Theorem swrdval 14604

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 14602	. . 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 776	. . 3 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∧ (𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝐿⟩)) → 𝑠 = 𝑆)
4		fveq2 6834	. . . . 5 ⊢ (𝑏 = ⟨𝐹, 𝐿⟩ → (1st ‘𝑏) = (1st ‘⟨𝐹, 𝐿⟩))
5	4	adantl 482	. . . 4 ⊢ ((𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝐿⟩) → (1st ‘𝑏) = (1st ‘⟨𝐹, 𝐿⟩))
6		op1stg 7950	. . . . 5 ⊢ ((𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (1st ‘⟨𝐹, 𝐿⟩) = 𝐹)
7	6	3adant1 1136	. . . 4 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (1st ‘⟨𝐹, 𝐿⟩) = 𝐹)
8	5, 7	sylan9eqr 2797	. . 3 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∧ (𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝐿⟩)) → (1st ‘𝑏) = 𝐹)
9		fveq2 6834	. . . . 5 ⊢ (𝑏 = ⟨𝐹, 𝐿⟩ → (2nd ‘𝑏) = (2nd ‘⟨𝐹, 𝐿⟩))
10	9	adantl 482	. . . 4 ⊢ ((𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝐿⟩) → (2nd ‘𝑏) = (2nd ‘⟨𝐹, 𝐿⟩))
11		op2ndg 7951	. . . . 5 ⊢ ((𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (2nd ‘⟨𝐹, 𝐿⟩) = 𝐿)
12	11	3adant1 1136	. . . 4 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (2nd ‘⟨𝐹, 𝐿⟩) = 𝐿)
13	10, 12	sylan9eqr 2797	. . 3 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∧ (𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝐿⟩)) → (2nd ‘𝑏) = 𝐿)
14		simp2 1143	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ (1st ‘𝑏) = 𝐹 ∧ (2nd ‘𝑏) = 𝐿) → (1st ‘𝑏) = 𝐹)
15		simp3 1144	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ (1st ‘𝑏) = 𝐹 ∧ (2nd ‘𝑏) = 𝐿) → (2nd ‘𝑏) = 𝐿)
16	14, 15	oveq12d 7381	. . . . 5 ⊢ ((𝑠 = 𝑆 ∧ (1st ‘𝑏) = 𝐹 ∧ (2nd ‘𝑏) = 𝐿) → ((1st ‘𝑏)..^(2nd ‘𝑏)) = (𝐹..^𝐿))
17		simp1 1142	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ (1st ‘𝑏) = 𝐹 ∧ (2nd ‘𝑏) = 𝐿) → 𝑠 = 𝑆)
18	17	dmeqd 5854	. . . . 5 ⊢ ((𝑠 = 𝑆 ∧ (1st ‘𝑏) = 𝐹 ∧ (2nd ‘𝑏) = 𝐿) → dom 𝑠 = dom 𝑆)
19	16, 18	sseq12d 3955	. . . 4 ⊢ ((𝑠 = 𝑆 ∧ (1st ‘𝑏) = 𝐹 ∧ (2nd ‘𝑏) = 𝐿) → (((1st ‘𝑏)..^(2nd ‘𝑏)) ⊆ dom 𝑠 ↔ (𝐹..^𝐿) ⊆ dom 𝑆))
20	15, 14	oveq12d 7381	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ (1st ‘𝑏) = 𝐹 ∧ (2nd ‘𝑏) = 𝐿) → ((2nd ‘𝑏) − (1st ‘𝑏)) = (𝐿 − 𝐹))
21	20	oveq2d 7379	. . . . 5 ⊢ ((𝑠 = 𝑆 ∧ (1st ‘𝑏) = 𝐹 ∧ (2nd ‘𝑏) = 𝐿) → (0..^((2nd ‘𝑏) − (1st ‘𝑏))) = (0..^(𝐿 − 𝐹)))
22	14	oveq2d 7379	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ (1st ‘𝑏) = 𝐹 ∧ (2nd ‘𝑏) = 𝐿) → (𝑥 + (1st ‘𝑏)) = (𝑥 + 𝐹))
23	17, 22	fveq12d 6841	. . . . 5 ⊢ ((𝑠 = 𝑆 ∧ (1st ‘𝑏) = 𝐹 ∧ (2nd ‘𝑏) = 𝐿) → (𝑠‘(𝑥 + (1st ‘𝑏))) = (𝑆‘(𝑥 + 𝐹)))
24	21, 23	mpteq12dv 5166	. . . 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 4486	. . 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 1379	. 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 3453	. . 3 ⊢ (𝑆 ∈ 𝑉 → 𝑆 ∈ V)
28	27	3ad2ant1 1139	. 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → 𝑆 ∈ V)
29		opelxpi 5662	. . 3 ⊢ ((𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → ⟨𝐹, 𝐿⟩ ∈ (ℤ × ℤ))
30	29	3adant1 1136	. 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → ⟨𝐹, 𝐿⟩ ∈ (ℤ × ℤ))
31		ovex 7396	. . . . 5 ⊢ (0..^(𝐿 − 𝐹)) ∈ V
32	31	mptex 7174	. . . 4 ⊢ (𝑥 ∈ (0..^(𝐿 − 𝐹)) ↦ (𝑆‘(𝑥 + 𝐹))) ∈ V
33		0ex 5236	. . . 4 ⊢ ∅ ∈ V
34	32, 33	ifex 4512	. . 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 7515	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 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  Vcvv 3432   ⊆ wss 3890  ∅c0 4268  ifcif 4461  ⟨cop 4568   ↦ cmpt 5160   × cxp 5623  dom cdm 5625  ‘cfv 6492  (class class class)co 7363   ∈ cmpo 7365  1^st c1st 7936  2^nd c2nd 7937  0cc0 11036   + caddc 11039   − cmin 11375  ℤcz 12522  ..^cfzo 13606   substr csubstr 14601

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7366  df-oprab 7367  df-mpo 7368  df-1st 7938  df-2nd 7939  df-substr 14602

This theorem is referenced by:  swrd00  14605  swrdcl  14606  swrdval2  14607  swrdlend  14614  swrdnd  14615  swrdnd2  14616  swrd0  14619  repswswrd  14744