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Mirrors > Home > MPE Home > Th. List > wwlkextproplem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for wwlkextprop 26272. (Contributed by Alexander van der Vekens, 31-Jul-2018.) |
Ref | Expression |
---|---|
wwlkextprop.x | ⊢ 𝑋 = ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) |
Ref | Expression |
---|---|
wwlkextproplem1 | ⊢ ((𝑊 ∈ 𝑋 ∧ 𝑁 ∈ ℕ0) → ((𝑊 substr 〈0, (𝑁 + 1)〉)‘0) = (𝑊‘0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wwlknimp 26215 | . . . . 5 ⊢ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) → (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = ((𝑁 + 1) + 1) ∧ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸)) | |
2 | 1zzd 11285 | . . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ0 → 1 ∈ ℤ) | |
3 | nn0z 11277 | . . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ) | |
4 | 3 | peano2zd 11361 | . . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℤ) |
5 | 4 | peano2zd 11361 | . . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ0 → ((𝑁 + 1) + 1) ∈ ℤ) |
6 | 2, 5, 4 | 3jca 1235 | . . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ0 → (1 ∈ ℤ ∧ ((𝑁 + 1) + 1) ∈ ℤ ∧ (𝑁 + 1) ∈ ℤ)) |
7 | nn0ge0 11195 | . . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ0 → 0 ≤ 𝑁) | |
8 | 1red 9934 | . . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ0 → 1 ∈ ℝ) | |
9 | nn0re 11178 | . . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ) | |
10 | 8, 9 | addge02d 10495 | . . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ0 → (0 ≤ 𝑁 ↔ 1 ≤ (𝑁 + 1))) |
11 | 7, 10 | mpbid 221 | . . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ0 → 1 ≤ (𝑁 + 1)) |
12 | peano2re 10088 | . . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℝ → (𝑁 + 1) ∈ ℝ) | |
13 | 9, 12 | syl 17 | . . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℝ) |
14 | 13 | lep1d 10834 | . . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ≤ ((𝑁 + 1) + 1)) |
15 | 11, 14 | jca 553 | . . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ0 → (1 ≤ (𝑁 + 1) ∧ (𝑁 + 1) ≤ ((𝑁 + 1) + 1))) |
16 | elfz2 12204 | . . . . . . . . . 10 ⊢ ((𝑁 + 1) ∈ (1...((𝑁 + 1) + 1)) ↔ ((1 ∈ ℤ ∧ ((𝑁 + 1) + 1) ∈ ℤ ∧ (𝑁 + 1) ∈ ℤ) ∧ (1 ≤ (𝑁 + 1) ∧ (𝑁 + 1) ≤ ((𝑁 + 1) + 1)))) | |
17 | 6, 15, 16 | sylanbrc 695 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ (1...((𝑁 + 1) + 1))) |
18 | oveq2 6557 | . . . . . . . . . 10 ⊢ ((#‘𝑊) = ((𝑁 + 1) + 1) → (1...(#‘𝑊)) = (1...((𝑁 + 1) + 1))) | |
19 | 18 | eleq2d 2673 | . . . . . . . . 9 ⊢ ((#‘𝑊) = ((𝑁 + 1) + 1) → ((𝑁 + 1) ∈ (1...(#‘𝑊)) ↔ (𝑁 + 1) ∈ (1...((𝑁 + 1) + 1)))) |
20 | 17, 19 | syl5ibr 235 | . . . . . . . 8 ⊢ ((#‘𝑊) = ((𝑁 + 1) + 1) → (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ (1...(#‘𝑊)))) |
21 | 20 | adantl 481 | . . . . . . 7 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) → (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ (1...(#‘𝑊)))) |
22 | simpl 472 | . . . . . . 7 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) → 𝑊 ∈ Word 𝑉) | |
23 | 21, 22 | jctild 564 | . . . . . 6 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) → (𝑁 ∈ ℕ0 → (𝑊 ∈ Word 𝑉 ∧ (𝑁 + 1) ∈ (1...(#‘𝑊))))) |
24 | 23 | 3adant3 1074 | . . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = ((𝑁 + 1) + 1) ∧ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸) → (𝑁 ∈ ℕ0 → (𝑊 ∈ Word 𝑉 ∧ (𝑁 + 1) ∈ (1...(#‘𝑊))))) |
25 | 1, 24 | syl 17 | . . . 4 ⊢ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) → (𝑁 ∈ ℕ0 → (𝑊 ∈ Word 𝑉 ∧ (𝑁 + 1) ∈ (1...(#‘𝑊))))) |
26 | wwlkextprop.x | . . . 4 ⊢ 𝑋 = ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) | |
27 | 25, 26 | eleq2s 2706 | . . 3 ⊢ (𝑊 ∈ 𝑋 → (𝑁 ∈ ℕ0 → (𝑊 ∈ Word 𝑉 ∧ (𝑁 + 1) ∈ (1...(#‘𝑊))))) |
28 | 27 | imp 444 | . 2 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝑁 ∈ ℕ0) → (𝑊 ∈ Word 𝑉 ∧ (𝑁 + 1) ∈ (1...(#‘𝑊)))) |
29 | swrd0fv0 13292 | . 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (𝑁 + 1) ∈ (1...(#‘𝑊))) → ((𝑊 substr 〈0, (𝑁 + 1)〉)‘0) = (𝑊‘0)) | |
30 | 28, 29 | syl 17 | 1 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝑁 ∈ ℕ0) → ((𝑊 substr 〈0, (𝑁 + 1)〉)‘0) = (𝑊‘0)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∀wral 2896 {cpr 4127 〈cop 4131 class class class wbr 4583 ran crn 5039 ‘cfv 5804 (class class class)co 6549 ℝcr 9814 0cc0 9815 1c1 9816 + caddc 9818 ≤ cle 9954 ℕ0cn0 11169 ℤcz 11254 ...cfz 12197 ..^cfzo 12334 #chash 12979 Word cword 13146 substr csubstr 13150 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-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-substr 13158 df-wwlk 26207 df-wwlkn 26208 |
This theorem is referenced by: wwlkextproplem3 26271 wwlkextprop 26272 |
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