![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > fzto1stfv1 | Structured version Visualization version GIF version |
Description: Value of our permutation 𝑃 at 1. (Contributed by Thierry Arnoux, 23-Aug-2020.) |
Ref | Expression |
---|---|
psgnfzto1st.d | ⊢ 𝐷 = (1...𝑁) |
psgnfzto1st.p | ⊢ 𝑃 = (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝐼, if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖))) |
Ref | Expression |
---|---|
fzto1stfv1 | ⊢ (𝐼 ∈ 𝐷 → (𝑃‘1) = 𝐼) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | psgnfzto1st.p | . 2 ⊢ 𝑃 = (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝐼, if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖))) | |
2 | iftrue 4419 | . 2 ⊢ (𝑖 = 1 → if(𝑖 = 1, 𝐼, if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖)) = 𝐼) | |
3 | elfzuz2 12946 | . . . 4 ⊢ (𝐼 ∈ (1...𝑁) → 𝑁 ∈ (ℤ≥‘1)) | |
4 | psgnfzto1st.d | . . . 4 ⊢ 𝐷 = (1...𝑁) | |
5 | 3, 4 | eleq2s 2869 | . . 3 ⊢ (𝐼 ∈ 𝐷 → 𝑁 ∈ (ℤ≥‘1)) |
6 | eluzfz1 12948 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘1) → 1 ∈ (1...𝑁)) | |
7 | 6, 4 | eleqtrrdi 2862 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘1) → 1 ∈ 𝐷) |
8 | 5, 7 | syl 17 | . 2 ⊢ (𝐼 ∈ 𝐷 → 1 ∈ 𝐷) |
9 | id 22 | . 2 ⊢ (𝐼 ∈ 𝐷 → 𝐼 ∈ 𝐷) | |
10 | 1, 2, 8, 9 | fvmptd3 6775 | 1 ⊢ (𝐼 ∈ 𝐷 → (𝑃‘1) = 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2112 ifcif 4413 class class class wbr 5025 ↦ cmpt 5105 ‘cfv 6328 (class class class)co 7143 1c1 10561 ≤ cle 10699 − cmin 10893 ℤ≥cuz 12267 ...cfz 12924 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-sep 5162 ax-nul 5169 ax-pow 5227 ax-pr 5291 ax-un 7452 ax-cnex 10616 ax-resscn 10617 ax-pre-lttri 10634 ax-pre-lttrn 10635 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3or 1086 df-3an 1087 df-tru 1542 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ne 2950 df-nel 3054 df-ral 3073 df-rex 3074 df-rab 3077 df-v 3409 df-sbc 3694 df-csb 3802 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-nul 4222 df-if 4414 df-pw 4489 df-sn 4516 df-pr 4518 df-op 4522 df-uni 4792 df-iun 4878 df-br 5026 df-opab 5088 df-mpt 5106 df-id 5423 df-xp 5523 df-rel 5524 df-cnv 5525 df-co 5526 df-dm 5527 df-rn 5528 df-res 5529 df-ima 5530 df-iota 6287 df-fun 6330 df-fn 6331 df-f 6332 df-f1 6333 df-fo 6334 df-f1o 6335 df-fv 6336 df-ov 7146 df-oprab 7147 df-mpo 7148 df-1st 7686 df-2nd 7687 df-er 8292 df-en 8521 df-dom 8522 df-sdom 8523 df-pnf 10700 df-mnf 10701 df-xr 10702 df-ltxr 10703 df-le 10704 df-neg 10896 df-z 12006 df-uz 12268 df-fz 12925 |
This theorem is referenced by: fzto1stinvn 30882 madjusmdetlem4 31286 |
Copyright terms: Public domain | W3C validator |