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Mirrors > Home > MPE Home > Th. List > gausslemma2dlem0d | Structured version Visualization version GIF version |
Description: Auxiliary lemma 4 for gausslemma2d 24899. (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 24881 | . . 3 ⊢ (𝜑 → 𝑃 ∈ ℕ) |
4 | nnre 10904 | . . . . 5 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℝ) | |
5 | 4re 10974 | . . . . . 6 ⊢ 4 ∈ ℝ | |
6 | 5 | a1i 11 | . . . . 5 ⊢ (𝑃 ∈ ℕ → 4 ∈ ℝ) |
7 | 4ne0 10994 | . . . . . 6 ⊢ 4 ≠ 0 | |
8 | 7 | a1i 11 | . . . . 5 ⊢ (𝑃 ∈ ℕ → 4 ≠ 0) |
9 | 4, 6, 8 | redivcld 10732 | . . . 4 ⊢ (𝑃 ∈ ℕ → (𝑃 / 4) ∈ ℝ) |
10 | nnnn0 11176 | . . . . . 6 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℕ0) | |
11 | 10 | nn0ge0d 11231 | . . . . 5 ⊢ (𝑃 ∈ ℕ → 0 ≤ 𝑃) |
12 | 4pos 10993 | . . . . . . 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 10771 | . . . . 5 ⊢ (((𝑃 ∈ ℝ ∧ 0 ≤ 𝑃) ∧ (4 ∈ ℝ ∧ 0 < 4)) → 0 ≤ (𝑃 / 4)) | |
16 | 4, 11, 14, 15 | syl21anc 1317 | . . . 4 ⊢ (𝑃 ∈ ℕ → 0 ≤ (𝑃 / 4)) |
17 | 9, 16 | jca 553 | . . 3 ⊢ (𝑃 ∈ ℕ → ((𝑃 / 4) ∈ ℝ ∧ 0 ≤ (𝑃 / 4))) |
18 | flge0nn0 12483 | . . 3 ⊢ (((𝑃 / 4) ∈ ℝ ∧ 0 ≤ (𝑃 / 4)) → (⌊‘(𝑃 / 4)) ∈ ℕ0) | |
19 | 3, 17, 18 | 3syl 18 | . 2 ⊢ (𝜑 → (⌊‘(𝑃 / 4)) ∈ ℕ0) |
20 | 1, 19 | syl5eqel 2692 | 1 ⊢ (𝜑 → 𝑀 ∈ ℕ0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∖ cdif 3537 {csn 4125 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 ℝcr 9814 0cc0 9815 < clt 9953 ≤ cle 9954 / cdiv 10563 ℕcn 10897 2c2 10947 4c4 10949 ℕ0cn0 11169 ⌊cfl 12453 ℙcprime 15223 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-fl 12455 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-dvds 14822 df-prm 15224 |
This theorem is referenced by: gausslemma2dlem0h 24888 gausslemma2dlem2 24892 gausslemma2dlem3 24893 gausslemma2dlem4 24894 gausslemma2dlem6 24897 |
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