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Mirrors > Home > MPE Home > Th. List > Mathboxes > fpprbasnn | Structured version Visualization version GIF version |
Description: The base of a Fermat pseudoprime is a positive integer. (Contributed by AV, 30-May-2023.) |
Ref | Expression |
---|---|
fpprbasnn | ⊢ (𝑋 ∈ ( FPPr ‘𝑁) → 𝑁 ∈ ℕ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-1 6 | . 2 ⊢ (𝑁 ∈ ℕ → (𝑋 ∈ ( FPPr ‘𝑁) → 𝑁 ∈ ℕ)) | |
2 | df-fppr 43939 | . . . 4 ⊢ FPPr = (𝑛 ∈ ℕ ↦ {𝑥 ∈ (ℤ≥‘4) ∣ (𝑥 ∉ ℙ ∧ 𝑥 ∥ ((𝑛↑(𝑥 − 1)) − 1))}) | |
3 | 2 | fvmptndm 6798 | . . 3 ⊢ (¬ 𝑁 ∈ ℕ → ( FPPr ‘𝑁) = ∅) |
4 | eleq2 2901 | . . . 4 ⊢ (( FPPr ‘𝑁) = ∅ → (𝑋 ∈ ( FPPr ‘𝑁) ↔ 𝑋 ∈ ∅)) | |
5 | noel 4296 | . . . . 5 ⊢ ¬ 𝑋 ∈ ∅ | |
6 | 5 | pm2.21i 119 | . . . 4 ⊢ (𝑋 ∈ ∅ → 𝑁 ∈ ℕ) |
7 | 4, 6 | syl6bi 255 | . . 3 ⊢ (( FPPr ‘𝑁) = ∅ → (𝑋 ∈ ( FPPr ‘𝑁) → 𝑁 ∈ ℕ)) |
8 | 3, 7 | syl 17 | . 2 ⊢ (¬ 𝑁 ∈ ℕ → (𝑋 ∈ ( FPPr ‘𝑁) → 𝑁 ∈ ℕ)) |
9 | 1, 8 | pm2.61i 184 | 1 ⊢ (𝑋 ∈ ( FPPr ‘𝑁) → 𝑁 ∈ ℕ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∉ wnel 3123 {crab 3142 ∅c0 4291 class class class wbr 5066 ‘cfv 6355 (class class class)co 7156 1c1 10538 − cmin 10870 ℕcn 11638 4c4 11695 ℤ≥cuz 12244 ↑cexp 13430 ∥ cdvds 15607 ℙcprime 16015 FPPr cfppr 43938 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-dm 5565 df-iota 6314 df-fv 6363 df-fppr 43939 |
This theorem is referenced by: fpprnn 43944 fpprwppr 43953 fpprwpprb 43954 |
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