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Mirrors > Home > MPE Home > Th. List > 1ewlk | Structured version Visualization version GIF version |
Description: A sequence of 1 edge is an s-walk of edges for all s. (Contributed by AV, 5-Jan-2021.) |
Ref | Expression |
---|---|
1ewlk | ⊢ ((𝐺 ∈ V ∧ 𝑆 ∈ ℕ0* ∧ 𝐼 ∈ dom (iEdg‘𝐺)) → 〈“𝐼”〉 ∈ (𝐺 EdgWalks 𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | s1cl 14548 | . . 3 ⊢ (𝐼 ∈ dom (iEdg‘𝐺) → 〈“𝐼”〉 ∈ Word dom (iEdg‘𝐺)) | |
2 | 1 | 3ad2ant3 1136 | . 2 ⊢ ((𝐺 ∈ V ∧ 𝑆 ∈ ℕ0* ∧ 𝐼 ∈ dom (iEdg‘𝐺)) → 〈“𝐼”〉 ∈ Word dom (iEdg‘𝐺)) |
3 | ral0 4511 | . . . 4 ⊢ ∀𝑘 ∈ ∅ 𝑆 ≤ (♯‘(((iEdg‘𝐺)‘(〈“𝐼”〉‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(〈“𝐼”〉‘𝑘)))) | |
4 | s1len 14552 | . . . . . . . 8 ⊢ (♯‘〈“𝐼”〉) = 1 | |
5 | 4 | oveq2i 7415 | . . . . . . 7 ⊢ (1..^(♯‘〈“𝐼”〉)) = (1..^1) |
6 | fzo0 13652 | . . . . . . 7 ⊢ (1..^1) = ∅ | |
7 | 5, 6 | eqtri 2761 | . . . . . 6 ⊢ (1..^(♯‘〈“𝐼”〉)) = ∅ |
8 | 7 | a1i 11 | . . . . 5 ⊢ (𝐼 ∈ dom (iEdg‘𝐺) → (1..^(♯‘〈“𝐼”〉)) = ∅) |
9 | 8 | raleqdv 3326 | . . . 4 ⊢ (𝐼 ∈ dom (iEdg‘𝐺) → (∀𝑘 ∈ (1..^(♯‘〈“𝐼”〉))𝑆 ≤ (♯‘(((iEdg‘𝐺)‘(〈“𝐼”〉‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(〈“𝐼”〉‘𝑘)))) ↔ ∀𝑘 ∈ ∅ 𝑆 ≤ (♯‘(((iEdg‘𝐺)‘(〈“𝐼”〉‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(〈“𝐼”〉‘𝑘)))))) |
10 | 3, 9 | mpbiri 258 | . . 3 ⊢ (𝐼 ∈ dom (iEdg‘𝐺) → ∀𝑘 ∈ (1..^(♯‘〈“𝐼”〉))𝑆 ≤ (♯‘(((iEdg‘𝐺)‘(〈“𝐼”〉‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(〈“𝐼”〉‘𝑘))))) |
11 | 10 | 3ad2ant3 1136 | . 2 ⊢ ((𝐺 ∈ V ∧ 𝑆 ∈ ℕ0* ∧ 𝐼 ∈ dom (iEdg‘𝐺)) → ∀𝑘 ∈ (1..^(♯‘〈“𝐼”〉))𝑆 ≤ (♯‘(((iEdg‘𝐺)‘(〈“𝐼”〉‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(〈“𝐼”〉‘𝑘))))) |
12 | eqid 2733 | . . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
13 | 12 | isewlk 28839 | . . 3 ⊢ ((𝐺 ∈ V ∧ 𝑆 ∈ ℕ0* ∧ 〈“𝐼”〉 ∈ Word dom (iEdg‘𝐺)) → (〈“𝐼”〉 ∈ (𝐺 EdgWalks 𝑆) ↔ (〈“𝐼”〉 ∈ Word dom (iEdg‘𝐺) ∧ ∀𝑘 ∈ (1..^(♯‘〈“𝐼”〉))𝑆 ≤ (♯‘(((iEdg‘𝐺)‘(〈“𝐼”〉‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(〈“𝐼”〉‘𝑘))))))) |
14 | 1, 13 | syl3an3 1166 | . 2 ⊢ ((𝐺 ∈ V ∧ 𝑆 ∈ ℕ0* ∧ 𝐼 ∈ dom (iEdg‘𝐺)) → (〈“𝐼”〉 ∈ (𝐺 EdgWalks 𝑆) ↔ (〈“𝐼”〉 ∈ Word dom (iEdg‘𝐺) ∧ ∀𝑘 ∈ (1..^(♯‘〈“𝐼”〉))𝑆 ≤ (♯‘(((iEdg‘𝐺)‘(〈“𝐼”〉‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(〈“𝐼”〉‘𝑘))))))) |
15 | 2, 11, 14 | mpbir2and 712 | 1 ⊢ ((𝐺 ∈ V ∧ 𝑆 ∈ ℕ0* ∧ 𝐼 ∈ dom (iEdg‘𝐺)) → 〈“𝐼”〉 ∈ (𝐺 EdgWalks 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∀wral 3062 Vcvv 3475 ∩ cin 3946 ∅c0 4321 class class class wbr 5147 dom cdm 5675 ‘cfv 6540 (class class class)co 7404 1c1 11107 ≤ cle 11245 − cmin 11440 ℕ0*cxnn0 12540 ..^cfzo 13623 ♯chash 14286 Word cword 14460 〈“cs1 14541 iEdgciedg 28237 EdgWalks cewlks 28832 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7851 df-1st 7970 df-2nd 7971 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-n0 12469 df-z 12555 df-uz 12819 df-fz 13481 df-fzo 13624 df-hash 14287 df-word 14461 df-s1 14542 df-ewlks 28835 |
This theorem is referenced by: (None) |
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