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Mirrors > Home > MPE Home > Th. List > 4sqlem6 | Structured version Visualization version GIF version |
Description: Lemma for 4sq 16902. (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 11222 | . . . 4 ⊢ (𝜑 → 0 ∈ ℝ) | |
2 | 4sqlem5.2 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℤ) | |
3 | 2 | zred 12671 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
4 | 4sqlem5.3 | . . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℕ) | |
5 | 4 | nnred 12232 | . . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℝ) |
6 | 5 | rehalfcld 12464 | . . . . . 6 ⊢ (𝜑 → (𝑀 / 2) ∈ ℝ) |
7 | 3, 6 | readdcld 11248 | . . . . 5 ⊢ (𝜑 → (𝐴 + (𝑀 / 2)) ∈ ℝ) |
8 | 4 | nnrpd 13019 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℝ+) |
9 | 7, 8 | modcld 13845 | . . . 4 ⊢ (𝜑 → ((𝐴 + (𝑀 / 2)) mod 𝑀) ∈ ℝ) |
10 | modge0 13849 | . . . . 5 ⊢ (((𝐴 + (𝑀 / 2)) ∈ ℝ ∧ 𝑀 ∈ ℝ+) → 0 ≤ ((𝐴 + (𝑀 / 2)) mod 𝑀)) | |
11 | 7, 8, 10 | syl2anc 583 | . . . 4 ⊢ (𝜑 → 0 ≤ ((𝐴 + (𝑀 / 2)) mod 𝑀)) |
12 | 1, 9, 6, 11 | lesub1dd 11835 | . . 3 ⊢ (𝜑 → (0 − (𝑀 / 2)) ≤ (((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2))) |
13 | df-neg 11452 | . . 3 ⊢ -(𝑀 / 2) = (0 − (𝑀 / 2)) | |
14 | 4sqlem5.4 | . . 3 ⊢ 𝐵 = (((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2)) | |
15 | 12, 13, 14 | 3brtr4g 5182 | . 2 ⊢ (𝜑 → -(𝑀 / 2) ≤ 𝐵) |
16 | modlt 13850 | . . . . . 6 ⊢ (((𝐴 + (𝑀 / 2)) ∈ ℝ ∧ 𝑀 ∈ ℝ+) → ((𝐴 + (𝑀 / 2)) mod 𝑀) < 𝑀) | |
17 | 7, 8, 16 | syl2anc 583 | . . . . 5 ⊢ (𝜑 → ((𝐴 + (𝑀 / 2)) mod 𝑀) < 𝑀) |
18 | 4 | nncnd 12233 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℂ) |
19 | 18 | 2halvesd 12463 | . . . . 5 ⊢ (𝜑 → ((𝑀 / 2) + (𝑀 / 2)) = 𝑀) |
20 | 17, 19 | breqtrrd 5176 | . . . 4 ⊢ (𝜑 → ((𝐴 + (𝑀 / 2)) mod 𝑀) < ((𝑀 / 2) + (𝑀 / 2))) |
21 | 9, 6, 6 | ltsubaddd 11815 | . . . 4 ⊢ (𝜑 → ((((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2)) < (𝑀 / 2) ↔ ((𝐴 + (𝑀 / 2)) mod 𝑀) < ((𝑀 / 2) + (𝑀 / 2)))) |
22 | 20, 21 | mpbird 257 | . . 3 ⊢ (𝜑 → (((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2)) < (𝑀 / 2)) |
23 | 14, 22 | eqbrtrid 5183 | . 2 ⊢ (𝜑 → 𝐵 < (𝑀 / 2)) |
24 | 15, 23 | jca 511 | 1 ⊢ (𝜑 → (-(𝑀 / 2) ≤ 𝐵 ∧ 𝐵 < (𝑀 / 2))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2105 class class class wbr 5148 (class class class)co 7412 ℝcr 11112 0cc0 11113 + caddc 11116 < clt 11253 ≤ cle 11254 − cmin 11449 -cneg 11450 / cdiv 11876 ℕcn 12217 2c2 12272 ℤcz 12563 ℝ+crp 12979 mod cmo 13839 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7728 ax-cnex 11169 ax-resscn 11170 ax-1cn 11171 ax-icn 11172 ax-addcl 11173 ax-addrcl 11174 ax-mulcl 11175 ax-mulrcl 11176 ax-mulcom 11177 ax-addass 11178 ax-mulass 11179 ax-distr 11180 ax-i2m1 11181 ax-1ne0 11182 ax-1rid 11183 ax-rnegex 11184 ax-rrecex 11185 ax-cnre 11186 ax-pre-lttri 11187 ax-pre-lttrn 11188 ax-pre-ltadd 11189 ax-pre-mulgt0 11190 ax-pre-sup 11191 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 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 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7859 df-2nd 7979 df-frecs 8269 df-wrecs 8300 df-recs 8374 df-rdg 8413 df-er 8706 df-en 8943 df-dom 8944 df-sdom 8945 df-sup 9440 df-inf 9441 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-div 11877 df-nn 12218 df-2 12280 df-n0 12478 df-z 12564 df-uz 12828 df-rp 12980 df-fl 13762 df-mod 13840 |
This theorem is referenced by: 4sqlem7 16882 4sqlem10 16885 |
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