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Mirrors > Home > MPE Home > Th. List > Mathboxes > signstfveq0a | Structured version Visualization version GIF version |
Description: Lemma for signstfveq0 29980. (Contributed by Thierry Arnoux, 11-Oct-2018.) |
Ref | Expression |
---|---|
signsv.p | ⊢ ⨣ = (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏)) |
signsv.w | ⊢ 𝑊 = {〈(Base‘ndx), {-1, 0, 1}〉, 〈(+g‘ndx), ⨣ 〉} |
signsv.t | ⊢ 𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(#‘𝑓)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓‘𝑖)))))) |
signsv.v | ⊢ 𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(#‘𝑓))if(((𝑇‘𝑓)‘𝑗) ≠ ((𝑇‘𝑓)‘(𝑗 − 1)), 1, 0)) |
signstfveq0.1 | ⊢ 𝑁 = (#‘𝐹) |
Ref | Expression |
---|---|
signstfveq0a | ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ (𝐹‘(𝑁 − 1)) = 0) → 𝑁 ∈ (ℤ≥‘2)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpll 786 | . . . . 5 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ (𝐹‘(𝑁 − 1)) = 0) → 𝐹 ∈ (Word ℝ ∖ {∅})) | |
2 | 1 | eldifad 3552 | . . . 4 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ (𝐹‘(𝑁 − 1)) = 0) → 𝐹 ∈ Word ℝ) |
3 | signstfveq0.1 | . . . . 5 ⊢ 𝑁 = (#‘𝐹) | |
4 | lencl 13179 | . . . . 5 ⊢ (𝐹 ∈ Word ℝ → (#‘𝐹) ∈ ℕ0) | |
5 | 3, 4 | syl5eqel 2692 | . . . 4 ⊢ (𝐹 ∈ Word ℝ → 𝑁 ∈ ℕ0) |
6 | 2, 5 | syl 17 | . . 3 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ (𝐹‘(𝑁 − 1)) = 0) → 𝑁 ∈ ℕ0) |
7 | eldifsn 4260 | . . . . 5 ⊢ (𝐹 ∈ (Word ℝ ∖ {∅}) ↔ (𝐹 ∈ Word ℝ ∧ 𝐹 ≠ ∅)) | |
8 | 1, 7 | sylib 207 | . . . 4 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ (𝐹‘(𝑁 − 1)) = 0) → (𝐹 ∈ Word ℝ ∧ 𝐹 ≠ ∅)) |
9 | hasheq0 13015 | . . . . . . 7 ⊢ (𝐹 ∈ Word ℝ → ((#‘𝐹) = 0 ↔ 𝐹 = ∅)) | |
10 | 9 | necon3bid 2826 | . . . . . 6 ⊢ (𝐹 ∈ Word ℝ → ((#‘𝐹) ≠ 0 ↔ 𝐹 ≠ ∅)) |
11 | 10 | biimpar 501 | . . . . 5 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐹 ≠ ∅) → (#‘𝐹) ≠ 0) |
12 | 3 | neeq1i 2846 | . . . . 5 ⊢ (𝑁 ≠ 0 ↔ (#‘𝐹) ≠ 0) |
13 | 11, 12 | sylibr 223 | . . . 4 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐹 ≠ ∅) → 𝑁 ≠ 0) |
14 | 8, 13 | syl 17 | . . 3 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ (𝐹‘(𝑁 − 1)) = 0) → 𝑁 ≠ 0) |
15 | elnnne0 11183 | . . 3 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℕ0 ∧ 𝑁 ≠ 0)) | |
16 | 6, 14, 15 | sylanbrc 695 | . 2 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ (𝐹‘(𝑁 − 1)) = 0) → 𝑁 ∈ ℕ) |
17 | simplr 788 | . . . . 5 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ (𝐹‘(𝑁 − 1)) = 0) → (𝐹‘0) ≠ 0) | |
18 | simpr 476 | . . . . 5 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ (𝐹‘(𝑁 − 1)) = 0) → (𝐹‘(𝑁 − 1)) = 0) | |
19 | 17, 18 | neeqtrrd 2856 | . . . 4 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ (𝐹‘(𝑁 − 1)) = 0) → (𝐹‘0) ≠ (𝐹‘(𝑁 − 1))) |
20 | 19 | necomd 2837 | . . 3 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ (𝐹‘(𝑁 − 1)) = 0) → (𝐹‘(𝑁 − 1)) ≠ (𝐹‘0)) |
21 | oveq1 6556 | . . . . . 6 ⊢ (𝑁 = 1 → (𝑁 − 1) = (1 − 1)) | |
22 | 1m1e0 10966 | . . . . . 6 ⊢ (1 − 1) = 0 | |
23 | 21, 22 | syl6eq 2660 | . . . . 5 ⊢ (𝑁 = 1 → (𝑁 − 1) = 0) |
24 | 23 | fveq2d 6107 | . . . 4 ⊢ (𝑁 = 1 → (𝐹‘(𝑁 − 1)) = (𝐹‘0)) |
25 | 24 | necon3i 2814 | . . 3 ⊢ ((𝐹‘(𝑁 − 1)) ≠ (𝐹‘0) → 𝑁 ≠ 1) |
26 | 20, 25 | syl 17 | . 2 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ (𝐹‘(𝑁 − 1)) = 0) → 𝑁 ≠ 1) |
27 | eluz2b3 11638 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘2) ↔ (𝑁 ∈ ℕ ∧ 𝑁 ≠ 1)) | |
28 | 16, 26, 27 | sylanbrc 695 | 1 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ (𝐹‘(𝑁 − 1)) = 0) → 𝑁 ∈ (ℤ≥‘2)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∖ cdif 3537 ∅c0 3874 ifcif 4036 {csn 4125 {cpr 4127 {ctp 4129 〈cop 4131 ↦ cmpt 4643 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 ℝcr 9814 0cc0 9815 1c1 9816 − cmin 10145 -cneg 10146 ℕcn 10897 2c2 10947 ℕ0cn0 11169 ℤ≥cuz 11563 ...cfz 12197 ..^cfzo 12334 #chash 12979 Word cword 13146 sgncsgn 13674 Σcsu 14264 ndxcnx 15692 Basecbs 15695 +gcplusg 15768 Σg cgsu 15924 |
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-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-2 10956 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 |
This theorem is referenced by: signstfveq0 29980 |
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