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Mirrors > Home > MPE Home > Th. List > 4sqlem6 | Structured version Visualization version GIF version |
Description: Lemma for 4sq 16879. (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 11199 | . . . 4 ⊢ (𝜑 → 0 ∈ ℝ) | |
2 | 4sqlem5.2 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℤ) | |
3 | 2 | zred 12648 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
4 | 4sqlem5.3 | . . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℕ) | |
5 | 4 | nnred 12209 | . . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℝ) |
6 | 5 | rehalfcld 12441 | . . . . . 6 ⊢ (𝜑 → (𝑀 / 2) ∈ ℝ) |
7 | 3, 6 | readdcld 11225 | . . . . 5 ⊢ (𝜑 → (𝐴 + (𝑀 / 2)) ∈ ℝ) |
8 | 4 | nnrpd 12996 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℝ+) |
9 | 7, 8 | modcld 13822 | . . . 4 ⊢ (𝜑 → ((𝐴 + (𝑀 / 2)) mod 𝑀) ∈ ℝ) |
10 | modge0 13826 | . . . . 5 ⊢ (((𝐴 + (𝑀 / 2)) ∈ ℝ ∧ 𝑀 ∈ ℝ+) → 0 ≤ ((𝐴 + (𝑀 / 2)) mod 𝑀)) | |
11 | 7, 8, 10 | syl2anc 584 | . . . 4 ⊢ (𝜑 → 0 ≤ ((𝐴 + (𝑀 / 2)) mod 𝑀)) |
12 | 1, 9, 6, 11 | lesub1dd 11812 | . . 3 ⊢ (𝜑 → (0 − (𝑀 / 2)) ≤ (((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2))) |
13 | df-neg 11429 | . . 3 ⊢ -(𝑀 / 2) = (0 − (𝑀 / 2)) | |
14 | 4sqlem5.4 | . . 3 ⊢ 𝐵 = (((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2)) | |
15 | 12, 13, 14 | 3brtr4g 5175 | . 2 ⊢ (𝜑 → -(𝑀 / 2) ≤ 𝐵) |
16 | modlt 13827 | . . . . . 6 ⊢ (((𝐴 + (𝑀 / 2)) ∈ ℝ ∧ 𝑀 ∈ ℝ+) → ((𝐴 + (𝑀 / 2)) mod 𝑀) < 𝑀) | |
17 | 7, 8, 16 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → ((𝐴 + (𝑀 / 2)) mod 𝑀) < 𝑀) |
18 | 4 | nncnd 12210 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℂ) |
19 | 18 | 2halvesd 12440 | . . . . 5 ⊢ (𝜑 → ((𝑀 / 2) + (𝑀 / 2)) = 𝑀) |
20 | 17, 19 | breqtrrd 5169 | . . . 4 ⊢ (𝜑 → ((𝐴 + (𝑀 / 2)) mod 𝑀) < ((𝑀 / 2) + (𝑀 / 2))) |
21 | 9, 6, 6 | ltsubaddd 11792 | . . . 4 ⊢ (𝜑 → ((((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2)) < (𝑀 / 2) ↔ ((𝐴 + (𝑀 / 2)) mod 𝑀) < ((𝑀 / 2) + (𝑀 / 2)))) |
22 | 20, 21 | mpbird 256 | . . 3 ⊢ (𝜑 → (((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2)) < (𝑀 / 2)) |
23 | 14, 22 | eqbrtrid 5176 | . 2 ⊢ (𝜑 → 𝐵 < (𝑀 / 2)) |
24 | 15, 23 | jca 512 | 1 ⊢ (𝜑 → (-(𝑀 / 2) ≤ 𝐵 ∧ 𝐵 < (𝑀 / 2))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 class class class wbr 5141 (class class class)co 7393 ℝcr 11091 0cc0 11092 + caddc 11095 < clt 11230 ≤ cle 11231 − cmin 11426 -cneg 11427 / cdiv 11853 ℕcn 12194 2c2 12249 ℤcz 12540 ℝ+crp 12956 mod cmo 13816 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-cnex 11148 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168 ax-pre-mulgt0 11169 ax-pre-sup 11170 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-om 7839 df-2nd 7958 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-er 8686 df-en 8923 df-dom 8924 df-sdom 8925 df-sup 9419 df-inf 9420 df-pnf 11232 df-mnf 11233 df-xr 11234 df-ltxr 11235 df-le 11236 df-sub 11428 df-neg 11429 df-div 11854 df-nn 12195 df-2 12257 df-n0 12455 df-z 12541 df-uz 12805 df-rp 12957 df-fl 13739 df-mod 13817 |
This theorem is referenced by: 4sqlem7 16859 4sqlem10 16862 |
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