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Mirrors > Home > MPE Home > Th. List > iswwlksnx | Structured version Visualization version GIF version |
Description: Properties of a word to represent a walk of a fixed length, definition of WWalks expanded. (Contributed by AV, 28-Apr-2021.) |
Ref | Expression |
---|---|
iswwlksnx.v | ⊢ 𝑉 = (Vtx‘𝐺) |
iswwlksnx.e | ⊢ 𝐸 = (Edg‘𝐺) |
Ref | Expression |
---|---|
iswwlksnx | ⊢ (𝑁 ∈ ℕ0 → (𝑊 ∈ (𝑁 WWalksN 𝐺) ↔ (𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((♯‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ∧ (♯‘𝑊) = (𝑁 + 1)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iswwlksn 29868 | . 2 ⊢ (𝑁 ∈ ℕ0 → (𝑊 ∈ (𝑁 WWalksN 𝐺) ↔ (𝑊 ∈ (WWalks‘𝐺) ∧ (♯‘𝑊) = (𝑁 + 1)))) | |
2 | iswwlksnx.v | . . . . . . 7 ⊢ 𝑉 = (Vtx‘𝐺) | |
3 | iswwlksnx.e | . . . . . . 7 ⊢ 𝐸 = (Edg‘𝐺) | |
4 | 2, 3 | iswwlks 29866 | . . . . . 6 ⊢ (𝑊 ∈ (WWalks‘𝐺) ↔ (𝑊 ≠ ∅ ∧ 𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((♯‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸)) |
5 | df-3an 1088 | . . . . . . 7 ⊢ ((𝑊 ≠ ∅ ∧ 𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((♯‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸) ↔ ((𝑊 ≠ ∅ ∧ 𝑊 ∈ Word 𝑉) ∧ ∀𝑖 ∈ (0..^((♯‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸)) | |
6 | nn0p1gt0 12553 | . . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ0 → 0 < (𝑁 + 1)) | |
7 | 6 | gt0ne0d 11825 | . . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ≠ 0) |
8 | 7 | adantr 480 | . . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ0 ∧ (♯‘𝑊) = (𝑁 + 1)) → (𝑁 + 1) ≠ 0) |
9 | neeq1 3001 | . . . . . . . . . . . . 13 ⊢ ((♯‘𝑊) = (𝑁 + 1) → ((♯‘𝑊) ≠ 0 ↔ (𝑁 + 1) ≠ 0)) | |
10 | 9 | adantl 481 | . . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ0 ∧ (♯‘𝑊) = (𝑁 + 1)) → ((♯‘𝑊) ≠ 0 ↔ (𝑁 + 1) ≠ 0)) |
11 | 8, 10 | mpbird 257 | . . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ0 ∧ (♯‘𝑊) = (𝑁 + 1)) → (♯‘𝑊) ≠ 0) |
12 | hasheq0 14399 | . . . . . . . . . . . 12 ⊢ (𝑊 ∈ Word 𝑉 → ((♯‘𝑊) = 0 ↔ 𝑊 = ∅)) | |
13 | 12 | necon3bid 2983 | . . . . . . . . . . 11 ⊢ (𝑊 ∈ Word 𝑉 → ((♯‘𝑊) ≠ 0 ↔ 𝑊 ≠ ∅)) |
14 | 11, 13 | syl5ibcom 245 | . . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ0 ∧ (♯‘𝑊) = (𝑁 + 1)) → (𝑊 ∈ Word 𝑉 → 𝑊 ≠ ∅)) |
15 | 14 | pm4.71rd 562 | . . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ (♯‘𝑊) = (𝑁 + 1)) → (𝑊 ∈ Word 𝑉 ↔ (𝑊 ≠ ∅ ∧ 𝑊 ∈ Word 𝑉))) |
16 | 15 | bicomd 223 | . . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ (♯‘𝑊) = (𝑁 + 1)) → ((𝑊 ≠ ∅ ∧ 𝑊 ∈ Word 𝑉) ↔ 𝑊 ∈ Word 𝑉)) |
17 | 16 | anbi1d 631 | . . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ (♯‘𝑊) = (𝑁 + 1)) → (((𝑊 ≠ ∅ ∧ 𝑊 ∈ Word 𝑉) ∧ ∀𝑖 ∈ (0..^((♯‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸) ↔ (𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((♯‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸))) |
18 | 5, 17 | bitrid 283 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ (♯‘𝑊) = (𝑁 + 1)) → ((𝑊 ≠ ∅ ∧ 𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((♯‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸) ↔ (𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((♯‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸))) |
19 | 4, 18 | bitrid 283 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ (♯‘𝑊) = (𝑁 + 1)) → (𝑊 ∈ (WWalks‘𝐺) ↔ (𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((♯‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸))) |
20 | 19 | ex 412 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → ((♯‘𝑊) = (𝑁 + 1) → (𝑊 ∈ (WWalks‘𝐺) ↔ (𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((♯‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸)))) |
21 | 20 | pm5.32rd 578 | . . 3 ⊢ (𝑁 ∈ ℕ0 → ((𝑊 ∈ (WWalks‘𝐺) ∧ (♯‘𝑊) = (𝑁 + 1)) ↔ ((𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((♯‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸) ∧ (♯‘𝑊) = (𝑁 + 1)))) |
22 | df-3an 1088 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((♯‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ∧ (♯‘𝑊) = (𝑁 + 1)) ↔ ((𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((♯‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸) ∧ (♯‘𝑊) = (𝑁 + 1))) | |
23 | 21, 22 | bitr4di 289 | . 2 ⊢ (𝑁 ∈ ℕ0 → ((𝑊 ∈ (WWalks‘𝐺) ∧ (♯‘𝑊) = (𝑁 + 1)) ↔ (𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((♯‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ∧ (♯‘𝑊) = (𝑁 + 1)))) |
24 | 1, 23 | bitrd 279 | 1 ⊢ (𝑁 ∈ ℕ0 → (𝑊 ∈ (𝑁 WWalksN 𝐺) ↔ (𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((♯‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ∧ (♯‘𝑊) = (𝑁 + 1)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∀wral 3059 ∅c0 4339 {cpr 4633 ‘cfv 6563 (class class class)co 7431 0cc0 11153 1c1 11154 + caddc 11156 − cmin 11490 ℕ0cn0 12524 ..^cfzo 13691 ♯chash 14366 Word cword 14549 Vtxcvtx 29028 Edgcedg 29079 WWalkscwwlks 29855 WWalksN cwwlksn 29856 |
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-int 4952 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-1o 8505 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-card 9977 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-hash 14367 df-word 14550 df-wwlks 29860 df-wwlksn 29861 |
This theorem is referenced by: clwwlknwwlksn 30067 wwlksubclwwlk 30087 |
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