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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 6557 | . . . . 5 ⊢ (𝑙 = 𝐿 → (0..^𝑙) = (0..^𝐿)) | |
2 | 1 | feq2d 5944 | . . . 4 ⊢ (𝑙 = 𝐿 → (𝑊:(0..^𝑙)⟶𝑆 ↔ 𝑊:(0..^𝐿)⟶𝑆)) |
3 | 2 | rspcev 3282 | . . 3 ⊢ ((𝐿 ∈ ℕ0 ∧ 𝑊:(0..^𝐿)⟶𝑆) → ∃𝑙 ∈ ℕ0 𝑊:(0..^𝑙)⟶𝑆) |
4 | 0nn0 11184 | . . . 4 ⊢ 0 ∈ ℕ0 | |
5 | fzo0n0 12387 | . . . . . . . . 9 ⊢ ((0..^𝐿) ≠ ∅ ↔ 𝐿 ∈ ℕ) | |
6 | nnnn0 11176 | . . . . . . . . 9 ⊢ (𝐿 ∈ ℕ → 𝐿 ∈ ℕ0) | |
7 | 5, 6 | sylbi 206 | . . . . . . . 8 ⊢ ((0..^𝐿) ≠ ∅ → 𝐿 ∈ ℕ0) |
8 | 7 | necon1bi 2810 | . . . . . . 7 ⊢ (¬ 𝐿 ∈ ℕ0 → (0..^𝐿) = ∅) |
9 | fzo0 12361 | . . . . . . 7 ⊢ (0..^0) = ∅ | |
10 | 8, 9 | syl6eqr 2662 | . . . . . 6 ⊢ (¬ 𝐿 ∈ ℕ0 → (0..^𝐿) = (0..^0)) |
11 | 10 | feq2d 5944 | . . . . 5 ⊢ (¬ 𝐿 ∈ ℕ0 → (𝑊:(0..^𝐿)⟶𝑆 ↔ 𝑊:(0..^0)⟶𝑆)) |
12 | 11 | biimpa 500 | . . . 4 ⊢ ((¬ 𝐿 ∈ ℕ0 ∧ 𝑊:(0..^𝐿)⟶𝑆) → 𝑊:(0..^0)⟶𝑆) |
13 | oveq2 6557 | . . . . . 6 ⊢ (𝑙 = 0 → (0..^𝑙) = (0..^0)) | |
14 | 13 | feq2d 5944 | . . . . 5 ⊢ (𝑙 = 0 → (𝑊:(0..^𝑙)⟶𝑆 ↔ 𝑊:(0..^0)⟶𝑆)) |
15 | 14 | rspcev 3282 | . . . 4 ⊢ ((0 ∈ ℕ0 ∧ 𝑊:(0..^0)⟶𝑆) → ∃𝑙 ∈ ℕ0 𝑊:(0..^𝑙)⟶𝑆) |
16 | 4, 12, 15 | sylancr 694 | . . 3 ⊢ ((¬ 𝐿 ∈ ℕ0 ∧ 𝑊:(0..^𝐿)⟶𝑆) → ∃𝑙 ∈ ℕ0 𝑊:(0..^𝑙)⟶𝑆) |
17 | 3, 16 | pm2.61ian 827 | . 2 ⊢ (𝑊:(0..^𝐿)⟶𝑆 → ∃𝑙 ∈ ℕ0 𝑊:(0..^𝑙)⟶𝑆) |
18 | iswrd 13162 | . 2 ⊢ (𝑊 ∈ Word 𝑆 ↔ ∃𝑙 ∈ ℕ0 𝑊:(0..^𝑙)⟶𝑆) | |
19 | 17, 18 | sylibr 223 | 1 ⊢ (𝑊:(0..^𝐿)⟶𝑆 → 𝑊 ∈ Word 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∃wrex 2897 ∅c0 3874 ⟶wf 5800 (class class class)co 6549 0cc0 9815 ℕcn 10897 ℕ0cn0 11169 ..^cfzo 12334 Word cword 13146 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-word 13154 |
This theorem is referenced by: iswrdb 13166 snopiswrd 13169 wrdv 13175 iswrdsymb 13177 iswrddm0 13184 ffz0iswrd 13187 wrdnval 13190 ccatcl 13212 swrdcl 13271 revcl 13361 repsw 13373 repsdf2 13376 cshf1 13407 wrdco 13428 wrdlen2i 13534 pmtrdifwrdellem1 17724 psgnunilem5 17737 ablfaclem2 18308 ablfac2 18311 wrdupgr 25752 wrdumgr 25763 wrdumgra 25845 wlkntrllem1 26089 is2wlk 26095 constr2wlk 26128 redwlk 26136 constr3trllem1 26178 wlkiswwlk2lem5 26223 clwlkisclwwlklem2a 26313 clwlkfclwwlk2wrd 26367 clwlkf1clwwlklem3 26375 eupatrl 26495 subiwrd 29774 sseqp1 29784 wrdres 29943 ofcccat 29946 signstf 29969 signshwrd 29992 wrdred1 40240 crctcshtrl 41026 1wlkiswwlks2lem5 41070 1wlkiswwlksupgr2 41074 clwlkclwwlklem2a 41207 clwlksfclwwlk2wrd 41265 clwlksf1clwwlklem3 41274 upgriseupth 41375 |
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