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Mirrors > Home > MPE Home > Th. List > msqge0 | Structured version Visualization version GIF version |
Description: A square is nonnegative. (Contributed by NM, 23-May-2007.) (Revised by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
msqge0 | ⊢ (𝐴 ∈ ℝ → 0 ≤ (𝐴 · 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq12 7144 | . . . . 5 ⊢ ((𝐴 = 0 ∧ 𝐴 = 0) → (𝐴 · 𝐴) = (0 · 0)) | |
2 | 1 | anidms 570 | . . . 4 ⊢ (𝐴 = 0 → (𝐴 · 𝐴) = (0 · 0)) |
3 | 0cn 10622 | . . . . 5 ⊢ 0 ∈ ℂ | |
4 | 3 | mul01i 10819 | . . . 4 ⊢ (0 · 0) = 0 |
5 | 2, 4 | eqtrdi 2849 | . . 3 ⊢ (𝐴 = 0 → (𝐴 · 𝐴) = 0) |
6 | 5 | breq2d 5042 | . 2 ⊢ (𝐴 = 0 → (0 ≤ (𝐴 · 𝐴) ↔ 0 ≤ 0)) |
7 | 0red 10633 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → 0 ∈ ℝ) | |
8 | simpl 486 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → 𝐴 ∈ ℝ) | |
9 | 8, 8 | remulcld 10660 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → (𝐴 · 𝐴) ∈ ℝ) |
10 | msqgt0 11149 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → 0 < (𝐴 · 𝐴)) | |
11 | 7, 9, 10 | ltled 10777 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → 0 ≤ (𝐴 · 𝐴)) |
12 | 0re 10632 | . . 3 ⊢ 0 ∈ ℝ | |
13 | leid 10725 | . . 3 ⊢ (0 ∈ ℝ → 0 ≤ 0) | |
14 | 12, 13 | mp1i 13 | . 2 ⊢ (𝐴 ∈ ℝ → 0 ≤ 0) |
15 | 6, 11, 14 | pm2.61ne 3072 | 1 ⊢ (𝐴 ∈ ℝ → 0 ≤ (𝐴 · 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 class class class wbr 5030 (class class class)co 7135 ℝcr 10525 0cc0 10526 · cmul 10531 ≤ cle 10665 |
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-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 |
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-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 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-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-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 |
This theorem is referenced by: msqge0i 11167 msqge0d 11197 recextlem2 11260 sqge0 13497 bernneq 13586 |
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