Mathbox for Thierry Arnoux |
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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 | . . 3 ⊢ 𝑃 = (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝐼, if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖))) | |
2 | 1 | a1i 11 | . 2 ⊢ (𝐼 ∈ 𝐷 → 𝑃 = (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝐼, if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖)))) |
3 | iftrue 4042 | . . 3 ⊢ (𝑖 = 1 → if(𝑖 = 1, 𝐼, if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖)) = 𝐼) | |
4 | 3 | adantl 481 | . 2 ⊢ ((𝐼 ∈ 𝐷 ∧ 𝑖 = 1) → if(𝑖 = 1, 𝐼, if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖)) = 𝐼) |
5 | elfzuz2 12217 | . . . 4 ⊢ (𝐼 ∈ (1...𝑁) → 𝑁 ∈ (ℤ≥‘1)) | |
6 | psgnfzto1st.d | . . . 4 ⊢ 𝐷 = (1...𝑁) | |
7 | 5, 6 | eleq2s 2706 | . . 3 ⊢ (𝐼 ∈ 𝐷 → 𝑁 ∈ (ℤ≥‘1)) |
8 | eluzfz1 12219 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘1) → 1 ∈ (1...𝑁)) | |
9 | 8, 6 | syl6eleqr 2699 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘1) → 1 ∈ 𝐷) |
10 | 7, 9 | syl 17 | . 2 ⊢ (𝐼 ∈ 𝐷 → 1 ∈ 𝐷) |
11 | id 22 | . 2 ⊢ (𝐼 ∈ 𝐷 → 𝐼 ∈ 𝐷) | |
12 | 2, 4, 10, 11 | fvmptd 6197 | 1 ⊢ (𝐼 ∈ 𝐷 → (𝑃‘1) = 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ifcif 4036 class class class wbr 4583 ↦ cmpt 4643 ‘cfv 5804 (class class class)co 6549 1c1 9816 ≤ cle 9954 − cmin 10145 ℤ≥cuz 11563 ...cfz 12197 |
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-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-pre-lttri 9889 ax-pre-lttrn 9890 |
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-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 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-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-neg 10148 df-z 11255 df-uz 11564 df-fz 12198 |
This theorem is referenced by: fzto1stinvn 29185 madjusmdetlem4 29224 |
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