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Mirrors > Home > MPE Home > Th. List > iswrdi | Structured version Visualization version GIF version |
Description: A zero-based sequence is a word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.) |
Ref | Expression |
---|---|
iswrdi | ⊢ (𝑊:(0..^𝐿)⟶𝑆 → 𝑊 ∈ Word 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 7167 | . . . . 5 ⊢ (𝑙 = 𝐿 → (0..^𝑙) = (0..^𝐿)) | |
2 | 1 | feq2d 6503 | . . . 4 ⊢ (𝑙 = 𝐿 → (𝑊:(0..^𝑙)⟶𝑆 ↔ 𝑊:(0..^𝐿)⟶𝑆)) |
3 | 2 | rspcev 3626 | . . 3 ⊢ ((𝐿 ∈ ℕ0 ∧ 𝑊:(0..^𝐿)⟶𝑆) → ∃𝑙 ∈ ℕ0 𝑊:(0..^𝑙)⟶𝑆) |
4 | 0nn0 11915 | . . . 4 ⊢ 0 ∈ ℕ0 | |
5 | fzo0n0 13092 | . . . . . . . . 9 ⊢ ((0..^𝐿) ≠ ∅ ↔ 𝐿 ∈ ℕ) | |
6 | nnnn0 11907 | . . . . . . . . 9 ⊢ (𝐿 ∈ ℕ → 𝐿 ∈ ℕ0) | |
7 | 5, 6 | sylbi 219 | . . . . . . . 8 ⊢ ((0..^𝐿) ≠ ∅ → 𝐿 ∈ ℕ0) |
8 | 7 | necon1bi 3047 | . . . . . . 7 ⊢ (¬ 𝐿 ∈ ℕ0 → (0..^𝐿) = ∅) |
9 | fzo0 13064 | . . . . . . 7 ⊢ (0..^0) = ∅ | |
10 | 8, 9 | syl6eqr 2877 | . . . . . 6 ⊢ (¬ 𝐿 ∈ ℕ0 → (0..^𝐿) = (0..^0)) |
11 | 10 | feq2d 6503 | . . . . 5 ⊢ (¬ 𝐿 ∈ ℕ0 → (𝑊:(0..^𝐿)⟶𝑆 ↔ 𝑊:(0..^0)⟶𝑆)) |
12 | 11 | biimpa 479 | . . . 4 ⊢ ((¬ 𝐿 ∈ ℕ0 ∧ 𝑊:(0..^𝐿)⟶𝑆) → 𝑊:(0..^0)⟶𝑆) |
13 | oveq2 7167 | . . . . . 6 ⊢ (𝑙 = 0 → (0..^𝑙) = (0..^0)) | |
14 | 13 | feq2d 6503 | . . . . 5 ⊢ (𝑙 = 0 → (𝑊:(0..^𝑙)⟶𝑆 ↔ 𝑊:(0..^0)⟶𝑆)) |
15 | 14 | rspcev 3626 | . . . 4 ⊢ ((0 ∈ ℕ0 ∧ 𝑊:(0..^0)⟶𝑆) → ∃𝑙 ∈ ℕ0 𝑊:(0..^𝑙)⟶𝑆) |
16 | 4, 12, 15 | sylancr 589 | . . 3 ⊢ ((¬ 𝐿 ∈ ℕ0 ∧ 𝑊:(0..^𝐿)⟶𝑆) → ∃𝑙 ∈ ℕ0 𝑊:(0..^𝑙)⟶𝑆) |
17 | 3, 16 | pm2.61ian 810 | . 2 ⊢ (𝑊:(0..^𝐿)⟶𝑆 → ∃𝑙 ∈ ℕ0 𝑊:(0..^𝑙)⟶𝑆) |
18 | iswrd 13866 | . 2 ⊢ (𝑊 ∈ Word 𝑆 ↔ ∃𝑙 ∈ ℕ0 𝑊:(0..^𝑙)⟶𝑆) | |
19 | 17, 18 | sylibr 236 | 1 ⊢ (𝑊:(0..^𝐿)⟶𝑆 → 𝑊 ∈ Word 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ≠ wne 3019 ∃wrex 3142 ∅c0 4294 ⟶wf 6354 (class class class)co 7159 0cc0 10540 ℕcn 11641 ℕ0cn0 11900 ..^cfzo 13036 Word cword 13864 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-fzo 13037 df-word 13865 |
This theorem is referenced by: iswrdb 13870 snopiswrd 13873 wrdvOLD 13881 iswrdsymb 13884 iswrddm0 13891 ffz0iswrd 13894 ffz0iswrdOLD 13895 wrdnval 13899 wrdred1 13915 ccatcl 13929 swrdcl 14010 revcl 14126 repsw 14140 repsdf2 14143 cshf1 14175 wrdco 14196 wrdlen2i 14307 pmtrdifwrdellem1 18612 psgnunilem5 18625 ablfaclem2 19211 ablfac2 19214 wrdupgr 26873 wrdumgr 26885 crctcshtrl 27604 wlkiswwlks2lem5 27654 wlkiswwlksupgr2 27658 clwlkclwwlklem2a 27779 upgriseupth 27989 wrdres 30617 cycpmconjslem1 30800 subiwrd 31647 sseqp1 31657 ofcccat 31817 signstf 31840 signshwrd 31863 lpadlem1 31952 frlmfzowrd 39147 frlmvscadiccat 39151 |
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