3adant1 1042	. 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → ⟨𝐹, 𝐿⟩ ∈ (ℤ × ℤ))
31		0zd 9535	. . . . 5 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → 0 ∈ ℤ)
32		simp3 1026	. . . . . 6 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → 𝐿 ∈ ℤ)
33		simp2 1025	. . . . . 6 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → 𝐹 ∈ ℤ)
34	32, 33	zsubcld 9651	. . . . 5 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (𝐿 − 𝐹) ∈ ℤ)
35		fzofig 10740	. . . . 5 ⊢ ((0 ∈ ℤ ∧ (𝐿 − 𝐹) ∈ ℤ) → (0..^(𝐿 − 𝐹)) ∈ Fin)
36	31, 34, 35	syl2anc 411	. . . 4 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (0..^(𝐿 − 𝐹)) ∈ Fin)
37	36	mptexd 5891	. . 3 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (𝑥 ∈ (0..^(𝐿 − 𝐹)) ↦ (𝑆‘(𝑥 + 𝐹))) ∈ V)
38		0ex 4221	. . . 4 ⊢ ∅ ∈ V
39	38	a1i 9	. . 3 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → ∅ ∈ V)
40	37, 39	ifexd 4587	. 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → if((𝐹..^𝐿) ⊆ dom 𝑆, (𝑥 ∈ (0..^(𝐿 − 𝐹)) ↦ (𝑆‘(𝑥 + 𝐹))), ∅) ∈ V)
41	2, 26, 28, 30, 40	ovmpod 6159	1 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (𝑆 substr ⟨𝐹, 𝐿⟩) = if((𝐹..^𝐿) ⊆ dom 𝑆, (𝑥 ∈ (0..^(𝐿 − 𝐹)) ↦ (𝑆‘(𝑥 + 𝐹))), ∅))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1005   = wceq 1398   ∈ wcel 2202  Vcvv 2803   ⊆ wss 3201  ∅c0 3496  ifcif 3607  ⟨cop 3676   ↦ cmpt 4155   × cxp 4729  dom cdm 4731  ‘cfv 5333  (class class class)co 6028   ∈ cmpo 6030  1^st c1st 6310  2^nd c2nd 6311  Fincfn 6952  0cc0 8075   + caddc 8078   − cmin 8392  ℤcz 9523  ..^cfzo 10422   substr csubstr 11275

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-addcom 8175  ax-addass 8177  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-0id 8183  ax-rnegex 8184  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-1o 6625  df-er 6745  df-en 6953  df-fin 6955  df-pnf 8258  df-mnf 8259  df-xr 8260  df-ltxr 8261  df-le 8262  df-sub 8394  df-neg 8395  df-inn 9186  df-n0 9445  df-z 9524  df-uz 9800  df-fz 10289  df-fzo 10423  df-substr 11276

This theorem is referenced by:  swrd00g  11279  swrdclg  11280  swrdval2  11281  swrdlend  11288  swrdnd  11289  swrd0g  11290  pfxval  11304  fnpfx  11307