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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 4526 | . 2 ⊢ (𝑖 = 1 → if(𝑖 = 1, 𝐼, if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖)) = 𝐼) | |
3 | elfzuz2 13503 | . . . 4 ⊢ (𝐼 ∈ (1...𝑁) → 𝑁 ∈ (ℤ≥‘1)) | |
4 | psgnfzto1st.d | . . . 4 ⊢ 𝐷 = (1...𝑁) | |
5 | 3, 4 | eleq2s 2843 | . . 3 ⊢ (𝐼 ∈ 𝐷 → 𝑁 ∈ (ℤ≥‘1)) |
6 | eluzfz1 13505 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘1) → 1 ∈ (1...𝑁)) | |
7 | 6, 4 | eleqtrrdi 2836 | . . 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 7011 | 1 ⊢ (𝐼 ∈ 𝐷 → (𝑃‘1) = 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ifcif 4520 class class class wbr 5138 ↦ cmpt 5221 ‘cfv 6533 (class class class)co 7401 1c1 11107 ≤ cle 11246 − cmin 11441 ℤ≥cuz 12819 ...cfz 13481 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11162 ax-resscn 11163 ax-pre-lttri 11180 ax-pre-lttrn 11181 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-ov 7404 df-oprab 7405 df-mpo 7406 df-1st 7968 df-2nd 7969 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-neg 11444 df-z 12556 df-uz 12820 df-fz 13482 |
This theorem is referenced by: fzto1stinvn 32731 madjusmdetlem4 33299 |
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