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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 7416 | . . . . 5 ⊢ (𝑙 = 𝐿 → (0..^𝑙) = (0..^𝐿)) | |
2 | 1 | feq2d 6703 | . . . 4 ⊢ (𝑙 = 𝐿 → (𝑊:(0..^𝑙)⟶𝑆 ↔ 𝑊:(0..^𝐿)⟶𝑆)) |
3 | 2 | rspcev 3612 | . . 3 ⊢ ((𝐿 ∈ ℕ0 ∧ 𝑊:(0..^𝐿)⟶𝑆) → ∃𝑙 ∈ ℕ0 𝑊:(0..^𝑙)⟶𝑆) |
4 | 0nn0 12486 | . . . 4 ⊢ 0 ∈ ℕ0 | |
5 | fzo0n0 13683 | . . . . . . . . 9 ⊢ ((0..^𝐿) ≠ ∅ ↔ 𝐿 ∈ ℕ) | |
6 | nnnn0 12478 | . . . . . . . . 9 ⊢ (𝐿 ∈ ℕ → 𝐿 ∈ ℕ0) | |
7 | 5, 6 | sylbi 216 | . . . . . . . 8 ⊢ ((0..^𝐿) ≠ ∅ → 𝐿 ∈ ℕ0) |
8 | 7 | necon1bi 2969 | . . . . . . 7 ⊢ (¬ 𝐿 ∈ ℕ0 → (0..^𝐿) = ∅) |
9 | fzo0 13655 | . . . . . . 7 ⊢ (0..^0) = ∅ | |
10 | 8, 9 | eqtr4di 2790 | . . . . . 6 ⊢ (¬ 𝐿 ∈ ℕ0 → (0..^𝐿) = (0..^0)) |
11 | 10 | feq2d 6703 | . . . . 5 ⊢ (¬ 𝐿 ∈ ℕ0 → (𝑊:(0..^𝐿)⟶𝑆 ↔ 𝑊:(0..^0)⟶𝑆)) |
12 | 11 | biimpa 477 | . . . 4 ⊢ ((¬ 𝐿 ∈ ℕ0 ∧ 𝑊:(0..^𝐿)⟶𝑆) → 𝑊:(0..^0)⟶𝑆) |
13 | oveq2 7416 | . . . . . 6 ⊢ (𝑙 = 0 → (0..^𝑙) = (0..^0)) | |
14 | 13 | feq2d 6703 | . . . . 5 ⊢ (𝑙 = 0 → (𝑊:(0..^𝑙)⟶𝑆 ↔ 𝑊:(0..^0)⟶𝑆)) |
15 | 14 | rspcev 3612 | . . . 4 ⊢ ((0 ∈ ℕ0 ∧ 𝑊:(0..^0)⟶𝑆) → ∃𝑙 ∈ ℕ0 𝑊:(0..^𝑙)⟶𝑆) |
16 | 4, 12, 15 | sylancr 587 | . . 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 14465 | . 2 ⊢ (𝑊 ∈ Word 𝑆 ↔ ∃𝑙 ∈ ℕ0 𝑊:(0..^𝑙)⟶𝑆) | |
19 | 17, 18 | sylibr 233 | 1 ⊢ (𝑊:(0..^𝐿)⟶𝑆 → 𝑊 ∈ Word 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∃wrex 3070 ∅c0 4322 ⟶wf 6539 (class class class)co 7408 0cc0 11109 ℕcn 12211 ℕ0cn0 12471 ..^cfzo 13626 Word cword 14463 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-1st 7974 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-n0 12472 df-z 12558 df-uz 12822 df-fz 13484 df-fzo 13627 df-word 14464 |
This theorem is referenced by: iswrdb 14469 snopiswrd 14472 iswrdsymb 14480 iswrddm0 14487 ffz0iswrd 14490 wrdnval 14494 wrdred1 14509 ccatcl 14523 swrdcl 14594 revcl 14710 repsw 14724 repsdf2 14727 cshf1 14759 wrdco 14781 wrdlen2i 14892 pmtrdifwrdellem1 19348 psgnunilem5 19361 ablfaclem2 19955 ablfac2 19958 wrdupgr 28342 wrdumgr 28354 crctcshtrl 29074 wlkiswwlks2lem5 29124 wlkiswwlksupgr2 29128 clwlkclwwlklem2a 29248 upgriseupth 29457 wrdres 32098 cycpmconjslem1 32308 subiwrd 33379 sseqp1 33389 ofcccat 33549 signstf 33572 signshwrd 33595 lpadlem1 33684 frlmfzowrd 41078 frlmvscadiccat 41082 |
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