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Mirrors > Home > MPE Home > Th. List > 4sqlem6 | Structured version Visualization version GIF version |
Description: Lemma for 4sq 16514. (Contributed by Mario Carneiro, 15-Jul-2014.) |
Ref | Expression |
---|---|
4sqlem5.2 | ⊢ (𝜑 → 𝐴 ∈ ℤ) |
4sqlem5.3 | ⊢ (𝜑 → 𝑀 ∈ ℕ) |
4sqlem5.4 | ⊢ 𝐵 = (((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2)) |
Ref | Expression |
---|---|
4sqlem6 | ⊢ (𝜑 → (-(𝑀 / 2) ≤ 𝐵 ∧ 𝐵 < (𝑀 / 2))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0red 10833 | . . . 4 ⊢ (𝜑 → 0 ∈ ℝ) | |
2 | 4sqlem5.2 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℤ) | |
3 | 2 | zred 12279 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
4 | 4sqlem5.3 | . . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℕ) | |
5 | 4 | nnred 11842 | . . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℝ) |
6 | 5 | rehalfcld 12074 | . . . . . 6 ⊢ (𝜑 → (𝑀 / 2) ∈ ℝ) |
7 | 3, 6 | readdcld 10859 | . . . . 5 ⊢ (𝜑 → (𝐴 + (𝑀 / 2)) ∈ ℝ) |
8 | 4 | nnrpd 12623 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℝ+) |
9 | 7, 8 | modcld 13445 | . . . 4 ⊢ (𝜑 → ((𝐴 + (𝑀 / 2)) mod 𝑀) ∈ ℝ) |
10 | modge0 13449 | . . . . 5 ⊢ (((𝐴 + (𝑀 / 2)) ∈ ℝ ∧ 𝑀 ∈ ℝ+) → 0 ≤ ((𝐴 + (𝑀 / 2)) mod 𝑀)) | |
11 | 7, 8, 10 | syl2anc 587 | . . . 4 ⊢ (𝜑 → 0 ≤ ((𝐴 + (𝑀 / 2)) mod 𝑀)) |
12 | 1, 9, 6, 11 | lesub1dd 11445 | . . 3 ⊢ (𝜑 → (0 − (𝑀 / 2)) ≤ (((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2))) |
13 | df-neg 11062 | . . 3 ⊢ -(𝑀 / 2) = (0 − (𝑀 / 2)) | |
14 | 4sqlem5.4 | . . 3 ⊢ 𝐵 = (((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2)) | |
15 | 12, 13, 14 | 3brtr4g 5084 | . 2 ⊢ (𝜑 → -(𝑀 / 2) ≤ 𝐵) |
16 | modlt 13450 | . . . . . 6 ⊢ (((𝐴 + (𝑀 / 2)) ∈ ℝ ∧ 𝑀 ∈ ℝ+) → ((𝐴 + (𝑀 / 2)) mod 𝑀) < 𝑀) | |
17 | 7, 8, 16 | syl2anc 587 | . . . . 5 ⊢ (𝜑 → ((𝐴 + (𝑀 / 2)) mod 𝑀) < 𝑀) |
18 | 4 | nncnd 11843 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℂ) |
19 | 18 | 2halvesd 12073 | . . . . 5 ⊢ (𝜑 → ((𝑀 / 2) + (𝑀 / 2)) = 𝑀) |
20 | 17, 19 | breqtrrd 5078 | . . . 4 ⊢ (𝜑 → ((𝐴 + (𝑀 / 2)) mod 𝑀) < ((𝑀 / 2) + (𝑀 / 2))) |
21 | 9, 6, 6 | ltsubaddd 11425 | . . . 4 ⊢ (𝜑 → ((((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2)) < (𝑀 / 2) ↔ ((𝐴 + (𝑀 / 2)) mod 𝑀) < ((𝑀 / 2) + (𝑀 / 2)))) |
22 | 20, 21 | mpbird 260 | . . 3 ⊢ (𝜑 → (((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2)) < (𝑀 / 2)) |
23 | 14, 22 | eqbrtrid 5085 | . 2 ⊢ (𝜑 → 𝐵 < (𝑀 / 2)) |
24 | 15, 23 | jca 515 | 1 ⊢ (𝜑 → (-(𝑀 / 2) ≤ 𝐵 ∧ 𝐵 < (𝑀 / 2))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 class class class wbr 5050 (class class class)co 7210 ℝcr 10725 0cc0 10726 + caddc 10729 < clt 10864 ≤ cle 10865 − cmin 11059 -cneg 11060 / cdiv 11486 ℕcn 11827 2c2 11882 ℤcz 12173 ℝ+crp 12583 mod cmo 13439 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5189 ax-nul 5196 ax-pow 5255 ax-pr 5319 ax-un 7520 ax-cnex 10782 ax-resscn 10783 ax-1cn 10784 ax-icn 10785 ax-addcl 10786 ax-addrcl 10787 ax-mulcl 10788 ax-mulrcl 10789 ax-mulcom 10790 ax-addass 10791 ax-mulass 10792 ax-distr 10793 ax-i2m1 10794 ax-1ne0 10795 ax-1rid 10796 ax-rnegex 10797 ax-rrecex 10798 ax-cnre 10799 ax-pre-lttri 10800 ax-pre-lttrn 10801 ax-pre-ltadd 10802 ax-pre-mulgt0 10803 ax-pre-sup 10804 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2940 df-nel 3044 df-ral 3063 df-rex 3064 df-reu 3065 df-rmo 3066 df-rab 3067 df-v 3407 df-sbc 3692 df-csb 3809 df-dif 3866 df-un 3868 df-in 3870 df-ss 3880 df-pss 3882 df-nul 4235 df-if 4437 df-pw 4512 df-sn 4539 df-pr 4541 df-tp 4543 df-op 4545 df-uni 4817 df-iun 4903 df-br 5051 df-opab 5113 df-mpt 5133 df-tr 5159 df-id 5452 df-eprel 5457 df-po 5465 df-so 5466 df-fr 5506 df-we 5508 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6157 df-ord 6213 df-on 6214 df-lim 6215 df-suc 6216 df-iota 6335 df-fun 6379 df-fn 6380 df-f 6381 df-f1 6382 df-fo 6383 df-f1o 6384 df-fv 6385 df-riota 7167 df-ov 7213 df-oprab 7214 df-mpo 7215 df-om 7642 df-wrecs 8044 df-recs 8105 df-rdg 8143 df-er 8388 df-en 8624 df-dom 8625 df-sdom 8626 df-sup 9055 df-inf 9056 df-pnf 10866 df-mnf 10867 df-xr 10868 df-ltxr 10869 df-le 10870 df-sub 11061 df-neg 11062 df-div 11487 df-nn 11828 df-2 11890 df-n0 12088 df-z 12174 df-uz 12436 df-rp 12584 df-fl 13364 df-mod 13440 |
This theorem is referenced by: 4sqlem7 16494 4sqlem10 16497 |
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