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Mirrors > Home > MPE Home > Th. List > clwlkcompim | Structured version Visualization version GIF version |
Description: Implications for the properties of the components of a closed walk. (Contributed by Alexander van der Vekens, 24-Jun-2018.) |
Ref | Expression |
---|---|
clwlkcomp.1 | ⊢ 𝐹 = (1st ‘𝑊) |
clwlkcomp.2 | ⊢ 𝑃 = (2nd ‘𝑊) |
Ref | Expression |
---|---|
clwlkcompim | ⊢ (𝑊 ∈ (𝑉 ClWalks 𝐸) → ((𝐹 ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉) ∧ (∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ∧ (𝑃‘0) = (𝑃‘(#‘𝐹))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-clwlk 26278 | . . . 4 ⊢ ClWalks = (𝑣 ∈ V, 𝑒 ∈ V ↦ {〈𝑓, 𝑝〉 ∣ (𝑓(𝑣 Walks 𝑒)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))}) | |
2 | vex 3176 | . . . . 5 ⊢ 𝑣 ∈ V | |
3 | vex 3176 | . . . . 5 ⊢ 𝑒 ∈ V | |
4 | clwlk 26281 | . . . . . 6 ⊢ ((𝑣 ∈ V ∧ 𝑒 ∈ V) → (𝑣 ClWalks 𝑒) = {〈𝑓, 𝑝〉 ∣ (𝑓(𝑣 Walks 𝑒)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))}) | |
5 | ovex 6577 | . . . . . 6 ⊢ (𝑣 ClWalks 𝑒) ∈ V | |
6 | 4, 5 | syl6eqelr 2697 | . . . . 5 ⊢ ((𝑣 ∈ V ∧ 𝑒 ∈ V) → {〈𝑓, 𝑝〉 ∣ (𝑓(𝑣 Walks 𝑒)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))} ∈ V) |
7 | 2, 3, 6 | mp2an 704 | . . . 4 ⊢ {〈𝑓, 𝑝〉 ∣ (𝑓(𝑣 Walks 𝑒)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))} ∈ V |
8 | oveq12 6558 | . . . . . . 7 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝑣 Walks 𝑒) = (𝑉 Walks 𝐸)) | |
9 | 8 | breqd 4594 | . . . . . 6 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝑓(𝑣 Walks 𝑒)𝑝 ↔ 𝑓(𝑉 Walks 𝐸)𝑝)) |
10 | 9 | anbi1d 737 | . . . . 5 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → ((𝑓(𝑣 Walks 𝑒)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓))) ↔ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓))))) |
11 | 10 | opabbidv 4648 | . . . 4 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → {〈𝑓, 𝑝〉 ∣ (𝑓(𝑣 Walks 𝑒)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))} = {〈𝑓, 𝑝〉 ∣ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))}) |
12 | 1, 7, 11 | elovmpt2 6777 | . . 3 ⊢ (𝑊 ∈ (𝑉 ClWalks 𝐸) ↔ (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑊 ∈ {〈𝑓, 𝑝〉 ∣ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))})) |
13 | elopaelxp 5114 | . . . 4 ⊢ (𝑊 ∈ {〈𝑓, 𝑝〉 ∣ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))} → 𝑊 ∈ (V × V)) | |
14 | clwlkcomp.1 | . . . . . 6 ⊢ 𝐹 = (1st ‘𝑊) | |
15 | clwlkcomp.2 | . . . . . 6 ⊢ 𝑃 = (2nd ‘𝑊) | |
16 | 14, 15 | clwlkcomp 26291 | . . . . 5 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑊 ∈ (V × V)) → (𝑊 ∈ (𝑉 ClWalks 𝐸) ↔ ((𝐹 ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉) ∧ (∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ∧ (𝑃‘0) = (𝑃‘(#‘𝐹)))))) |
17 | 16 | biimpd 218 | . . . 4 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑊 ∈ (V × V)) → (𝑊 ∈ (𝑉 ClWalks 𝐸) → ((𝐹 ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉) ∧ (∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ∧ (𝑃‘0) = (𝑃‘(#‘𝐹)))))) |
18 | 13, 17 | syl3an3 1353 | . . 3 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑊 ∈ {〈𝑓, 𝑝〉 ∣ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))}) → (𝑊 ∈ (𝑉 ClWalks 𝐸) → ((𝐹 ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉) ∧ (∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ∧ (𝑃‘0) = (𝑃‘(#‘𝐹)))))) |
19 | 12, 18 | sylbi 206 | . 2 ⊢ (𝑊 ∈ (𝑉 ClWalks 𝐸) → (𝑊 ∈ (𝑉 ClWalks 𝐸) → ((𝐹 ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉) ∧ (∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ∧ (𝑃‘0) = (𝑃‘(#‘𝐹)))))) |
20 | 19 | pm2.43i 50 | 1 ⊢ (𝑊 ∈ (𝑉 ClWalks 𝐸) → ((𝐹 ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉) ∧ (∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ∧ (𝑃‘0) = (𝑃‘(#‘𝐹))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∀wral 2896 Vcvv 3173 {cpr 4127 class class class wbr 4583 {copab 4642 × cxp 5036 dom cdm 5038 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 1st c1st 7057 2nd c2nd 7058 0cc0 9815 1c1 9816 + caddc 9818 ...cfz 12197 ..^cfzo 12334 #chash 12979 Word cword 13146 Walks cwalk 26026 ClWalks cclwlk 26275 |
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-oadd 7451 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-wlk 26036 df-clwlk 26278 |
This theorem is referenced by: clwlkfclwwlk2wrd 26367 clwlkfclwwlk1hash 26369 clwlkfclwwlk 26371 clwlkf1clwwlklem1 26373 clwlkf1clwwlklem2 26374 clwlkf1clwwlklem3 26375 |
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