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Mirrors > Home > MPE Home > Th. List > wwlkn0 | Structured version Visualization version GIF version |
Description: A walk of length 0 is represented by a singleton word. (Contributed by Alexander van der Vekens, 20-Jul-2018.) |
Ref | Expression |
---|---|
wwlkn0 | ⊢ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘0) → ∃𝑣 ∈ 𝑉 𝑊 = 〈“𝑣”〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wwlknprop 26214 | . 2 ⊢ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘0) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (0 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉))) | |
2 | simpl 472 | . . . . . 6 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → 𝑉 ∈ V) | |
3 | simpr 476 | . . . . . 6 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → 𝐸 ∈ V) | |
4 | 0nn0 11184 | . . . . . . 7 ⊢ 0 ∈ ℕ0 | |
5 | 4 | a1i 11 | . . . . . 6 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → 0 ∈ ℕ0) |
6 | 2, 3, 5 | 3jca 1235 | . . . . 5 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 0 ∈ ℕ0)) |
7 | 6 | adantr 480 | . . . 4 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (0 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉)) → (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 0 ∈ ℕ0)) |
8 | iswwlkn 26212 | . . . 4 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 0 ∈ ℕ0) → (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘0) ↔ (𝑊 ∈ (𝑉 WWalks 𝐸) ∧ (#‘𝑊) = (0 + 1)))) | |
9 | 7, 8 | syl 17 | . . 3 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (0 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉)) → (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘0) ↔ (𝑊 ∈ (𝑉 WWalks 𝐸) ∧ (#‘𝑊) = (0 + 1)))) |
10 | 0p1e1 11009 | . . . . . . . . 9 ⊢ (0 + 1) = 1 | |
11 | 10 | eqeq2i 2622 | . . . . . . . 8 ⊢ ((#‘𝑊) = (0 + 1) ↔ (#‘𝑊) = 1) |
12 | eqs1 13245 | . . . . . . . . . 10 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 1) → 𝑊 = 〈“(𝑊‘0)”〉) | |
13 | wrdlen1 13198 | . . . . . . . . . . . 12 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 1) → ∃𝑣 ∈ 𝑉 (𝑊‘0) = 𝑣) | |
14 | s1eq 13233 | . . . . . . . . . . . . 13 ⊢ ((𝑊‘0) = 𝑣 → 〈“(𝑊‘0)”〉 = 〈“𝑣”〉) | |
15 | 14 | reximi 2994 | . . . . . . . . . . . 12 ⊢ (∃𝑣 ∈ 𝑉 (𝑊‘0) = 𝑣 → ∃𝑣 ∈ 𝑉 〈“(𝑊‘0)”〉 = 〈“𝑣”〉) |
16 | 13, 15 | syl 17 | . . . . . . . . . . 11 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 1) → ∃𝑣 ∈ 𝑉 〈“(𝑊‘0)”〉 = 〈“𝑣”〉) |
17 | eqeq1 2614 | . . . . . . . . . . . . 13 ⊢ (〈“(𝑊‘0)”〉 = 𝑊 → (〈“(𝑊‘0)”〉 = 〈“𝑣”〉 ↔ 𝑊 = 〈“𝑣”〉)) | |
18 | 17 | eqcoms 2618 | . . . . . . . . . . . 12 ⊢ (𝑊 = 〈“(𝑊‘0)”〉 → (〈“(𝑊‘0)”〉 = 〈“𝑣”〉 ↔ 𝑊 = 〈“𝑣”〉)) |
19 | 18 | rexbidv 3034 | . . . . . . . . . . 11 ⊢ (𝑊 = 〈“(𝑊‘0)”〉 → (∃𝑣 ∈ 𝑉 〈“(𝑊‘0)”〉 = 〈“𝑣”〉 ↔ ∃𝑣 ∈ 𝑉 𝑊 = 〈“𝑣”〉)) |
20 | 16, 19 | syl5ib 233 | . . . . . . . . . 10 ⊢ (𝑊 = 〈“(𝑊‘0)”〉 → ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 1) → ∃𝑣 ∈ 𝑉 𝑊 = 〈“𝑣”〉)) |
21 | 12, 20 | mpcom 37 | . . . . . . . . 9 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 1) → ∃𝑣 ∈ 𝑉 𝑊 = 〈“𝑣”〉) |
22 | 21 | expcom 450 | . . . . . . . 8 ⊢ ((#‘𝑊) = 1 → (𝑊 ∈ Word 𝑉 → ∃𝑣 ∈ 𝑉 𝑊 = 〈“𝑣”〉)) |
23 | 11, 22 | sylbi 206 | . . . . . . 7 ⊢ ((#‘𝑊) = (0 + 1) → (𝑊 ∈ Word 𝑉 → ∃𝑣 ∈ 𝑉 𝑊 = 〈“𝑣”〉)) |
24 | 23 | adantl 481 | . . . . . 6 ⊢ ((𝑊 ∈ (𝑉 WWalks 𝐸) ∧ (#‘𝑊) = (0 + 1)) → (𝑊 ∈ Word 𝑉 → ∃𝑣 ∈ 𝑉 𝑊 = 〈“𝑣”〉)) |
25 | 24 | com12 32 | . . . . 5 ⊢ (𝑊 ∈ Word 𝑉 → ((𝑊 ∈ (𝑉 WWalks 𝐸) ∧ (#‘𝑊) = (0 + 1)) → ∃𝑣 ∈ 𝑉 𝑊 = 〈“𝑣”〉)) |
26 | 25 | adantl 481 | . . . 4 ⊢ ((0 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉) → ((𝑊 ∈ (𝑉 WWalks 𝐸) ∧ (#‘𝑊) = (0 + 1)) → ∃𝑣 ∈ 𝑉 𝑊 = 〈“𝑣”〉)) |
27 | 26 | adantl 481 | . . 3 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (0 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉)) → ((𝑊 ∈ (𝑉 WWalks 𝐸) ∧ (#‘𝑊) = (0 + 1)) → ∃𝑣 ∈ 𝑉 𝑊 = 〈“𝑣”〉)) |
28 | 9, 27 | sylbid 229 | . 2 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (0 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉)) → (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘0) → ∃𝑣 ∈ 𝑉 𝑊 = 〈“𝑣”〉)) |
29 | 1, 28 | mpcom 37 | 1 ⊢ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘0) → ∃𝑣 ∈ 𝑉 𝑊 = 〈“𝑣”〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 Vcvv 3173 ‘cfv 5804 (class class class)co 6549 0cc0 9815 1c1 9816 + caddc 9818 ℕ0cn0 11169 #chash 12979 Word cword 13146 〈“cs1 13149 WWalks cwwlk 26205 WWalksN cwwlkn 26206 |
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-s1 13157 df-wwlk 26207 df-wwlkn 26208 |
This theorem is referenced by: (None) |
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