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Mirrors > Home > MPE Home > Th. List > swrd00 | Structured version Visualization version GIF version |
Description: A zero length substring. (Contributed by Stefan O'Rear, 27-Aug-2015.) |
Ref | Expression |
---|---|
swrd00 | ⊢ (𝑆 substr ⟨𝑋, 𝑋⟩) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelxp 5714 | . . . 4 ⊢ (⟨𝑆, ⟨𝑋, 𝑋⟩⟩ ∈ (V × (ℤ × ℤ)) ↔ (𝑆 ∈ V ∧ ⟨𝑋, 𝑋⟩ ∈ (ℤ × ℤ))) | |
2 | opelxp 5714 | . . . . 5 ⊢ (⟨𝑋, 𝑋⟩ ∈ (ℤ × ℤ) ↔ (𝑋 ∈ ℤ ∧ 𝑋 ∈ ℤ)) | |
3 | swrdval 14626 | . . . . . . 7 ⊢ ((𝑆 ∈ V ∧ 𝑋 ∈ ℤ ∧ 𝑋 ∈ ℤ) → (𝑆 substr ⟨𝑋, 𝑋⟩) = if((𝑋..^𝑋) ⊆ dom 𝑆, (𝑥 ∈ (0..^(𝑋 − 𝑋)) ↦ (𝑆‘(𝑥 + 𝑋))), ∅)) | |
4 | fzo0 13689 | . . . . . . . . . 10 ⊢ (𝑋..^𝑋) = ∅ | |
5 | 0ss 4397 | . . . . . . . . . 10 ⊢ ∅ ⊆ dom 𝑆 | |
6 | 4, 5 | eqsstri 4014 | . . . . . . . . 9 ⊢ (𝑋..^𝑋) ⊆ dom 𝑆 |
7 | 6 | iftruei 4536 | . . . . . . . 8 ⊢ if((𝑋..^𝑋) ⊆ dom 𝑆, (𝑥 ∈ (0..^(𝑋 − 𝑋)) ↦ (𝑆‘(𝑥 + 𝑋))), ∅) = (𝑥 ∈ (0..^(𝑋 − 𝑋)) ↦ (𝑆‘(𝑥 + 𝑋))) |
8 | zcn 12594 | . . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ ℤ → 𝑋 ∈ ℂ) | |
9 | 8 | subidd 11590 | . . . . . . . . . . . . 13 ⊢ (𝑋 ∈ ℤ → (𝑋 − 𝑋) = 0) |
10 | 9 | oveq2d 7436 | . . . . . . . . . . . 12 ⊢ (𝑋 ∈ ℤ → (0..^(𝑋 − 𝑋)) = (0..^0)) |
11 | 10 | 3ad2ant2 1132 | . . . . . . . . . . 11 ⊢ ((𝑆 ∈ V ∧ 𝑋 ∈ ℤ ∧ 𝑋 ∈ ℤ) → (0..^(𝑋 − 𝑋)) = (0..^0)) |
12 | fzo0 13689 | . . . . . . . . . . 11 ⊢ (0..^0) = ∅ | |
13 | 11, 12 | eqtrdi 2784 | . . . . . . . . . 10 ⊢ ((𝑆 ∈ V ∧ 𝑋 ∈ ℤ ∧ 𝑋 ∈ ℤ) → (0..^(𝑋 − 𝑋)) = ∅) |
14 | 13 | mpteq1d 5243 | . . . . . . . . 9 ⊢ ((𝑆 ∈ V ∧ 𝑋 ∈ ℤ ∧ 𝑋 ∈ ℤ) → (𝑥 ∈ (0..^(𝑋 − 𝑋)) ↦ (𝑆‘(𝑥 + 𝑋))) = (𝑥 ∈ ∅ ↦ (𝑆‘(𝑥 + 𝑋)))) |
15 | mpt0 6697 | . . . . . . . . 9 ⊢ (𝑥 ∈ ∅ ↦ (𝑆‘(𝑥 + 𝑋))) = ∅ | |
16 | 14, 15 | eqtrdi 2784 | . . . . . . . 8 ⊢ ((𝑆 ∈ V ∧ 𝑋 ∈ ℤ ∧ 𝑋 ∈ ℤ) → (𝑥 ∈ (0..^(𝑋 − 𝑋)) ↦ (𝑆‘(𝑥 + 𝑋))) = ∅) |
17 | 7, 16 | eqtrid 2780 | . . . . . . 7 ⊢ ((𝑆 ∈ V ∧ 𝑋 ∈ ℤ ∧ 𝑋 ∈ ℤ) → if((𝑋..^𝑋) ⊆ dom 𝑆, (𝑥 ∈ (0..^(𝑋 − 𝑋)) ↦ (𝑆‘(𝑥 + 𝑋))), ∅) = ∅) |
18 | 3, 17 | eqtrd 2768 | . . . . . 6 ⊢ ((𝑆 ∈ V ∧ 𝑋 ∈ ℤ ∧ 𝑋 ∈ ℤ) → (𝑆 substr ⟨𝑋, 𝑋⟩) = ∅) |
19 | 18 | 3expb 1118 | . . . . 5 ⊢ ((𝑆 ∈ V ∧ (𝑋 ∈ ℤ ∧ 𝑋 ∈ ℤ)) → (𝑆 substr ⟨𝑋, 𝑋⟩) = ∅) |
20 | 2, 19 | sylan2b 593 | . . . 4 ⊢ ((𝑆 ∈ V ∧ ⟨𝑋, 𝑋⟩ ∈ (ℤ × ℤ)) → (𝑆 substr ⟨𝑋, 𝑋⟩) = ∅) |
21 | 1, 20 | sylbi 216 | . . 3 ⊢ (⟨𝑆, ⟨𝑋, 𝑋⟩⟩ ∈ (V × (ℤ × ℤ)) → (𝑆 substr ⟨𝑋, 𝑋⟩) = ∅) |
22 | df-substr 14624 | . . . 4 ⊢ substr = (𝑠 ∈ V, 𝑏 ∈ (ℤ × ℤ) ↦ if(((1st ‘𝑏)..^(2nd ‘𝑏)) ⊆ dom 𝑠, (𝑥 ∈ (0..^((2nd ‘𝑏) − (1st ‘𝑏))) ↦ (𝑠‘(𝑥 + (1st ‘𝑏)))), ∅)) | |
23 | ovex 7453 | . . . . . 6 ⊢ (0..^((2nd ‘𝑏) − (1st ‘𝑏))) ∈ V | |
24 | 23 | mptex 7235 | . . . . 5 ⊢ (𝑥 ∈ (0..^((2nd ‘𝑏) − (1st ‘𝑏))) ↦ (𝑠‘(𝑥 + (1st ‘𝑏)))) ∈ V |
25 | 0ex 5307 | . . . . 5 ⊢ ∅ ∈ V | |
26 | 24, 25 | ifex 4579 | . . . 4 ⊢ if(((1st ‘𝑏)..^(2nd ‘𝑏)) ⊆ dom 𝑠, (𝑥 ∈ (0..^((2nd ‘𝑏) − (1st ‘𝑏))) ↦ (𝑠‘(𝑥 + (1st ‘𝑏)))), ∅) ∈ V |
27 | 22, 26 | dmmpo 8075 | . . 3 ⊢ dom substr = (V × (ℤ × ℤ)) |
28 | 21, 27 | eleq2s 2847 | . 2 ⊢ (⟨𝑆, ⟨𝑋, 𝑋⟩⟩ ∈ dom substr → (𝑆 substr ⟨𝑋, 𝑋⟩) = ∅) |
29 | df-ov 7423 | . . 3 ⊢ (𝑆 substr ⟨𝑋, 𝑋⟩) = ( substr ‘⟨𝑆, ⟨𝑋, 𝑋⟩⟩) | |
30 | ndmfv 6932 | . . 3 ⊢ (¬ ⟨𝑆, ⟨𝑋, 𝑋⟩⟩ ∈ dom substr → ( substr ‘⟨𝑆, ⟨𝑋, 𝑋⟩⟩) = ∅) | |
31 | 29, 30 | eqtrid 2780 | . 2 ⊢ (¬ ⟨𝑆, ⟨𝑋, 𝑋⟩⟩ ∈ dom substr → (𝑆 substr ⟨𝑋, 𝑋⟩) = ∅) |
32 | 28, 31 | pm2.61i 182 | 1 ⊢ (𝑆 substr ⟨𝑋, 𝑋⟩) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 Vcvv 3471 ⊆ wss 3947 ∅c0 4323 ifcif 4529 ⟨cop 4635 ↦ cmpt 5231 × cxp 5676 dom cdm 5678 ‘cfv 6548 (class class class)co 7420 1st c1st 7991 2nd c2nd 7992 0cc0 11139 + caddc 11142 − cmin 11475 ℤcz 12589 ..^cfzo 13660 substr csubstr 14623 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-n0 12504 df-z 12590 df-uz 12854 df-fz 13518 df-fzo 13661 df-substr 14624 |
This theorem is referenced by: pfx00 14657 swrdccatin1 14708 swrdccat3blem 14722 |
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