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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 4531 | . 2 ⊢ (𝑖 = 1 → if(𝑖 = 1, 𝐼, if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖)) = 𝐼) | |
| 3 | elfzuz2 13569 | . . . 4 ⊢ (𝐼 ∈ (1...𝑁) → 𝑁 ∈ (ℤ≥‘1)) | |
| 4 | psgnfzto1st.d | . . . 4 ⊢ 𝐷 = (1...𝑁) | |
| 5 | 3, 4 | eleq2s 2859 | . . 3 ⊢ (𝐼 ∈ 𝐷 → 𝑁 ∈ (ℤ≥‘1)) |
| 6 | eluzfz1 13571 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘1) → 1 ∈ (1...𝑁)) | |
| 7 | 6, 4 | eleqtrrdi 2852 | . . 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 7039 | 1 ⊢ (𝐼 ∈ 𝐷 → (𝑃‘1) = 𝐼) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ifcif 4525 class class class wbr 5143 ↦ cmpt 5225 ‘cfv 6561 (class class class)co 7431 1c1 11156 ≤ cle 11296 − cmin 11492 ℤ≥cuz 12878 ...cfz 13547 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-pre-lttri 11229 ax-pre-lttrn 11230 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-neg 11495 df-z 12614 df-uz 12879 df-fz 13548 |
| This theorem is referenced by: fzto1stinvn 33124 madjusmdetlem4 33829 |
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