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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 | . 2 ⊢ 𝑃 = (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝐼, if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖))) | |
2 | iftrue 4535 | . 2 ⊢ (𝑖 = 1 → if(𝑖 = 1, 𝐼, if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖)) = 𝐼) | |
3 | elfzuz2 13511 | . . . 4 ⊢ (𝐼 ∈ (1...𝑁) → 𝑁 ∈ (ℤ≥‘1)) | |
4 | psgnfzto1st.d | . . . 4 ⊢ 𝐷 = (1...𝑁) | |
5 | 3, 4 | eleq2s 2850 | . . 3 ⊢ (𝐼 ∈ 𝐷 → 𝑁 ∈ (ℤ≥‘1)) |
6 | eluzfz1 13513 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘1) → 1 ∈ (1...𝑁)) | |
7 | 6, 4 | eleqtrrdi 2843 | . . 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 7022 | 1 ⊢ (𝐼 ∈ 𝐷 → (𝑃‘1) = 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2105 ifcif 4529 class class class wbr 5149 ↦ cmpt 5232 ‘cfv 6544 (class class class)co 7412 1c1 11114 ≤ cle 11254 − cmin 11449 ℤ≥cuz 12827 ...cfz 13489 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7728 ax-cnex 11169 ax-resscn 11170 ax-pre-lttri 11187 ax-pre-lttrn 11188 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 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-rab 3432 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7415 df-oprab 7416 df-mpo 7417 df-1st 7978 df-2nd 7979 df-er 8706 df-en 8943 df-dom 8944 df-sdom 8945 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-neg 11452 df-z 12564 df-uz 12828 df-fz 13490 |
This theorem is referenced by: fzto1stinvn 32530 madjusmdetlem4 33105 |
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