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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 5586 | . . . 4 ⊢ (〈𝑆, 〈𝑋, 𝑋〉〉 ∈ (V × (ℤ × ℤ)) ↔ (𝑆 ∈ V ∧ 〈𝑋, 𝑋〉 ∈ (ℤ × ℤ))) | |
2 | opelxp 5586 | . . . . 5 ⊢ (〈𝑋, 𝑋〉 ∈ (ℤ × ℤ) ↔ (𝑋 ∈ ℤ ∧ 𝑋 ∈ ℤ)) | |
3 | swrdval 13999 | . . . . . . 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 13055 | . . . . . . . . . 10 ⊢ (𝑋..^𝑋) = ∅ | |
5 | 0ss 4350 | . . . . . . . . . 10 ⊢ ∅ ⊆ dom 𝑆 | |
6 | 4, 5 | eqsstri 4001 | . . . . . . . . 9 ⊢ (𝑋..^𝑋) ⊆ dom 𝑆 |
7 | 6 | iftruei 4474 | . . . . . . . 8 ⊢ if((𝑋..^𝑋) ⊆ dom 𝑆, (𝑥 ∈ (0..^(𝑋 − 𝑋)) ↦ (𝑆‘(𝑥 + 𝑋))), ∅) = (𝑥 ∈ (0..^(𝑋 − 𝑋)) ↦ (𝑆‘(𝑥 + 𝑋))) |
8 | zcn 11980 | . . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ ℤ → 𝑋 ∈ ℂ) | |
9 | 8 | subidd 10979 | . . . . . . . . . . . . 13 ⊢ (𝑋 ∈ ℤ → (𝑋 − 𝑋) = 0) |
10 | 9 | oveq2d 7166 | . . . . . . . . . . . 12 ⊢ (𝑋 ∈ ℤ → (0..^(𝑋 − 𝑋)) = (0..^0)) |
11 | 10 | 3ad2ant2 1130 | . . . . . . . . . . 11 ⊢ ((𝑆 ∈ V ∧ 𝑋 ∈ ℤ ∧ 𝑋 ∈ ℤ) → (0..^(𝑋 − 𝑋)) = (0..^0)) |
12 | fzo0 13055 | . . . . . . . . . . 11 ⊢ (0..^0) = ∅ | |
13 | 11, 12 | syl6eq 2872 | . . . . . . . . . 10 ⊢ ((𝑆 ∈ V ∧ 𝑋 ∈ ℤ ∧ 𝑋 ∈ ℤ) → (0..^(𝑋 − 𝑋)) = ∅) |
14 | 13 | mpteq1d 5148 | . . . . . . . . 9 ⊢ ((𝑆 ∈ V ∧ 𝑋 ∈ ℤ ∧ 𝑋 ∈ ℤ) → (𝑥 ∈ (0..^(𝑋 − 𝑋)) ↦ (𝑆‘(𝑥 + 𝑋))) = (𝑥 ∈ ∅ ↦ (𝑆‘(𝑥 + 𝑋)))) |
15 | mpt0 6485 | . . . . . . . . 9 ⊢ (𝑥 ∈ ∅ ↦ (𝑆‘(𝑥 + 𝑋))) = ∅ | |
16 | 14, 15 | syl6eq 2872 | . . . . . . . 8 ⊢ ((𝑆 ∈ V ∧ 𝑋 ∈ ℤ ∧ 𝑋 ∈ ℤ) → (𝑥 ∈ (0..^(𝑋 − 𝑋)) ↦ (𝑆‘(𝑥 + 𝑋))) = ∅) |
17 | 7, 16 | syl5eq 2868 | . . . . . . 7 ⊢ ((𝑆 ∈ V ∧ 𝑋 ∈ ℤ ∧ 𝑋 ∈ ℤ) → if((𝑋..^𝑋) ⊆ dom 𝑆, (𝑥 ∈ (0..^(𝑋 − 𝑋)) ↦ (𝑆‘(𝑥 + 𝑋))), ∅) = ∅) |
18 | 3, 17 | eqtrd 2856 | . . . . . 6 ⊢ ((𝑆 ∈ V ∧ 𝑋 ∈ ℤ ∧ 𝑋 ∈ ℤ) → (𝑆 substr 〈𝑋, 𝑋〉) = ∅) |
19 | 18 | 3expb 1116 | . . . . 5 ⊢ ((𝑆 ∈ V ∧ (𝑋 ∈ ℤ ∧ 𝑋 ∈ ℤ)) → (𝑆 substr 〈𝑋, 𝑋〉) = ∅) |
20 | 2, 19 | sylan2b 595 | . . . 4 ⊢ ((𝑆 ∈ V ∧ 〈𝑋, 𝑋〉 ∈ (ℤ × ℤ)) → (𝑆 substr 〈𝑋, 𝑋〉) = ∅) |
21 | 1, 20 | sylbi 219 | . . 3 ⊢ (〈𝑆, 〈𝑋, 𝑋〉〉 ∈ (V × (ℤ × ℤ)) → (𝑆 substr 〈𝑋, 𝑋〉) = ∅) |
22 | df-substr 13997 | . . . 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 7183 | . . . . . 6 ⊢ (0..^((2nd ‘𝑏) − (1st ‘𝑏))) ∈ V | |
24 | 23 | mptex 6980 | . . . . 5 ⊢ (𝑥 ∈ (0..^((2nd ‘𝑏) − (1st ‘𝑏))) ↦ (𝑠‘(𝑥 + (1st ‘𝑏)))) ∈ V |
25 | 0ex 5204 | . . . . 5 ⊢ ∅ ∈ V | |
26 | 24, 25 | ifex 4515 | . . . 4 ⊢ if(((1st ‘𝑏)..^(2nd ‘𝑏)) ⊆ dom 𝑠, (𝑥 ∈ (0..^((2nd ‘𝑏) − (1st ‘𝑏))) ↦ (𝑠‘(𝑥 + (1st ‘𝑏)))), ∅) ∈ V |
27 | 22, 26 | dmmpo 7763 | . . 3 ⊢ dom substr = (V × (ℤ × ℤ)) |
28 | 21, 27 | eleq2s 2931 | . 2 ⊢ (〈𝑆, 〈𝑋, 𝑋〉〉 ∈ dom substr → (𝑆 substr 〈𝑋, 𝑋〉) = ∅) |
29 | df-ov 7153 | . . 3 ⊢ (𝑆 substr 〈𝑋, 𝑋〉) = ( substr ‘〈𝑆, 〈𝑋, 𝑋〉〉) | |
30 | ndmfv 6695 | . . 3 ⊢ (¬ 〈𝑆, 〈𝑋, 𝑋〉〉 ∈ dom substr → ( substr ‘〈𝑆, 〈𝑋, 𝑋〉〉) = ∅) | |
31 | 29, 30 | syl5eq 2868 | . 2 ⊢ (¬ 〈𝑆, 〈𝑋, 𝑋〉〉 ∈ dom substr → (𝑆 substr 〈𝑋, 𝑋〉) = ∅) |
32 | 28, 31 | pm2.61i 184 | 1 ⊢ (𝑆 substr 〈𝑋, 𝑋〉) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 Vcvv 3495 ⊆ wss 3936 ∅c0 4291 ifcif 4467 〈cop 4567 ↦ cmpt 5139 × cxp 5548 dom cdm 5550 ‘cfv 6350 (class class class)co 7150 1st c1st 7681 2nd c2nd 7682 0cc0 10531 + caddc 10534 − cmin 10864 ℤcz 11975 ..^cfzo 13027 substr csubstr 13996 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 df-fzo 13028 df-substr 13997 |
This theorem is referenced by: pfx00 14030 swrdccatin1 14081 swrdccat3blem 14095 |
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