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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 7439 | . . . . 5 ⊢ (𝑙 = 𝐿 → (0..^𝑙) = (0..^𝐿)) | |
2 | 1 | feq2d 6723 | . . . 4 ⊢ (𝑙 = 𝐿 → (𝑊:(0..^𝑙)⟶𝑆 ↔ 𝑊:(0..^𝐿)⟶𝑆)) |
3 | 2 | rspcev 3622 | . . 3 ⊢ ((𝐿 ∈ ℕ0 ∧ 𝑊:(0..^𝐿)⟶𝑆) → ∃𝑙 ∈ ℕ0 𝑊:(0..^𝑙)⟶𝑆) |
4 | 0nn0 12539 | . . . 4 ⊢ 0 ∈ ℕ0 | |
5 | fzo0n0 13752 | . . . . . . . . 9 ⊢ ((0..^𝐿) ≠ ∅ ↔ 𝐿 ∈ ℕ) | |
6 | nnnn0 12531 | . . . . . . . . 9 ⊢ (𝐿 ∈ ℕ → 𝐿 ∈ ℕ0) | |
7 | 5, 6 | sylbi 217 | . . . . . . . 8 ⊢ ((0..^𝐿) ≠ ∅ → 𝐿 ∈ ℕ0) |
8 | 7 | necon1bi 2967 | . . . . . . 7 ⊢ (¬ 𝐿 ∈ ℕ0 → (0..^𝐿) = ∅) |
9 | fzo0 13720 | . . . . . . 7 ⊢ (0..^0) = ∅ | |
10 | 8, 9 | eqtr4di 2793 | . . . . . 6 ⊢ (¬ 𝐿 ∈ ℕ0 → (0..^𝐿) = (0..^0)) |
11 | 10 | feq2d 6723 | . . . . 5 ⊢ (¬ 𝐿 ∈ ℕ0 → (𝑊:(0..^𝐿)⟶𝑆 ↔ 𝑊:(0..^0)⟶𝑆)) |
12 | 11 | biimpa 476 | . . . 4 ⊢ ((¬ 𝐿 ∈ ℕ0 ∧ 𝑊:(0..^𝐿)⟶𝑆) → 𝑊:(0..^0)⟶𝑆) |
13 | oveq2 7439 | . . . . . 6 ⊢ (𝑙 = 0 → (0..^𝑙) = (0..^0)) | |
14 | 13 | feq2d 6723 | . . . . 5 ⊢ (𝑙 = 0 → (𝑊:(0..^𝑙)⟶𝑆 ↔ 𝑊:(0..^0)⟶𝑆)) |
15 | 14 | rspcev 3622 | . . . 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 812 | . 2 ⊢ (𝑊:(0..^𝐿)⟶𝑆 → ∃𝑙 ∈ ℕ0 𝑊:(0..^𝑙)⟶𝑆) |
18 | iswrd 14551 | . 2 ⊢ (𝑊 ∈ Word 𝑆 ↔ ∃𝑙 ∈ ℕ0 𝑊:(0..^𝑙)⟶𝑆) | |
19 | 17, 18 | sylibr 234 | 1 ⊢ (𝑊:(0..^𝐿)⟶𝑆 → 𝑊 ∈ Word 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∃wrex 3068 ∅c0 4339 ⟶wf 6559 (class class class)co 7431 0cc0 11153 ℕcn 12264 ℕ0cn0 12524 ..^cfzo 13691 Word cword 14549 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-fzo 13692 df-word 14550 |
This theorem is referenced by: iswrdb 14555 snopiswrd 14558 iswrdsymb 14566 iswrddm0 14573 ffz0iswrd 14576 wrdnval 14580 wrdred1 14595 ccatcl 14609 swrdcl 14680 revcl 14796 repsw 14810 repsdf2 14813 cshf1 14845 wrdco 14867 wrdlen2i 14978 pmtrdifwrdellem1 19514 psgnunilem5 19527 ablfaclem2 20121 ablfac2 20124 wrdupgr 29117 wrdumgr 29129 crctcshtrl 29853 wlkiswwlks2lem5 29903 wlkiswwlksupgr2 29907 clwlkclwwlklem2a 30027 upgriseupth 30236 wrdres 32904 wrdpmcl 32907 ccatws1f1o 32921 ccatws1f1olast 32922 wrdpmtrlast 33096 cycpmconjslem1 33157 1arithidomlem1 33543 1arithidomlem2 33544 1arithidom 33545 subiwrd 34367 sseqp1 34377 ofcccat 34537 signstf 34560 signshwrd 34583 lpadlem1 34671 frlmfzowrd 42489 frlmvscadiccat 42493 grtriclwlk3 47850 |
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