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Theorem wwlknbp1 28195

Description: Other basic properties of a walk of a fixed length as word. (Contributed by AV, 8-Mar-2022.)

**Assertion**
Ref	Expression
wwlknbp1	⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝑁 ∈ ℕ₀ ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑁 + 1)))

Proof of Theorem wwlknbp1 Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2738	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2	1	wwlknbp 28193	. 2 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝐺 ∈ V ∧ 𝑁 ∈ ℕ₀ ∧ 𝑊 ∈ Word (Vtx‘𝐺)))
3		eqid 2738	. . 3 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
4	1, 3	wwlknp 28194	. 2 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
5		simpl 483	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺))) → 𝑁 ∈ ℕ₀)
6		simpr1 1193	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺))) → 𝑊 ∈ Word (Vtx‘𝐺))
7		simpr2 1194	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺))) → (♯‘𝑊) = (𝑁 + 1))
8	5, 6, 7	3jca 1127	. . . 4 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺))) → (𝑁 ∈ ℕ₀ ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑁 + 1)))
9	8	ex 413	. . 3 ⊢ (𝑁 ∈ ℕ₀ → ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺)) → (𝑁 ∈ ℕ₀ ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑁 + 1))))
10	9	3ad2ant2 1133	. 2 ⊢ ((𝐺 ∈ V ∧ 𝑁 ∈ ℕ₀ ∧ 𝑊 ∈ Word (Vtx‘𝐺)) → ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺)) → (𝑁 ∈ ℕ₀ ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑁 + 1))))
11	2, 4, 10	sylc 65	1 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝑁 ∈ ℕ₀ ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑁 + 1)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ∀wral 3064 Vcvv 3430 {cpr 4564 ‘cfv 6427 (class class class)co 7268 0cc0 10859 1c1 10860 + caddc 10862 ℕ₀cn0 12221 ..^cfzo 13370 ♯chash 14032 Word cword 14205 Vtxcvtx 27354 Edgcedg 27405 WWalksN cwwlksn 28177

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-rep 5209 ax-sep 5222 ax-nul 5229 ax-pow 5287 ax-pr 5351 ax-un 7579 ax-cnex 10915 ax-resscn 10916 ax-1cn 10917 ax-icn 10918 ax-addcl 10919 ax-addrcl 10920 ax-mulcl 10921 ax-mulrcl 10922 ax-mulcom 10923 ax-addass 10924 ax-mulass 10925 ax-distr 10926 ax-i2m1 10927 ax-1ne0 10928 ax-1rid 10929 ax-rnegex 10930 ax-rrecex 10931 ax-cnre 10932 ax-pre-lttri 10933 ax-pre-lttrn 10934 ax-pre-ltadd 10935 ax-pre-mulgt0 10936

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rab 3073 df-v 3432 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4258 df-if 4461 df-pw 4536 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4841 df-int 4881 df-iun 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5485 df-eprel 5491 df-po 5499 df-so 5500 df-fr 5540 df-we 5542 df-xp 5591 df-rel 5592 df-cnv 5593 df-co 5594 df-dm 5595 df-rn 5596 df-res 5597 df-ima 5598 df-pred 6196 df-ord 6263 df-on 6264 df-lim 6265 df-suc 6266 df-iota 6385 df-fun 6429 df-fn 6430 df-f 6431 df-f1 6432 df-fo 6433 df-f1o 6434 df-fv 6435 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-om 7704 df-1st 7821 df-2nd 7822 df-frecs 8085 df-wrecs 8116 df-recs 8190 df-rdg 8229 df-1o 8285 df-er 8486 df-map 8605 df-en 8722 df-dom 8723 df-sdom 8724 df-fin 8725 df-card 9685 df-pnf 10999 df-mnf 11000 df-xr 11001 df-ltxr 11002 df-le 11003 df-sub 11195 df-neg 11196 df-nn 11962 df-n0 12222 df-z 12308 df-uz 12571 df-fz 13228 df-fzo 13371 df-hash 14033 df-word 14206 df-wwlks 28181 df-wwlksn 28182

This theorem is referenced by: wwlknvtx 28196 wwlknllvtx 28197 wwlknlsw 28198 wwlksnextproplem1 28260 wwlksnwwlksnon 28266 wspthsnwspthsnon 28267 wspn0 28275 elwwlks2s3 28302 wwlks2onv 28304 elwwlks2ons3im 28305 wwlksext2clwwlk 28407 numclwwlk2lem1lem 28692 numclwlk2lem2f1o 28729

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