Theorem swrdval 14543

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 14541	. . 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 11	. 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 770	. . 3 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∧ (𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝐿⟩)) → 𝑠 = 𝑆)
4		fveq2 6817	. . . . 5 ⊢ (𝑏 = ⟨𝐹, 𝐿⟩ → (1st ‘𝑏) = (1st ‘⟨𝐹, 𝐿⟩))
5	4	adantl 481	. . . 4 ⊢ ((𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝐿⟩) → (1st ‘𝑏) = (1st ‘⟨𝐹, 𝐿⟩))
6		op1stg 7928	. . . . 5 ⊢ ((𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (1st ‘⟨𝐹, 𝐿⟩) = 𝐹)
7	6	3adant1 1130	. . . 4 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (1st ‘⟨𝐹, 𝐿⟩) = 𝐹)
8	5, 7	sylan9eqr 2787	. . 3 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∧ (𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝐿⟩)) → (1st ‘𝑏) = 𝐹)
9		fveq2 6817	. . . . 5 ⊢ (𝑏 = ⟨𝐹, 𝐿⟩ → (2nd ‘𝑏) = (2nd ‘⟨𝐹, 𝐿⟩))
10	9	adantl 481	. . . 4 ⊢ ((𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝐿⟩) → (2nd ‘𝑏) = (2nd ‘⟨𝐹, 𝐿⟩))
11		op2ndg 7929	. . . . 5 ⊢ ((𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (2nd ‘⟨𝐹, 𝐿⟩) = 𝐿)
12	11	3adant1 1130	. . . 4 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (2nd ‘⟨𝐹, 𝐿⟩) = 𝐿)
13	10, 12	sylan9eqr 2787	. . 3 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∧ (𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝐿⟩)) → (2nd ‘𝑏) = 𝐿)
14		simp2 1137	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ (1st ‘𝑏) = 𝐹 ∧ (2nd ‘𝑏) = 𝐿) → (1st ‘𝑏) = 𝐹)
15		simp3 1138	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ (1st ‘𝑏) = 𝐹 ∧ (2nd ‘𝑏) = 𝐿) → (2nd ‘𝑏) = 𝐿)
16	14, 15	oveq12d 7359	. . . . 5 ⊢ ((𝑠 = 𝑆 ∧ (1st ‘𝑏) = 𝐹 ∧ (2nd ‘𝑏) = 𝐿) → ((1st ‘𝑏)..^(2nd ‘𝑏)) = (𝐹..^𝐿))
17		simp1 1136	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ (1st ‘𝑏) = 𝐹 ∧ (2nd ‘𝑏) = 𝐿) → 𝑠 = 𝑆)
18	17	dmeqd 5843	. . . . 5 ⊢ ((𝑠 = 𝑆 ∧ (1st ‘𝑏) = 𝐹 ∧ (2nd ‘𝑏) = 𝐿) → dom 𝑠 = dom 𝑆)
19	16, 18	sseq12d 3966	. . . 4 ⊢ ((𝑠 = 𝑆 ∧ (1st ‘𝑏) = 𝐹 ∧ (2nd ‘𝑏) = 𝐿) → (((1st ‘𝑏)..^(2nd ‘𝑏)) ⊆ dom 𝑠 ↔ (𝐹..^𝐿) ⊆ dom 𝑆))
20	15, 14	oveq12d 7359	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ (1st ‘𝑏) = 𝐹 ∧ (2nd ‘𝑏) = 𝐿) → ((2nd ‘𝑏) − (1st ‘𝑏)) = (𝐿 − 𝐹))
21	20	oveq2d 7357	. . . . 5 ⊢ ((𝑠 = 𝑆 ∧ (1st ‘𝑏) = 𝐹 ∧ (2nd ‘𝑏) = 𝐿) → (0..^((2nd ‘𝑏) − (1st ‘𝑏))) = (0..^(𝐿 − 𝐹)))
22	14	oveq2d 7357	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ (1st ‘𝑏) = 𝐹 ∧ (2nd ‘𝑏) = 𝐿) → (𝑥 + (1st ‘𝑏)) = (𝑥 + 𝐹))
23	17, 22	fveq12d 6824	. . . . 5 ⊢ ((𝑠 = 𝑆 ∧ (1st ‘𝑏) = 𝐹 ∧ (2nd ‘𝑏) = 𝐿) → (𝑠‘(𝑥 + (1st ‘𝑏))) = (𝑆‘(𝑥 + 𝐹)))
24	21, 23	mpteq12dv 5176	. . . 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 4498	. . 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 1373	. 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 3455	. . 3 ⊢ (𝑆 ∈ 𝑉 → 𝑆 ∈ V)
28	27	3ad2ant1 1133	. 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → 𝑆 ∈ V)
29		opelxpi 5651	. . 3 ⊢ ((𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → ⟨𝐹, 𝐿⟩ ∈ (ℤ × ℤ))
30	29	3adant1 1130	. 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → ⟨𝐹, 𝐿⟩ ∈ (ℤ × ℤ))
31		ovex 7374	. . . . 5 ⊢ (0..^(𝐿 − 𝐹)) ∈ V
32	31	mptex 7152	. . . 4 ⊢ (𝑥 ∈ (0..^(𝐿 − 𝐹)) ↦ (𝑆‘(𝑥 + 𝐹))) ∈ V
33		0ex 5243	. . . 4 ⊢ ∅ ∈ V
34	32, 33	ifex 4524	. . 3 ⊢ if((𝐹..^𝐿) ⊆ dom 𝑆, (𝑥 ∈ (0..^(𝐿 − 𝐹)) ↦ (𝑆‘(𝑥 + 𝐹))), ∅) ∈ V
35	34	a1i 11	. 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → if((𝐹..^𝐿) ⊆ dom 𝑆, (𝑥 ∈ (0..^(𝐿 − 𝐹)) ↦ (𝑆‘(𝑥 + 𝐹))), ∅) ∈ V)
36	2, 26, 28, 30, 35	ovmpod 7493	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 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2110  Vcvv 3434   ⊆ wss 3900  ∅c0 4281  ifcif 4473  ⟨cop 4580   ↦ cmpt 5170   × cxp 5612  dom cdm 5614  ‘cfv 6477  (class class class)co 7341   ∈ cmpo 7343  1^st c1st 7914  2^nd c2nd 7915  0cc0 10998   + caddc 11001   − cmin 11336  ℤcz 12460  ..^cfzo 13546   substr csubstr 14540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pr 5368  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3345  df-rab 3394  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4282  df-if 4474  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6433  df-fun 6479  df-fn 6480  df-f 6481  df-f1 6482  df-fo 6483  df-f1o 6484  df-fv 6485  df-ov 7344  df-oprab 7345  df-mpo 7346  df-1st 7916  df-2nd 7917  df-substr 14541

This theorem is referenced by:  swrd00  14544  swrdcl  14545  swrdval2  14546  swrdlend  14553  swrdnd  14554  swrdnd2  14555  swrd0  14558  repswswrd  14683