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Mirrors > Home > MPE Home > Th. List > 4sqlem7 | Structured version Visualization version GIF version |
Description: Lemma for 4sq 16290. (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 |
---|---|
4sqlem7 | ⊢ (𝜑 → (𝐵↑2) ≤ (((𝑀↑2) / 2) / 2)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 4sqlem5.2 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℤ) | |
2 | 4sqlem5.3 | . . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℕ) | |
3 | 4sqlem5.4 | . . . . . . 7 ⊢ 𝐵 = (((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2)) | |
4 | 1, 2, 3 | 4sqlem5 16268 | . . . . . 6 ⊢ (𝜑 → (𝐵 ∈ ℤ ∧ ((𝐴 − 𝐵) / 𝑀) ∈ ℤ)) |
5 | 4 | simpld 498 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℤ) |
6 | 5 | zred 12075 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ) |
7 | 2 | nnrpd 12417 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℝ+) |
8 | 7 | rphalfcld 12431 | . . . . 5 ⊢ (𝜑 → (𝑀 / 2) ∈ ℝ+) |
9 | 8 | rpred 12419 | . . . 4 ⊢ (𝜑 → (𝑀 / 2) ∈ ℝ) |
10 | 1, 2, 3 | 4sqlem6 16269 | . . . . 5 ⊢ (𝜑 → (-(𝑀 / 2) ≤ 𝐵 ∧ 𝐵 < (𝑀 / 2))) |
11 | 10 | simprd 499 | . . . 4 ⊢ (𝜑 → 𝐵 < (𝑀 / 2)) |
12 | 6, 9, 11 | ltled 10777 | . . 3 ⊢ (𝜑 → 𝐵 ≤ (𝑀 / 2)) |
13 | 10 | simpld 498 | . . . 4 ⊢ (𝜑 → -(𝑀 / 2) ≤ 𝐵) |
14 | 9, 6, 13 | lenegcon1d 11211 | . . 3 ⊢ (𝜑 → -𝐵 ≤ (𝑀 / 2)) |
15 | 8 | rpge0d 12423 | . . . 4 ⊢ (𝜑 → 0 ≤ (𝑀 / 2)) |
16 | lenegsq 14672 | . . . 4 ⊢ ((𝐵 ∈ ℝ ∧ (𝑀 / 2) ∈ ℝ ∧ 0 ≤ (𝑀 / 2)) → ((𝐵 ≤ (𝑀 / 2) ∧ -𝐵 ≤ (𝑀 / 2)) ↔ (𝐵↑2) ≤ ((𝑀 / 2)↑2))) | |
17 | 6, 9, 15, 16 | syl3anc 1368 | . . 3 ⊢ (𝜑 → ((𝐵 ≤ (𝑀 / 2) ∧ -𝐵 ≤ (𝑀 / 2)) ↔ (𝐵↑2) ≤ ((𝑀 / 2)↑2))) |
18 | 12, 14, 17 | mpbi2and 711 | . 2 ⊢ (𝜑 → (𝐵↑2) ≤ ((𝑀 / 2)↑2)) |
19 | 2cnd 11703 | . . . . 5 ⊢ (𝜑 → 2 ∈ ℂ) | |
20 | 19 | sqvald 13503 | . . . 4 ⊢ (𝜑 → (2↑2) = (2 · 2)) |
21 | 20 | oveq2d 7151 | . . 3 ⊢ (𝜑 → ((𝑀↑2) / (2↑2)) = ((𝑀↑2) / (2 · 2))) |
22 | 2 | nncnd 11641 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ℂ) |
23 | 2ne0 11729 | . . . . 5 ⊢ 2 ≠ 0 | |
24 | 23 | a1i 11 | . . . 4 ⊢ (𝜑 → 2 ≠ 0) |
25 | 22, 19, 24 | sqdivd 13519 | . . 3 ⊢ (𝜑 → ((𝑀 / 2)↑2) = ((𝑀↑2) / (2↑2))) |
26 | 22 | sqcld 13504 | . . . 4 ⊢ (𝜑 → (𝑀↑2) ∈ ℂ) |
27 | 26, 19, 19, 24, 24 | divdiv1d 11436 | . . 3 ⊢ (𝜑 → (((𝑀↑2) / 2) / 2) = ((𝑀↑2) / (2 · 2))) |
28 | 21, 25, 27 | 3eqtr4d 2843 | . 2 ⊢ (𝜑 → ((𝑀 / 2)↑2) = (((𝑀↑2) / 2) / 2)) |
29 | 18, 28 | breqtrd 5056 | 1 ⊢ (𝜑 → (𝐵↑2) ≤ (((𝑀↑2) / 2) / 2)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 class class class wbr 5030 (class class class)co 7135 ℝcr 10525 0cc0 10526 + caddc 10529 · cmul 10531 < clt 10664 ≤ cle 10665 − cmin 10859 -cneg 10860 / cdiv 11286 ℕcn 11625 2c2 11680 ℤcz 11969 mod cmo 13232 ↑cexp 13425 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-sup 8890 df-inf 8891 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-fl 13157 df-mod 13233 df-seq 13365 df-exp 13426 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 |
This theorem is referenced by: 4sqlem15 16285 4sqlem16 16286 2sqlem8 26010 |
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