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Mirrors > Home > MPE Home > Th. List > Mathboxes > aks6d1c2p1 | Structured version Visualization version GIF version |
Description: In the AKS-theorem the subset defined by 𝐸 takes values in the positive integers. (Contributed by metakunt, 7-Jan-2025.) |
Ref | Expression |
---|---|
aks6d1c2p1.1 | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
aks6d1c2p1.2 | ⊢ (𝜑 → 𝑃 ∈ ℙ) |
aks6d1c2p1.3 | ⊢ (𝜑 → 𝑃 ∥ 𝑁) |
aks6d1c2p1.4 | ⊢ 𝐸 = (𝑘 ∈ ℕ0, 𝑙 ∈ ℕ0 ↦ ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙))) |
Ref | Expression |
---|---|
aks6d1c2p1 | ⊢ (𝜑 → 𝐸:(ℕ0 × ℕ0)⟶ℕ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | aks6d1c2p1.2 | . . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℙ) | |
2 | prmnn 16708 | . . . . . 6 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ) | |
3 | 1, 2 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℕ) |
4 | 3 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (ℕ0 × ℕ0)) → 𝑃 ∈ ℕ) |
5 | simpr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ (ℕ0 × ℕ0)) → 𝑎 ∈ (ℕ0 × ℕ0)) | |
6 | xp1st 8045 | . . . . 5 ⊢ (𝑎 ∈ (ℕ0 × ℕ0) → (1st ‘𝑎) ∈ ℕ0) | |
7 | 5, 6 | syl 17 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (ℕ0 × ℕ0)) → (1st ‘𝑎) ∈ ℕ0) |
8 | 4, 7 | nnexpcld 14281 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ (ℕ0 × ℕ0)) → (𝑃↑(1st ‘𝑎)) ∈ ℕ) |
9 | aks6d1c2p1.3 | . . . . . 6 ⊢ (𝜑 → 𝑃 ∥ 𝑁) | |
10 | aks6d1c2p1.1 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
11 | 10, 3 | jca 511 | . . . . . . 7 ⊢ (𝜑 → (𝑁 ∈ ℕ ∧ 𝑃 ∈ ℕ)) |
12 | nndivdvds 16296 | . . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℕ) → (𝑃 ∥ 𝑁 ↔ (𝑁 / 𝑃) ∈ ℕ)) | |
13 | 11, 12 | syl 17 | . . . . . 6 ⊢ (𝜑 → (𝑃 ∥ 𝑁 ↔ (𝑁 / 𝑃) ∈ ℕ)) |
14 | 9, 13 | mpbid 232 | . . . . 5 ⊢ (𝜑 → (𝑁 / 𝑃) ∈ ℕ) |
15 | 14 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (ℕ0 × ℕ0)) → (𝑁 / 𝑃) ∈ ℕ) |
16 | xp2nd 8046 | . . . . 5 ⊢ (𝑎 ∈ (ℕ0 × ℕ0) → (2nd ‘𝑎) ∈ ℕ0) | |
17 | 5, 16 | syl 17 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (ℕ0 × ℕ0)) → (2nd ‘𝑎) ∈ ℕ0) |
18 | 15, 17 | nnexpcld 14281 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ (ℕ0 × ℕ0)) → ((𝑁 / 𝑃)↑(2nd ‘𝑎)) ∈ ℕ) |
19 | 8, 18 | nnmulcld 12317 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ (ℕ0 × ℕ0)) → ((𝑃↑(1st ‘𝑎)) · ((𝑁 / 𝑃)↑(2nd ‘𝑎))) ∈ ℕ) |
20 | aks6d1c2p1.4 | . . 3 ⊢ 𝐸 = (𝑘 ∈ ℕ0, 𝑙 ∈ ℕ0 ↦ ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙))) | |
21 | vex 3482 | . . . . . . . 8 ⊢ 𝑘 ∈ V | |
22 | vex 3482 | . . . . . . . 8 ⊢ 𝑙 ∈ V | |
23 | 21, 22 | op1std 8023 | . . . . . . 7 ⊢ (𝑎 = 〈𝑘, 𝑙〉 → (1st ‘𝑎) = 𝑘) |
24 | 23 | oveq2d 7447 | . . . . . 6 ⊢ (𝑎 = 〈𝑘, 𝑙〉 → (𝑃↑(1st ‘𝑎)) = (𝑃↑𝑘)) |
25 | 21, 22 | op2ndd 8024 | . . . . . . 7 ⊢ (𝑎 = 〈𝑘, 𝑙〉 → (2nd ‘𝑎) = 𝑙) |
26 | 25 | oveq2d 7447 | . . . . . 6 ⊢ (𝑎 = 〈𝑘, 𝑙〉 → ((𝑁 / 𝑃)↑(2nd ‘𝑎)) = ((𝑁 / 𝑃)↑𝑙)) |
27 | 24, 26 | oveq12d 7449 | . . . . 5 ⊢ (𝑎 = 〈𝑘, 𝑙〉 → ((𝑃↑(1st ‘𝑎)) · ((𝑁 / 𝑃)↑(2nd ‘𝑎))) = ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙))) |
28 | 27 | mpompt 7547 | . . . 4 ⊢ (𝑎 ∈ (ℕ0 × ℕ0) ↦ ((𝑃↑(1st ‘𝑎)) · ((𝑁 / 𝑃)↑(2nd ‘𝑎)))) = (𝑘 ∈ ℕ0, 𝑙 ∈ ℕ0 ↦ ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙))) |
29 | 28 | eqcomi 2744 | . . 3 ⊢ (𝑘 ∈ ℕ0, 𝑙 ∈ ℕ0 ↦ ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙))) = (𝑎 ∈ (ℕ0 × ℕ0) ↦ ((𝑃↑(1st ‘𝑎)) · ((𝑁 / 𝑃)↑(2nd ‘𝑎)))) |
30 | 20, 29 | eqtri 2763 | . 2 ⊢ 𝐸 = (𝑎 ∈ (ℕ0 × ℕ0) ↦ ((𝑃↑(1st ‘𝑎)) · ((𝑁 / 𝑃)↑(2nd ‘𝑎)))) |
31 | 19, 30 | fmptd 7134 | 1 ⊢ (𝜑 → 𝐸:(ℕ0 × ℕ0)⟶ℕ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 〈cop 4637 class class class wbr 5148 ↦ cmpt 5231 × cxp 5687 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ∈ cmpo 7433 1st c1st 8011 2nd c2nd 8012 · cmul 11158 / cdiv 11918 ℕcn 12264 ℕ0cn0 12524 ↑cexp 14099 ∥ cdvds 16287 ℙcprime 16705 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-n0 12525 df-z 12612 df-uz 12877 df-seq 14040 df-exp 14100 df-dvds 16288 df-prm 16706 |
This theorem is referenced by: aks6d1c2p2 42101 aks6d1c2lem4 42109 aks6d1c6lem2 42153 aks6d1c6lem4 42155 aks6d1c6lem5 42159 aks6d1c7lem1 42162 |
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