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Mirrors > Home > MPE Home > Th. List > clwlkclwwlklem2a2 | Structured version Visualization version GIF version |
Description: Lemma 2 for clwlkclwwlklem2a 29116. (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 14465 | . . . 4 ⊢ (𝑃 ∈ Word 𝑉 → (♯‘𝑃) ∈ ℕ0) | |
2 | nn0z 12565 | . . . . . 6 ⊢ ((♯‘𝑃) ∈ ℕ0 → (♯‘𝑃) ∈ ℤ) | |
3 | 2 | adantr 481 | . . . . 5 ⊢ (((♯‘𝑃) ∈ ℕ0 ∧ 2 ≤ (♯‘𝑃)) → (♯‘𝑃) ∈ ℤ) |
4 | 0red 11199 | . . . . . 6 ⊢ (((♯‘𝑃) ∈ ℕ0 ∧ 2 ≤ (♯‘𝑃)) → 0 ∈ ℝ) | |
5 | 2re 12268 | . . . . . . 7 ⊢ 2 ∈ ℝ | |
6 | 5 | a1i 11 | . . . . . 6 ⊢ (((♯‘𝑃) ∈ ℕ0 ∧ 2 ≤ (♯‘𝑃)) → 2 ∈ ℝ) |
7 | nn0re 12463 | . . . . . . 7 ⊢ ((♯‘𝑃) ∈ ℕ0 → (♯‘𝑃) ∈ ℝ) | |
8 | 7 | adantr 481 | . . . . . 6 ⊢ (((♯‘𝑃) ∈ ℕ0 ∧ 2 ≤ (♯‘𝑃)) → (♯‘𝑃) ∈ ℝ) |
9 | 2pos 12297 | . . . . . . 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 11356 | . . . . 5 ⊢ (((♯‘𝑃) ∈ ℕ0 ∧ 2 ≤ (♯‘𝑃)) → 0 < (♯‘𝑃)) |
13 | elnnz 12550 | . . . . 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 12495 | . . 3 ⊢ ((♯‘𝑃) ∈ ℕ → ((♯‘𝑃) − 1) ∈ ℕ0) | |
17 | 15, 16 | syl 17 | . 2 ⊢ ((𝑃 ∈ Word 𝑉 ∧ 2 ≤ (♯‘𝑃)) → ((♯‘𝑃) − 1) ∈ ℕ0) |
18 | fvex 6891 | . . . 4 ⊢ (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}) ∈ V | |
19 | fvex 6891 | . . . 4 ⊢ (◡𝐸‘{(𝑃‘𝑥), (𝑃‘0)}) ∈ V | |
20 | 18, 19 | ifex 4572 | . . 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 6680 | . 2 ⊢ 𝐹 Fn (0..^((♯‘𝑃) − 1)) |
23 | ffzo0hash 14390 | . 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 4522 {cpr 4624 class class class wbr 5141 ↦ cmpt 5224 ◡ccnv 5668 Fn wfn 6527 ‘cfv 6532 (class class class)co 7393 ℝcr 11091 0cc0 11092 1c1 11093 + caddc 11095 < clt 11230 ≤ cle 11231 − cmin 11426 ℕcn 12194 2c2 12249 ℕ0cn0 12454 ℤcz 12540 ..^cfzo 13609 ♯chash 14272 Word cword 14446 |
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 2702 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-cnex 11148 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168 ax-pre-mulgt0 11169 |
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 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-om 7839 df-1st 7957 df-2nd 7958 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-1o 8448 df-er 8686 df-en 8923 df-dom 8924 df-sdom 8925 df-fin 8926 df-card 9916 df-pnf 11232 df-mnf 11233 df-xr 11234 df-ltxr 11235 df-le 11236 df-sub 11428 df-neg 11429 df-nn 12195 df-2 12257 df-n0 12455 df-z 12541 df-uz 12805 df-fz 13467 df-fzo 13610 df-hash 14273 df-word 14447 |
This theorem is referenced by: clwlkclwwlklem2a3 29112 clwlkclwwlklem2a 29116 |
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