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Mirrors > Home > MPE Home > Th. List > gausslemma2dlem0d | Structured version Visualization version GIF version |
Description: Auxiliary lemma 4 for gausslemma2d 27395. (Contributed by AV, 9-Jul-2021.) |
Ref | Expression |
---|---|
gausslemma2dlem0.p | ⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2})) |
gausslemma2dlem0.m | ⊢ 𝑀 = (⌊‘(𝑃 / 4)) |
Ref | Expression |
---|---|
gausslemma2dlem0d | ⊢ (𝜑 → 𝑀 ∈ ℕ0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gausslemma2dlem0.m | . 2 ⊢ 𝑀 = (⌊‘(𝑃 / 4)) | |
2 | gausslemma2dlem0.p | . . . 4 ⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2})) | |
3 | 2 | gausslemma2dlem0a 27377 | . . 3 ⊢ (𝜑 → 𝑃 ∈ ℕ) |
4 | nnre 12266 | . . . . 5 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℝ) | |
5 | 4re 12343 | . . . . . 6 ⊢ 4 ∈ ℝ | |
6 | 5 | a1i 11 | . . . . 5 ⊢ (𝑃 ∈ ℕ → 4 ∈ ℝ) |
7 | 4ne0 12367 | . . . . . 6 ⊢ 4 ≠ 0 | |
8 | 7 | a1i 11 | . . . . 5 ⊢ (𝑃 ∈ ℕ → 4 ≠ 0) |
9 | 4, 6, 8 | redivcld 12089 | . . . 4 ⊢ (𝑃 ∈ ℕ → (𝑃 / 4) ∈ ℝ) |
10 | nnnn0 12526 | . . . . . 6 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℕ0) | |
11 | 10 | nn0ge0d 12582 | . . . . 5 ⊢ (𝑃 ∈ ℕ → 0 ≤ 𝑃) |
12 | 4pos 12366 | . . . . . . 7 ⊢ 0 < 4 | |
13 | 5, 12 | pm3.2i 469 | . . . . . 6 ⊢ (4 ∈ ℝ ∧ 0 < 4) |
14 | 13 | a1i 11 | . . . . 5 ⊢ (𝑃 ∈ ℕ → (4 ∈ ℝ ∧ 0 < 4)) |
15 | divge0 12130 | . . . . 5 ⊢ (((𝑃 ∈ ℝ ∧ 0 ≤ 𝑃) ∧ (4 ∈ ℝ ∧ 0 < 4)) → 0 ≤ (𝑃 / 4)) | |
16 | 4, 11, 14, 15 | syl21anc 836 | . . . 4 ⊢ (𝑃 ∈ ℕ → 0 ≤ (𝑃 / 4)) |
17 | 9, 16 | jca 510 | . . 3 ⊢ (𝑃 ∈ ℕ → ((𝑃 / 4) ∈ ℝ ∧ 0 ≤ (𝑃 / 4))) |
18 | flge0nn0 13835 | . . 3 ⊢ (((𝑃 / 4) ∈ ℝ ∧ 0 ≤ (𝑃 / 4)) → (⌊‘(𝑃 / 4)) ∈ ℕ0) | |
19 | 3, 17, 18 | 3syl 18 | . 2 ⊢ (𝜑 → (⌊‘(𝑃 / 4)) ∈ ℕ0) |
20 | 1, 19 | eqeltrid 2829 | 1 ⊢ (𝜑 → 𝑀 ∈ ℕ0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ≠ wne 2929 ∖ cdif 3943 {csn 4632 class class class wbr 5152 ‘cfv 6553 (class class class)co 7423 ℝcr 11153 0cc0 11154 < clt 11294 ≤ cle 11295 / cdiv 11917 ℕcn 12259 2c2 12314 4c4 12316 ℕ0cn0 12519 ⌊cfl 13805 ℙcprime 16667 |
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 2166 ax-ext 2696 ax-sep 5303 ax-nul 5310 ax-pow 5368 ax-pr 5432 ax-un 7745 ax-cnex 11210 ax-resscn 11211 ax-1cn 11212 ax-icn 11213 ax-addcl 11214 ax-addrcl 11215 ax-mulcl 11216 ax-mulrcl 11217 ax-mulcom 11218 ax-addass 11219 ax-mulass 11220 ax-distr 11221 ax-i2m1 11222 ax-1ne0 11223 ax-1rid 11224 ax-rnegex 11225 ax-rrecex 11226 ax-cnre 11227 ax-pre-lttri 11228 ax-pre-lttrn 11229 ax-pre-ltadd 11230 ax-pre-mulgt0 11231 ax-pre-sup 11232 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4325 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5579 df-eprel 5585 df-po 5593 df-so 5594 df-fr 5636 df-we 5638 df-xp 5687 df-rel 5688 df-cnv 5689 df-co 5690 df-dm 5691 df-rn 5692 df-res 5693 df-ima 5694 df-pred 6311 df-ord 6378 df-on 6379 df-lim 6380 df-suc 6381 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7379 df-ov 7426 df-oprab 7427 df-mpo 7428 df-om 7876 df-2nd 8003 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-2o 8496 df-er 8733 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-sup 9481 df-inf 9482 df-pnf 11296 df-mnf 11297 df-xr 11298 df-ltxr 11299 df-le 11300 df-sub 11492 df-neg 11493 df-div 11918 df-nn 12260 df-2 12322 df-3 12323 df-4 12324 df-n0 12520 df-z 12606 df-uz 12870 df-rp 13024 df-fl 13807 df-seq 14017 df-exp 14077 df-cj 15099 df-re 15100 df-im 15101 df-sqrt 15235 df-abs 15236 df-dvds 16252 df-prm 16668 |
This theorem is referenced by: gausslemma2dlem0h 27384 gausslemma2dlem2 27388 gausslemma2dlem3 27389 gausslemma2dlem4 27390 gausslemma2dlem6 27393 |
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