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Mirrors > Home > MPE Home > Th. List > Mathboxes > 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 41041 | . 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 41039 | . . . . . 6 ⊢ (𝑊 ∈ (WWalkS‘𝐺) ↔ (𝑊 ≠ ∅ ∧ 𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸)) |
5 | df-3an 1033 | . . . . . . 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 11199 | . . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ0 → 0 < (𝑁 + 1)) | |
7 | 6 | gt0ne0d 10471 | . . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ≠ 0) |
8 | 7 | adantr 480 | . . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ0 ∧ (#‘𝑊) = (𝑁 + 1)) → (𝑁 + 1) ≠ 0) |
9 | neeq1 2844 | . . . . . . . . . . . . 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 246 | . . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ0 ∧ (#‘𝑊) = (𝑁 + 1)) → (#‘𝑊) ≠ 0) |
12 | hasheq0 13015 | . . . . . . . . . . . 12 ⊢ (𝑊 ∈ Word 𝑉 → ((#‘𝑊) = 0 ↔ 𝑊 = ∅)) | |
13 | 12 | necon3bid 2826 | . . . . . . . . . . 11 ⊢ (𝑊 ∈ Word 𝑉 → ((#‘𝑊) ≠ 0 ↔ 𝑊 ≠ ∅)) |
14 | 11, 13 | syl5ibcom 234 | . . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ0 ∧ (#‘𝑊) = (𝑁 + 1)) → (𝑊 ∈ Word 𝑉 → 𝑊 ≠ ∅)) |
15 | 14 | pm4.71rd 665 | . . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ (#‘𝑊) = (𝑁 + 1)) → (𝑊 ∈ Word 𝑉 ↔ (𝑊 ≠ ∅ ∧ 𝑊 ∈ Word 𝑉))) |
16 | 15 | bicomd 212 | . . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ (#‘𝑊) = (𝑁 + 1)) → ((𝑊 ≠ ∅ ∧ 𝑊 ∈ Word 𝑉) ↔ 𝑊 ∈ Word 𝑉)) |
17 | 16 | anbi1d 737 | . . . . . . 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 | syl5bb 271 | . . . . . 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 | syl5bb 271 | . . . . 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 449 | . . . 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 670 | . . 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 1033 | . . 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 | syl6bbr 277 | . 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 267 | 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 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ∅c0 3874 {cpr 4127 ‘cfv 5804 (class class class)co 6549 0cc0 9815 1c1 9816 + caddc 9818 − cmin 10145 ℕ0cn0 11169 ..^cfzo 12334 #chash 12979 Word cword 13146 Vtxcvtx 25673 Edgcedga 25792 WWalkScwwlks 41028 WWalkSN cwwlksn 41029 |
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-int 4411 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-1o 7447 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 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-hash 12980 df-word 13154 df-wwlks 41033 df-wwlksn 41034 |
This theorem is referenced by: wwlksubclwwlks 41232 |
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