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Mirrors > Home > MPE Home > Th. List > 4sqlem7 | Structured version Visualization version GIF version |
Description: Lemma for 4sq 15506. (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 15484 | . . . . . 6 ⊢ (𝜑 → (𝐵 ∈ ℤ ∧ ((𝐴 − 𝐵) / 𝑀) ∈ ℤ)) |
5 | 4 | simpld 474 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℤ) |
6 | 5 | zred 11358 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ) |
7 | 2 | nnrpd 11746 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℝ+) |
8 | 7 | rphalfcld 11760 | . . . . 5 ⊢ (𝜑 → (𝑀 / 2) ∈ ℝ+) |
9 | 8 | rpred 11748 | . . . 4 ⊢ (𝜑 → (𝑀 / 2) ∈ ℝ) |
10 | 1, 2, 3 | 4sqlem6 15485 | . . . . 5 ⊢ (𝜑 → (-(𝑀 / 2) ≤ 𝐵 ∧ 𝐵 < (𝑀 / 2))) |
11 | 10 | simprd 478 | . . . 4 ⊢ (𝜑 → 𝐵 < (𝑀 / 2)) |
12 | 6, 9, 11 | ltled 10064 | . . 3 ⊢ (𝜑 → 𝐵 ≤ (𝑀 / 2)) |
13 | 10 | simpld 474 | . . . 4 ⊢ (𝜑 → -(𝑀 / 2) ≤ 𝐵) |
14 | 9, 6, 13 | lenegcon1d 10488 | . . 3 ⊢ (𝜑 → -𝐵 ≤ (𝑀 / 2)) |
15 | 8 | rpge0d 11752 | . . . 4 ⊢ (𝜑 → 0 ≤ (𝑀 / 2)) |
16 | lenegsq 13908 | . . . 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 1318 | . . 3 ⊢ (𝜑 → ((𝐵 ≤ (𝑀 / 2) ∧ -𝐵 ≤ (𝑀 / 2)) ↔ (𝐵↑2) ≤ ((𝑀 / 2)↑2))) |
18 | 12, 14, 17 | mpbi2and 958 | . 2 ⊢ (𝜑 → (𝐵↑2) ≤ ((𝑀 / 2)↑2)) |
19 | 2cnd 10970 | . . . . 5 ⊢ (𝜑 → 2 ∈ ℂ) | |
20 | 19 | sqvald 12867 | . . . 4 ⊢ (𝜑 → (2↑2) = (2 · 2)) |
21 | 20 | oveq2d 6565 | . . 3 ⊢ (𝜑 → ((𝑀↑2) / (2↑2)) = ((𝑀↑2) / (2 · 2))) |
22 | 2 | nncnd 10913 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ℂ) |
23 | 2ne0 10990 | . . . . 5 ⊢ 2 ≠ 0 | |
24 | 23 | a1i 11 | . . . 4 ⊢ (𝜑 → 2 ≠ 0) |
25 | 22, 19, 24 | sqdivd 12883 | . . 3 ⊢ (𝜑 → ((𝑀 / 2)↑2) = ((𝑀↑2) / (2↑2))) |
26 | 22 | sqcld 12868 | . . . 4 ⊢ (𝜑 → (𝑀↑2) ∈ ℂ) |
27 | 26, 19, 19, 24, 24 | divdiv1d 10711 | . . 3 ⊢ (𝜑 → (((𝑀↑2) / 2) / 2) = ((𝑀↑2) / (2 · 2))) |
28 | 21, 25, 27 | 3eqtr4d 2654 | . 2 ⊢ (𝜑 → ((𝑀 / 2)↑2) = (((𝑀↑2) / 2) / 2)) |
29 | 18, 28 | breqtrd 4609 | 1 ⊢ (𝜑 → (𝐵↑2) ≤ (((𝑀↑2) / 2) / 2)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 class class class wbr 4583 (class class class)co 6549 ℝcr 9814 0cc0 9815 + caddc 9818 · cmul 9820 < clt 9953 ≤ cle 9954 − cmin 10145 -cneg 10146 / cdiv 10563 ℕcn 10897 2c2 10947 ℤcz 11254 mod cmo 12530 ↑cexp 12722 |
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-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-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 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-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-fl 12455 df-mod 12531 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 |
This theorem is referenced by: 4sqlem15 15501 4sqlem16 15502 2sqlem8 24951 |
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