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Mirrors > Home > MPE Home > Th. List > sqrt0 | Structured version Visualization version GIF version |
Description: The square root of zero is zero. (Contributed by Mario Carneiro, 9-Jul-2013.) |
Ref | Expression |
---|---|
sqrt0 | ⊢ (√‘0) = 0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0cn 11155 | . . 3 ⊢ 0 ∈ ℂ | |
2 | sqrtval 15131 | . . 3 ⊢ (0 ∈ ℂ → (√‘0) = (℩𝑥 ∈ ℂ ((𝑥↑2) = 0 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ+))) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ (√‘0) = (℩𝑥 ∈ ℂ ((𝑥↑2) = 0 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ+)) |
4 | id 22 | . . . 4 ⊢ (0 ∈ ℂ → 0 ∈ ℂ) | |
5 | sqeq0 14034 | . . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → ((𝑥↑2) = 0 ↔ 𝑥 = 0)) | |
6 | 5 | biimpa 478 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ (𝑥↑2) = 0) → 𝑥 = 0) |
7 | 6 | 3ad2antr1 1189 | . . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ ((𝑥↑2) = 0 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ+)) → 𝑥 = 0) |
8 | 7 | ex 414 | . . . . . 6 ⊢ (𝑥 ∈ ℂ → (((𝑥↑2) = 0 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ+) → 𝑥 = 0)) |
9 | sq0i 14106 | . . . . . . 7 ⊢ (𝑥 = 0 → (𝑥↑2) = 0) | |
10 | 0le0 12262 | . . . . . . . 8 ⊢ 0 ≤ 0 | |
11 | fveq2 6846 | . . . . . . . . 9 ⊢ (𝑥 = 0 → (ℜ‘𝑥) = (ℜ‘0)) | |
12 | re0 15046 | . . . . . . . . 9 ⊢ (ℜ‘0) = 0 | |
13 | 11, 12 | eqtrdi 2789 | . . . . . . . 8 ⊢ (𝑥 = 0 → (ℜ‘𝑥) = 0) |
14 | 10, 13 | breqtrrid 5147 | . . . . . . 7 ⊢ (𝑥 = 0 → 0 ≤ (ℜ‘𝑥)) |
15 | 0re 11165 | . . . . . . . . 9 ⊢ 0 ∈ ℝ | |
16 | eleq1 2822 | . . . . . . . . 9 ⊢ (𝑥 = 0 → (𝑥 ∈ ℝ ↔ 0 ∈ ℝ)) | |
17 | 15, 16 | mpbiri 258 | . . . . . . . 8 ⊢ (𝑥 = 0 → 𝑥 ∈ ℝ) |
18 | rennim 15133 | . . . . . . . 8 ⊢ (𝑥 ∈ ℝ → (i · 𝑥) ∉ ℝ+) | |
19 | 17, 18 | syl 17 | . . . . . . 7 ⊢ (𝑥 = 0 → (i · 𝑥) ∉ ℝ+) |
20 | 9, 14, 19 | 3jca 1129 | . . . . . 6 ⊢ (𝑥 = 0 → ((𝑥↑2) = 0 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ+)) |
21 | 8, 20 | impbid1 224 | . . . . 5 ⊢ (𝑥 ∈ ℂ → (((𝑥↑2) = 0 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ+) ↔ 𝑥 = 0)) |
22 | 21 | adantl 483 | . . . 4 ⊢ ((0 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (((𝑥↑2) = 0 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ+) ↔ 𝑥 = 0)) |
23 | 4, 22 | riota5 7347 | . . 3 ⊢ (0 ∈ ℂ → (℩𝑥 ∈ ℂ ((𝑥↑2) = 0 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ+)) = 0) |
24 | 1, 23 | ax-mp 5 | . 2 ⊢ (℩𝑥 ∈ ℂ ((𝑥↑2) = 0 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ+)) = 0 |
25 | 3, 24 | eqtri 2761 | 1 ⊢ (√‘0) = 0 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∉ wnel 3046 class class class wbr 5109 ‘cfv 6500 ℩crio 7316 (class class class)co 7361 ℂcc 11057 ℝcr 11058 0cc0 11059 ici 11061 · cmul 11064 ≤ cle 11198 2c2 12216 ℝ+crp 12923 ↑cexp 13976 ℜcre 14991 √csqrt 15127 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-er 8654 df-en 8890 df-dom 8891 df-sdom 8892 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-div 11821 df-nn 12162 df-2 12224 df-n0 12422 df-z 12508 df-uz 12772 df-rp 12924 df-seq 13916 df-exp 13977 df-cj 14993 df-re 14994 df-im 14995 df-sqrt 15129 |
This theorem is referenced by: sqrt00 15157 abs0 15179 cnsqrt00 15286 cphsqrtcl2 24573 cxpsqrt 26081 cxpsqrtth 26107 loglesqrt 26134 asin1 26267 normgt0 30118 norm0 30119 ftc1anclem3 36203 areacirc 36221 rrncmslem 36341 sqrtcval 42005 |
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