Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > isupwlk | Structured version Visualization version GIF version |
Description: Properties of a pair of functions to be a simple walk. (Contributed by Alexander van der Vekens, 20-Oct-2017.) (Revised by AV, 28-Dec-2020.) |
Ref | Expression |
---|---|
upwlksfval.v | ⊢ 𝑉 = (Vtx‘𝐺) |
upwlksfval.i | ⊢ 𝐼 = (iEdg‘𝐺) |
Ref | Expression |
---|---|
isupwlk | ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐹 ∈ 𝑈 ∧ 𝑃 ∈ 𝑍) → (𝐹(UPWalks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-br 5069 | . . 3 ⊢ (𝐹(UPWalks‘𝐺)𝑃 ↔ 〈𝐹, 𝑃〉 ∈ (UPWalks‘𝐺)) | |
2 | upwlksfval.v | . . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺) | |
3 | upwlksfval.i | . . . . . 6 ⊢ 𝐼 = (iEdg‘𝐺) | |
4 | 2, 3 | upwlksfval 44017 | . . . . 5 ⊢ (𝐺 ∈ 𝑊 → (UPWalks‘𝐺) = {〈𝑓, 𝑝〉 ∣ (𝑓 ∈ Word dom 𝐼 ∧ 𝑝:(0...(♯‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))(𝐼‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))})}) |
5 | 4 | 3ad2ant1 1129 | . . . 4 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐹 ∈ 𝑈 ∧ 𝑃 ∈ 𝑍) → (UPWalks‘𝐺) = {〈𝑓, 𝑝〉 ∣ (𝑓 ∈ Word dom 𝐼 ∧ 𝑝:(0...(♯‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))(𝐼‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))})}) |
6 | 5 | eleq2d 2900 | . . 3 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐹 ∈ 𝑈 ∧ 𝑃 ∈ 𝑍) → (〈𝐹, 𝑃〉 ∈ (UPWalks‘𝐺) ↔ 〈𝐹, 𝑃〉 ∈ {〈𝑓, 𝑝〉 ∣ (𝑓 ∈ Word dom 𝐼 ∧ 𝑝:(0...(♯‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))(𝐼‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))})})) |
7 | 1, 6 | syl5bb 285 | . 2 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐹 ∈ 𝑈 ∧ 𝑃 ∈ 𝑍) → (𝐹(UPWalks‘𝐺)𝑃 ↔ 〈𝐹, 𝑃〉 ∈ {〈𝑓, 𝑝〉 ∣ (𝑓 ∈ Word dom 𝐼 ∧ 𝑝:(0...(♯‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))(𝐼‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))})})) |
8 | eleq1 2902 | . . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓 ∈ Word dom 𝐼 ↔ 𝐹 ∈ Word dom 𝐼)) | |
9 | 8 | adantr 483 | . . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (𝑓 ∈ Word dom 𝐼 ↔ 𝐹 ∈ Word dom 𝐼)) |
10 | simpr 487 | . . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → 𝑝 = 𝑃) | |
11 | fveq2 6672 | . . . . . . . 8 ⊢ (𝑓 = 𝐹 → (♯‘𝑓) = (♯‘𝐹)) | |
12 | 11 | oveq2d 7174 | . . . . . . 7 ⊢ (𝑓 = 𝐹 → (0...(♯‘𝑓)) = (0...(♯‘𝐹))) |
13 | 12 | adantr 483 | . . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (0...(♯‘𝑓)) = (0...(♯‘𝐹))) |
14 | 10, 13 | feq12d 6504 | . . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (𝑝:(0...(♯‘𝑓))⟶𝑉 ↔ 𝑃:(0...(♯‘𝐹))⟶𝑉)) |
15 | 11 | oveq2d 7174 | . . . . . . 7 ⊢ (𝑓 = 𝐹 → (0..^(♯‘𝑓)) = (0..^(♯‘𝐹))) |
16 | 15 | adantr 483 | . . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (0..^(♯‘𝑓)) = (0..^(♯‘𝐹))) |
17 | fveq1 6671 | . . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑘) = (𝐹‘𝑘)) | |
18 | 17 | fveq2d 6676 | . . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝐼‘(𝑓‘𝑘)) = (𝐼‘(𝐹‘𝑘))) |
19 | fveq1 6671 | . . . . . . . 8 ⊢ (𝑝 = 𝑃 → (𝑝‘𝑘) = (𝑃‘𝑘)) | |
20 | fveq1 6671 | . . . . . . . 8 ⊢ (𝑝 = 𝑃 → (𝑝‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1))) | |
21 | 19, 20 | preq12d 4679 | . . . . . . 7 ⊢ (𝑝 = 𝑃 → {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))} = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) |
22 | 18, 21 | eqeqan12d 2840 | . . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → ((𝐼‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))} ↔ (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})) |
23 | 16, 22 | raleqbidv 3403 | . . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (∀𝑘 ∈ (0..^(♯‘𝑓))(𝐼‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))} ↔ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})) |
24 | 9, 14, 23 | 3anbi123d 1432 | . . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → ((𝑓 ∈ Word dom 𝐼 ∧ 𝑝:(0...(♯‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))(𝐼‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))}) ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))) |
25 | 24 | opelopabga 5422 | . . 3 ⊢ ((𝐹 ∈ 𝑈 ∧ 𝑃 ∈ 𝑍) → (〈𝐹, 𝑃〉 ∈ {〈𝑓, 𝑝〉 ∣ (𝑓 ∈ Word dom 𝐼 ∧ 𝑝:(0...(♯‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))(𝐼‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))})} ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))) |
26 | 25 | 3adant1 1126 | . 2 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐹 ∈ 𝑈 ∧ 𝑃 ∈ 𝑍) → (〈𝐹, 𝑃〉 ∈ {〈𝑓, 𝑝〉 ∣ (𝑓 ∈ Word dom 𝐼 ∧ 𝑝:(0...(♯‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))(𝐼‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))})} ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))) |
27 | 7, 26 | bitrd 281 | 1 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐹 ∈ 𝑈 ∧ 𝑃 ∈ 𝑍) → (𝐹(UPWalks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∀wral 3140 {cpr 4571 〈cop 4575 class class class wbr 5068 {copab 5130 dom cdm 5557 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 0cc0 10539 1c1 10540 + caddc 10542 ...cfz 12895 ..^cfzo 13036 ♯chash 13693 Word cword 13864 Vtxcvtx 26783 iEdgciedg 26784 UPWalkscupwlks 44015 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-fzo 13037 df-hash 13694 df-word 13865 df-upwlks 44016 |
This theorem is referenced by: isupwlkg 44019 upwlkwlk 44021 upgrwlkupwlk 44022 upgrisupwlkALT 44024 |
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