Theorem swrdval 14568

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 14566	. . 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 6826	. . . . 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 7943	. . . . 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 2786	. . 3 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∧ (𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝐿⟩)) → (1st ‘𝑏) = 𝐹)
9		fveq2 6826	. . . . 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 7944	. . . . 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 2786	. . 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 7371	. . . . 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 5852	. . . . 5 ⊢ ((𝑠 = 𝑆 ∧ (1st ‘𝑏) = 𝐹 ∧ (2nd ‘𝑏) = 𝐿) → dom 𝑠 = dom 𝑆)
19	16, 18	sseq12d 3971	. . . 4 ⊢ ((𝑠 = 𝑆 ∧ (1st ‘𝑏) = 𝐹 ∧ (2nd ‘𝑏) = 𝐿) → (((1st ‘𝑏)..^(2nd ‘𝑏)) ⊆ dom 𝑠 ↔ (𝐹..^𝐿) ⊆ dom 𝑆))
20	15, 14	oveq12d 7371	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ (1st ‘𝑏) = 𝐹 ∧ (2nd ‘𝑏) = 𝐿) → ((2nd ‘𝑏) − (1st ‘𝑏)) = (𝐿 − 𝐹))
21	20	oveq2d 7369	. . . . 5 ⊢ ((𝑠 = 𝑆 ∧ (1st ‘𝑏) = 𝐹 ∧ (2nd ‘𝑏) = 𝐿) → (0..^((2nd ‘𝑏) − (1st ‘𝑏))) = (0..^(𝐿 − 𝐹)))
22	14	oveq2d 7369	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ (1st ‘𝑏) = 𝐹 ∧ (2nd ‘𝑏) = 𝐿) → (𝑥 + (1st ‘𝑏)) = (𝑥 + 𝐹))
23	17, 22	fveq12d 6833	. . . . 5 ⊢ ((𝑠 = 𝑆 ∧ (1st ‘𝑏) = 𝐹 ∧ (2nd ‘𝑏) = 𝐿) → (𝑠‘(𝑥 + (1st ‘𝑏))) = (𝑆‘(𝑥 + 𝐹)))
24	21, 23	mpteq12dv 5182	. . . 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 4503	. . 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 3459	. . 3 ⊢ (𝑆 ∈ 𝑉 → 𝑆 ∈ V)
28	27	3ad2ant1 1133	. 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → 𝑆 ∈ V)
29		opelxpi 5660	. . 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 7386	. . . . 5 ⊢ (0..^(𝐿 − 𝐹)) ∈ V
32	31	mptex 7163	. . . 4 ⊢ (𝑥 ∈ (0..^(𝐿 − 𝐹)) ↦ (𝑆‘(𝑥 + 𝐹))) ∈ V
33		0ex 5249	. . . 4 ⊢ ∅ ∈ V
34	32, 33	ifex 4529	. . 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 7505	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 1540   ∈ wcel 2109  Vcvv 3438   ⊆ wss 3905  ∅c0 4286  ifcif 4478  ⟨cop 4585   ↦ cmpt 5176   × cxp 5621  dom cdm 5623  ‘cfv 6486  (class class class)co 7353   ∈ cmpo 7355  1^st c1st 7929  2^nd c2nd 7930  0cc0 11028   + caddc 11031   − cmin 11365  ℤcz 12489  ..^cfzo 13575   substr csubstr 14565

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7356  df-oprab 7357  df-mpo 7358  df-1st 7931  df-2nd 7932  df-substr 14566

This theorem is referenced by:  swrd00  14569  swrdcl  14570  swrdval2  14571  swrdlend  14578  swrdnd  14579  swrdnd2  14580  swrd0  14583  repswswrd  14708