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Mathbox for Alexander van der Vekens |
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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 44243 | . . . 4 ⊢ FPPr = (𝑛 ∈ ℕ ↦ {𝑥 ∈ (ℤ≥‘4) ∣ (𝑥 ∉ ℙ ∧ 𝑥 ∥ ((𝑛↑(𝑥 − 1)) − 1))}) | |
3 | 2 | fvmptndm 6775 | . . 3 ⊢ (¬ 𝑁 ∈ ℕ → ( FPPr ‘𝑁) = ∅) |
4 | eleq2 2878 | . . . 4 ⊢ (( FPPr ‘𝑁) = ∅ → (𝑋 ∈ ( FPPr ‘𝑁) ↔ 𝑋 ∈ ∅)) | |
5 | noel 4247 | . . . . 5 ⊢ ¬ 𝑋 ∈ ∅ | |
6 | 5 | pm2.21i 119 | . . . 4 ⊢ (𝑋 ∈ ∅ → 𝑁 ∈ ℕ) |
7 | 4, 6 | syl6bi 256 | . . 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 185 | 1 ⊢ (𝑋 ∈ ( FPPr ‘𝑁) → 𝑁 ∈ ℕ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∉ wnel 3091 {crab 3110 ∅c0 4243 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 1c1 10527 − cmin 10859 ℕcn 11625 4c4 11682 ℤ≥cuz 12231 ↑cexp 13425 ∥ cdvds 15599 ℙcprime 16005 FPPr cfppr 44242 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-dm 5529 df-iota 6283 df-fv 6332 df-fppr 44243 |
This theorem is referenced by: fpprnn 44248 fpprwppr 44257 fpprwpprb 44258 |
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