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Mirrors > Home > MPE Home > Th. List > gausslemma2dlem0d | Structured version Visualization version GIF version |
Description: Auxiliary lemma 4 for gausslemma2d 27294. (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 27276 | . . 3 ⊢ (𝜑 → 𝑃 ∈ ℕ) |
4 | nnre 12241 | . . . . 5 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℝ) | |
5 | 4re 12318 | . . . . . 6 ⊢ 4 ∈ ℝ | |
6 | 5 | a1i 11 | . . . . 5 ⊢ (𝑃 ∈ ℕ → 4 ∈ ℝ) |
7 | 4ne0 12342 | . . . . . 6 ⊢ 4 ≠ 0 | |
8 | 7 | a1i 11 | . . . . 5 ⊢ (𝑃 ∈ ℕ → 4 ≠ 0) |
9 | 4, 6, 8 | redivcld 12064 | . . . 4 ⊢ (𝑃 ∈ ℕ → (𝑃 / 4) ∈ ℝ) |
10 | nnnn0 12501 | . . . . . 6 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℕ0) | |
11 | 10 | nn0ge0d 12557 | . . . . 5 ⊢ (𝑃 ∈ ℕ → 0 ≤ 𝑃) |
12 | 4pos 12341 | . . . . . . 7 ⊢ 0 < 4 | |
13 | 5, 12 | pm3.2i 470 | . . . . . 6 ⊢ (4 ∈ ℝ ∧ 0 < 4) |
14 | 13 | a1i 11 | . . . . 5 ⊢ (𝑃 ∈ ℕ → (4 ∈ ℝ ∧ 0 < 4)) |
15 | divge0 12105 | . . . . 5 ⊢ (((𝑃 ∈ ℝ ∧ 0 ≤ 𝑃) ∧ (4 ∈ ℝ ∧ 0 < 4)) → 0 ≤ (𝑃 / 4)) | |
16 | 4, 11, 14, 15 | syl21anc 837 | . . . 4 ⊢ (𝑃 ∈ ℕ → 0 ≤ (𝑃 / 4)) |
17 | 9, 16 | jca 511 | . . 3 ⊢ (𝑃 ∈ ℕ → ((𝑃 / 4) ∈ ℝ ∧ 0 ≤ (𝑃 / 4))) |
18 | flge0nn0 13809 | . . 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 2832 | 1 ⊢ (𝜑 → 𝑀 ∈ ℕ0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ≠ wne 2935 ∖ cdif 3941 {csn 4624 class class class wbr 5142 ‘cfv 6542 (class class class)co 7414 ℝcr 11129 0cc0 11130 < clt 11270 ≤ cle 11271 / cdiv 11893 ℕcn 12234 2c2 12289 4c4 12291 ℕ0cn0 12494 ⌊cfl 13779 ℙcprime 16633 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 ax-pre-sup 11208 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-er 8718 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-sup 9457 df-inf 9458 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-div 11894 df-nn 12235 df-2 12297 df-3 12298 df-4 12299 df-n0 12495 df-z 12581 df-uz 12845 df-rp 12999 df-fl 13781 df-seq 13991 df-exp 14051 df-cj 15070 df-re 15071 df-im 15072 df-sqrt 15206 df-abs 15207 df-dvds 16223 df-prm 16634 |
This theorem is referenced by: gausslemma2dlem0h 27283 gausslemma2dlem2 27287 gausslemma2dlem3 27288 gausslemma2dlem4 27289 gausslemma2dlem6 27292 |
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