| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| 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 11253 | . . 3 ⊢ 0 ∈ ℂ | |
| 2 | sqrtval 15276 | . . 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 14160 | . . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → ((𝑥↑2) = 0 ↔ 𝑥 = 0)) | |
| 6 | 5 | biimpa 476 | . . . . . . . 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 412 | . . . . . 6 ⊢ (𝑥 ∈ ℂ → (((𝑥↑2) = 0 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ+) → 𝑥 = 0)) |
| 9 | sq0i 14232 | . . . . . . 7 ⊢ (𝑥 = 0 → (𝑥↑2) = 0) | |
| 10 | 0le0 12367 | . . . . . . . 8 ⊢ 0 ≤ 0 | |
| 11 | fveq2 6906 | . . . . . . . . 9 ⊢ (𝑥 = 0 → (ℜ‘𝑥) = (ℜ‘0)) | |
| 12 | re0 15191 | . . . . . . . . 9 ⊢ (ℜ‘0) = 0 | |
| 13 | 11, 12 | eqtrdi 2793 | . . . . . . . 8 ⊢ (𝑥 = 0 → (ℜ‘𝑥) = 0) |
| 14 | 10, 13 | breqtrrid 5181 | . . . . . . 7 ⊢ (𝑥 = 0 → 0 ≤ (ℜ‘𝑥)) |
| 15 | 0re 11263 | . . . . . . . . 9 ⊢ 0 ∈ ℝ | |
| 16 | eleq1 2829 | . . . . . . . . 9 ⊢ (𝑥 = 0 → (𝑥 ∈ ℝ ↔ 0 ∈ ℝ)) | |
| 17 | 15, 16 | mpbiri 258 | . . . . . . . 8 ⊢ (𝑥 = 0 → 𝑥 ∈ ℝ) |
| 18 | rennim 15278 | . . . . . . . 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 225 | . . . . 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 7417 | . . 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 2765 | 1 ⊢ (√‘0) = 0 |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∉ wnel 3046 class class class wbr 5143 ‘cfv 6561 ℩crio 7387 (class class class)co 7431 ℂcc 11153 ℝcr 11154 0cc0 11155 ici 11157 · cmul 11160 ≤ cle 11296 2c2 12321 ℝ+crp 13034 ↑cexp 14102 ℜcre 15136 √csqrt 15272 |
| 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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-seq 14043 df-exp 14103 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 |
| This theorem is referenced by: sqrt00 15302 abs0 15324 cnsqrt00 15431 cphsqrtcl2 25220 cxpsqrt 26745 cxpsqrtth 26772 loglesqrt 26804 asin1 26937 normgt0 31146 norm0 31147 ftc1anclem3 37702 areacirc 37720 rrncmslem 37839 sqrtcval 43654 |
| Copyright terms: Public domain | W3C validator |