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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 11204 | . . 3 ⊢ 0 ∈ ℂ | |
2 | sqrtval 15182 | . . 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 14083 | . . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → ((𝑥↑2) = 0 ↔ 𝑥 = 0)) | |
6 | 5 | biimpa 476 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ (𝑥↑2) = 0) → 𝑥 = 0) |
7 | 6 | 3ad2antr1 1185 | . . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ ((𝑥↑2) = 0 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ+)) → 𝑥 = 0) |
8 | 7 | ex 412 | . . . . . 6 ⊢ (𝑥 ∈ ℂ → (((𝑥↑2) = 0 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ+) → 𝑥 = 0)) |
9 | sq0i 14155 | . . . . . . 7 ⊢ (𝑥 = 0 → (𝑥↑2) = 0) | |
10 | 0le0 12311 | . . . . . . . 8 ⊢ 0 ≤ 0 | |
11 | fveq2 6882 | . . . . . . . . 9 ⊢ (𝑥 = 0 → (ℜ‘𝑥) = (ℜ‘0)) | |
12 | re0 15097 | . . . . . . . . 9 ⊢ (ℜ‘0) = 0 | |
13 | 11, 12 | eqtrdi 2780 | . . . . . . . 8 ⊢ (𝑥 = 0 → (ℜ‘𝑥) = 0) |
14 | 10, 13 | breqtrrid 5177 | . . . . . . 7 ⊢ (𝑥 = 0 → 0 ≤ (ℜ‘𝑥)) |
15 | 0re 11214 | . . . . . . . . 9 ⊢ 0 ∈ ℝ | |
16 | eleq1 2813 | . . . . . . . . 9 ⊢ (𝑥 = 0 → (𝑥 ∈ ℝ ↔ 0 ∈ ℝ)) | |
17 | 15, 16 | mpbiri 258 | . . . . . . . 8 ⊢ (𝑥 = 0 → 𝑥 ∈ ℝ) |
18 | rennim 15184 | . . . . . . . 8 ⊢ (𝑥 ∈ ℝ → (i · 𝑥) ∉ ℝ+) | |
19 | 17, 18 | syl 17 | . . . . . . 7 ⊢ (𝑥 = 0 → (i · 𝑥) ∉ ℝ+) |
20 | 9, 14, 19 | 3jca 1125 | . . . . . 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 481 | . . . 4 ⊢ ((0 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (((𝑥↑2) = 0 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ+) ↔ 𝑥 = 0)) |
23 | 4, 22 | riota5 7388 | . . 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 2752 | 1 ⊢ (√‘0) = 0 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∉ wnel 3038 class class class wbr 5139 ‘cfv 6534 ℩crio 7357 (class class class)co 7402 ℂcc 11105 ℝcr 11106 0cc0 11107 ici 11109 · cmul 11112 ≤ cle 11247 2c2 12265 ℝ+crp 12972 ↑cexp 14025 ℜcre 15042 √csqrt 15178 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-pnf 11248 df-mnf 11249 df-xr 11250 df-ltxr 11251 df-le 11252 df-sub 11444 df-neg 11445 df-div 11870 df-nn 12211 df-2 12273 df-n0 12471 df-z 12557 df-uz 12821 df-rp 12973 df-seq 13965 df-exp 14026 df-cj 15044 df-re 15045 df-im 15046 df-sqrt 15180 |
This theorem is referenced by: sqrt00 15208 abs0 15230 cnsqrt00 15337 cphsqrtcl2 25038 cxpsqrt 26556 cxpsqrtth 26583 loglesqrt 26612 asin1 26745 normgt0 30852 norm0 30853 ftc1anclem3 37057 areacirc 37075 rrncmslem 37194 sqrtcval 42906 |
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