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Mirrors > Home > MPE Home > Th. List > sumsqeq0 | Structured version Visualization version GIF version |
Description: Two real numbers are equal to 0 iff their Euclidean norm is. (Contributed by NM, 29-Apr-2005.) (Revised by Stefan O'Rear, 5-Oct-2014.) (Proof shortened by Mario Carneiro, 28-May-2016.) |
Ref | Expression |
---|---|
sumsqeq0 | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 = 0 ∧ 𝐵 = 0) ↔ ((𝐴↑2) + (𝐵↑2)) = 0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resqcl 13886 | . . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴↑2) ∈ ℝ) | |
2 | sqge0 13897 | . . . 4 ⊢ (𝐴 ∈ ℝ → 0 ≤ (𝐴↑2)) | |
3 | 1, 2 | jca 513 | . . 3 ⊢ (𝐴 ∈ ℝ → ((𝐴↑2) ∈ ℝ ∧ 0 ≤ (𝐴↑2))) |
4 | resqcl 13886 | . . . 4 ⊢ (𝐵 ∈ ℝ → (𝐵↑2) ∈ ℝ) | |
5 | sqge0 13897 | . . . 4 ⊢ (𝐵 ∈ ℝ → 0 ≤ (𝐵↑2)) | |
6 | 4, 5 | jca 513 | . . 3 ⊢ (𝐵 ∈ ℝ → ((𝐵↑2) ∈ ℝ ∧ 0 ≤ (𝐵↑2))) |
7 | add20 11529 | . . 3 ⊢ ((((𝐴↑2) ∈ ℝ ∧ 0 ≤ (𝐴↑2)) ∧ ((𝐵↑2) ∈ ℝ ∧ 0 ≤ (𝐵↑2))) → (((𝐴↑2) + (𝐵↑2)) = 0 ↔ ((𝐴↑2) = 0 ∧ (𝐵↑2) = 0))) | |
8 | 3, 6, 7 | syl2an 597 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (((𝐴↑2) + (𝐵↑2)) = 0 ↔ ((𝐴↑2) = 0 ∧ (𝐵↑2) = 0))) |
9 | recn 11003 | . . . 4 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
10 | sqeq0 13882 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((𝐴↑2) = 0 ↔ 𝐴 = 0)) | |
11 | 9, 10 | syl 17 | . . 3 ⊢ (𝐴 ∈ ℝ → ((𝐴↑2) = 0 ↔ 𝐴 = 0)) |
12 | recn 11003 | . . . 4 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℂ) | |
13 | sqeq0 13882 | . . . 4 ⊢ (𝐵 ∈ ℂ → ((𝐵↑2) = 0 ↔ 𝐵 = 0)) | |
14 | 12, 13 | syl 17 | . . 3 ⊢ (𝐵 ∈ ℝ → ((𝐵↑2) = 0 ↔ 𝐵 = 0)) |
15 | 11, 14 | bi2anan9 637 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (((𝐴↑2) = 0 ∧ (𝐵↑2) = 0) ↔ (𝐴 = 0 ∧ 𝐵 = 0))) |
16 | 8, 15 | bitr2d 281 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 = 0 ∧ 𝐵 = 0) ↔ ((𝐴↑2) + (𝐵↑2)) = 0)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1539 ∈ wcel 2104 class class class wbr 5081 (class class class)co 7303 ℂcc 10911 ℝcr 10912 0cc0 10913 + caddc 10916 ≤ cle 11052 2c2 12070 ↑cexp 13824 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7616 ax-cnex 10969 ax-resscn 10970 ax-1cn 10971 ax-icn 10972 ax-addcl 10973 ax-addrcl 10974 ax-mulcl 10975 ax-mulrcl 10976 ax-mulcom 10977 ax-addass 10978 ax-mulass 10979 ax-distr 10980 ax-i2m1 10981 ax-1ne0 10982 ax-1rid 10983 ax-rnegex 10984 ax-rrecex 10985 ax-cnre 10986 ax-pre-lttri 10987 ax-pre-lttrn 10988 ax-pre-ltadd 10989 ax-pre-mulgt0 10990 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3286 df-rab 3287 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-op 4572 df-uni 4845 df-iun 4933 df-br 5082 df-opab 5144 df-mpt 5165 df-tr 5199 df-id 5496 df-eprel 5502 df-po 5510 df-so 5511 df-fr 5551 df-we 5553 df-xp 5602 df-rel 5603 df-cnv 5604 df-co 5605 df-dm 5606 df-rn 5607 df-res 5608 df-ima 5609 df-pred 6213 df-ord 6280 df-on 6281 df-lim 6282 df-suc 6283 df-iota 6406 df-fun 6456 df-fn 6457 df-f 6458 df-f1 6459 df-fo 6460 df-f1o 6461 df-fv 6462 df-riota 7260 df-ov 7306 df-oprab 7307 df-mpo 7308 df-om 7741 df-2nd 7860 df-frecs 8124 df-wrecs 8155 df-recs 8229 df-rdg 8268 df-er 8525 df-en 8761 df-dom 8762 df-sdom 8763 df-pnf 11053 df-mnf 11054 df-xr 11055 df-ltxr 11056 df-le 11057 df-sub 11249 df-neg 11250 df-nn 12016 df-2 12078 df-n0 12276 df-z 12362 df-uz 12625 df-seq 13764 df-exp 13825 |
This theorem is referenced by: crreczi 13985 diophin 40630 |
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