Theorem swrdval 13704

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 13702	. . 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 789	. . 3 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∧ (𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝐿⟩)) → 𝑠 = 𝑆)
4		fveq2 6434	. . . . 5 ⊢ (𝑏 = ⟨𝐹, 𝐿⟩ → (1st ‘𝑏) = (1st ‘⟨𝐹, 𝐿⟩))
5	4	adantl 475	. . . 4 ⊢ ((𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝐿⟩) → (1st ‘𝑏) = (1st ‘⟨𝐹, 𝐿⟩))
6		op1stg 7441	. . . . 5 ⊢ ((𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (1st ‘⟨𝐹, 𝐿⟩) = 𝐹)
7	6	3adant1 1166	. . . 4 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (1st ‘⟨𝐹, 𝐿⟩) = 𝐹)
8	5, 7	sylan9eqr 2884	. . 3 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∧ (𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝐿⟩)) → (1st ‘𝑏) = 𝐹)
9		fveq2 6434	. . . . 5 ⊢ (𝑏 = ⟨𝐹, 𝐿⟩ → (2nd ‘𝑏) = (2nd ‘⟨𝐹, 𝐿⟩))
10	9	adantl 475	. . . 4 ⊢ ((𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝐿⟩) → (2nd ‘𝑏) = (2nd ‘⟨𝐹, 𝐿⟩))
11		op2ndg 7442	. . . . 5 ⊢ ((𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (2nd ‘⟨𝐹, 𝐿⟩) = 𝐿)
12	11	3adant1 1166	. . . 4 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (2nd ‘⟨𝐹, 𝐿⟩) = 𝐿)
13	10, 12	sylan9eqr 2884	. . 3 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∧ (𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝐿⟩)) → (2nd ‘𝑏) = 𝐿)
14		simp2 1173	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ (1st ‘𝑏) = 𝐹 ∧ (2nd ‘𝑏) = 𝐿) → (1st ‘𝑏) = 𝐹)
15		simp3 1174	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ (1st ‘𝑏) = 𝐹 ∧ (2nd ‘𝑏) = 𝐿) → (2nd ‘𝑏) = 𝐿)
16	14, 15	oveq12d 6924	. . . . 5 ⊢ ((𝑠 = 𝑆 ∧ (1st ‘𝑏) = 𝐹 ∧ (2nd ‘𝑏) = 𝐿) → ((1st ‘𝑏)..^(2nd ‘𝑏)) = (𝐹..^𝐿))
17		simp1 1172	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ (1st ‘𝑏) = 𝐹 ∧ (2nd ‘𝑏) = 𝐿) → 𝑠 = 𝑆)
18	17	dmeqd 5559	. . . . 5 ⊢ ((𝑠 = 𝑆 ∧ (1st ‘𝑏) = 𝐹 ∧ (2nd ‘𝑏) = 𝐿) → dom 𝑠 = dom 𝑆)
19	16, 18	sseq12d 3860	. . . 4 ⊢ ((𝑠 = 𝑆 ∧ (1st ‘𝑏) = 𝐹 ∧ (2nd ‘𝑏) = 𝐿) → (((1st ‘𝑏)..^(2nd ‘𝑏)) ⊆ dom 𝑠 ↔ (𝐹..^𝐿) ⊆ dom 𝑆))
20	15, 14	oveq12d 6924	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ (1st ‘𝑏) = 𝐹 ∧ (2nd ‘𝑏) = 𝐿) → ((2nd ‘𝑏) − (1st ‘𝑏)) = (𝐿 − 𝐹))
21	20	oveq2d 6922	. . . . 5 ⊢ ((𝑠 = 𝑆 ∧ (1st ‘𝑏) = 𝐹 ∧ (2nd ‘𝑏) = 𝐿) → (0..^((2nd ‘𝑏) − (1st ‘𝑏))) = (0..^(𝐿 − 𝐹)))
22	14	oveq2d 6922	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ (1st ‘𝑏) = 𝐹 ∧ (2nd ‘𝑏) = 𝐿) → (𝑥 + (1st ‘𝑏)) = (𝑥 + 𝐹))
23	17, 22	fveq12d 6441	. . . . 5 ⊢ ((𝑠 = 𝑆 ∧ (1st ‘𝑏) = 𝐹 ∧ (2nd ‘𝑏) = 𝐿) → (𝑠‘(𝑥 + (1st ‘𝑏))) = (𝑆‘(𝑥 + 𝐹)))
24	21, 23	mpteq12dv 4957	. . . 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 4330	. . 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 1496	. 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 3430	. . 3 ⊢ (𝑆 ∈ 𝑉 → 𝑆 ∈ V)
28	27	3ad2ant1 1169	. 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → 𝑆 ∈ V)
29		opelxpi 5380	. . 3 ⊢ ((𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → ⟨𝐹, 𝐿⟩ ∈ (ℤ × ℤ))
30	29	3adant1 1166	. 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → ⟨𝐹, 𝐿⟩ ∈ (ℤ × ℤ))
31		ovex 6938	. . . . 5 ⊢ (0..^(𝐿 − 𝐹)) ∈ V
32	31	mptex 6743	. . . 4 ⊢ (𝑥 ∈ (0..^(𝐿 − 𝐹)) ↦ (𝑆‘(𝑥 + 𝐹))) ∈ V
33		0ex 5015	. . . 4 ⊢ ∅ ∈ V
34	32, 33	ifex 4355	. . 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	ovmpt2d 7049	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 386   ∧ w3a 1113   = wceq 1658   ∈ wcel 2166  Vcvv 3415   ⊆ wss 3799  ∅c0 4145  ifcif 4307  ⟨cop 4404   ↦ cmpt 4953   × cxp 5341  dom cdm 5343  ‘cfv 6124  (class class class)co 6906   ↦ cmpt2 6908  1^st c1st 7427  2^nd c2nd 7428  0cc0 10253   + caddc 10256   − cmin 10586  ℤcz 11705  ..^cfzo 12761   substr csubstr 13701

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-rep 4995  ax-sep 5006  ax-nul 5014  ax-pow 5066  ax-pr 5128  ax-un 7210

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-ral 3123  df-rex 3124  df-reu 3125  df-rab 3127  df-v 3417  df-sbc 3664  df-csb 3759  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4660  df-iun 4743  df-br 4875  df-opab 4937  df-mpt 4954  df-id 5251  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-res 5355  df-ima 5356  df-iota 6087  df-fun 6126  df-fn 6127  df-f 6128  df-f1 6129  df-fo 6130  df-f1o 6131  df-fv 6132  df-ov 6909  df-oprab 6910  df-mpt2 6911  df-1st 7429  df-2nd 7430  df-substr 13702

This theorem is referenced by:  swrd00  13705  swrdcl  13706  swrdval2  13707  swrdlend  13719  swrdnd  13720  swrdnd2  13721  swrd0  13724  repswswrd  13901