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Mirrors > Home > MPE Home > Th. List > clwlkclwwlklem2a2 | Structured version Visualization version GIF version |
Description: Lemma 2 for clwlkclwwlklem2a 29248. (Contributed by Alexander van der Vekens, 21-Jun-2018.) |
Ref | Expression |
---|---|
clwlkclwwlklem2.f | ⊢ 𝐹 = (𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ if(𝑥 < ((♯‘𝑃) − 2), (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}), (◡𝐸‘{(𝑃‘𝑥), (𝑃‘0)}))) |
Ref | Expression |
---|---|
clwlkclwwlklem2a2 | ⊢ ((𝑃 ∈ Word 𝑉 ∧ 2 ≤ (♯‘𝑃)) → (♯‘𝐹) = ((♯‘𝑃) − 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lencl 14482 | . . . 4 ⊢ (𝑃 ∈ Word 𝑉 → (♯‘𝑃) ∈ ℕ0) | |
2 | nn0z 12582 | . . . . . 6 ⊢ ((♯‘𝑃) ∈ ℕ0 → (♯‘𝑃) ∈ ℤ) | |
3 | 2 | adantr 481 | . . . . 5 ⊢ (((♯‘𝑃) ∈ ℕ0 ∧ 2 ≤ (♯‘𝑃)) → (♯‘𝑃) ∈ ℤ) |
4 | 0red 11216 | . . . . . 6 ⊢ (((♯‘𝑃) ∈ ℕ0 ∧ 2 ≤ (♯‘𝑃)) → 0 ∈ ℝ) | |
5 | 2re 12285 | . . . . . . 7 ⊢ 2 ∈ ℝ | |
6 | 5 | a1i 11 | . . . . . 6 ⊢ (((♯‘𝑃) ∈ ℕ0 ∧ 2 ≤ (♯‘𝑃)) → 2 ∈ ℝ) |
7 | nn0re 12480 | . . . . . . 7 ⊢ ((♯‘𝑃) ∈ ℕ0 → (♯‘𝑃) ∈ ℝ) | |
8 | 7 | adantr 481 | . . . . . 6 ⊢ (((♯‘𝑃) ∈ ℕ0 ∧ 2 ≤ (♯‘𝑃)) → (♯‘𝑃) ∈ ℝ) |
9 | 2pos 12314 | . . . . . . 7 ⊢ 0 < 2 | |
10 | 9 | a1i 11 | . . . . . 6 ⊢ (((♯‘𝑃) ∈ ℕ0 ∧ 2 ≤ (♯‘𝑃)) → 0 < 2) |
11 | simpr 485 | . . . . . 6 ⊢ (((♯‘𝑃) ∈ ℕ0 ∧ 2 ≤ (♯‘𝑃)) → 2 ≤ (♯‘𝑃)) | |
12 | 4, 6, 8, 10, 11 | ltletrd 11373 | . . . . 5 ⊢ (((♯‘𝑃) ∈ ℕ0 ∧ 2 ≤ (♯‘𝑃)) → 0 < (♯‘𝑃)) |
13 | elnnz 12567 | . . . . 5 ⊢ ((♯‘𝑃) ∈ ℕ ↔ ((♯‘𝑃) ∈ ℤ ∧ 0 < (♯‘𝑃))) | |
14 | 3, 12, 13 | sylanbrc 583 | . . . 4 ⊢ (((♯‘𝑃) ∈ ℕ0 ∧ 2 ≤ (♯‘𝑃)) → (♯‘𝑃) ∈ ℕ) |
15 | 1, 14 | sylan 580 | . . 3 ⊢ ((𝑃 ∈ Word 𝑉 ∧ 2 ≤ (♯‘𝑃)) → (♯‘𝑃) ∈ ℕ) |
16 | nnm1nn0 12512 | . . 3 ⊢ ((♯‘𝑃) ∈ ℕ → ((♯‘𝑃) − 1) ∈ ℕ0) | |
17 | 15, 16 | syl 17 | . 2 ⊢ ((𝑃 ∈ Word 𝑉 ∧ 2 ≤ (♯‘𝑃)) → ((♯‘𝑃) − 1) ∈ ℕ0) |
18 | fvex 6904 | . . . 4 ⊢ (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}) ∈ V | |
19 | fvex 6904 | . . . 4 ⊢ (◡𝐸‘{(𝑃‘𝑥), (𝑃‘0)}) ∈ V | |
20 | 18, 19 | ifex 4578 | . . 3 ⊢ if(𝑥 < ((♯‘𝑃) − 2), (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}), (◡𝐸‘{(𝑃‘𝑥), (𝑃‘0)})) ∈ V |
21 | clwlkclwwlklem2.f | . . 3 ⊢ 𝐹 = (𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ if(𝑥 < ((♯‘𝑃) − 2), (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}), (◡𝐸‘{(𝑃‘𝑥), (𝑃‘0)}))) | |
22 | 20, 21 | fnmpti 6693 | . 2 ⊢ 𝐹 Fn (0..^((♯‘𝑃) − 1)) |
23 | ffzo0hash 14407 | . 2 ⊢ ((((♯‘𝑃) − 1) ∈ ℕ0 ∧ 𝐹 Fn (0..^((♯‘𝑃) − 1))) → (♯‘𝐹) = ((♯‘𝑃) − 1)) | |
24 | 17, 22, 23 | sylancl 586 | 1 ⊢ ((𝑃 ∈ Word 𝑉 ∧ 2 ≤ (♯‘𝑃)) → (♯‘𝐹) = ((♯‘𝑃) − 1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ifcif 4528 {cpr 4630 class class class wbr 5148 ↦ cmpt 5231 ◡ccnv 5675 Fn wfn 6538 ‘cfv 6543 (class class class)co 7408 ℝcr 11108 0cc0 11109 1c1 11110 + caddc 11112 < clt 11247 ≤ cle 11248 − cmin 11443 ℕcn 12211 2c2 12266 ℕ0cn0 12471 ℤcz 12557 ..^cfzo 13626 ♯chash 14289 Word cword 14463 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-1st 7974 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-card 9933 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-n0 12472 df-z 12558 df-uz 12822 df-fz 13484 df-fzo 13627 df-hash 14290 df-word 14464 |
This theorem is referenced by: clwlkclwwlklem2a3 29244 clwlkclwwlklem2a 29248 |
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