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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 11230 | . . 3 ⊢ 0 ∈ ℂ | |
2 | sqrtval 15210 | . . 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 14110 | . . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → ((𝑥↑2) = 0 ↔ 𝑥 = 0)) | |
6 | 5 | biimpa 476 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ (𝑥↑2) = 0) → 𝑥 = 0) |
7 | 6 | 3ad2antr1 1186 | . . . . . . 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 14182 | . . . . . . 7 ⊢ (𝑥 = 0 → (𝑥↑2) = 0) | |
10 | 0le0 12337 | . . . . . . . 8 ⊢ 0 ≤ 0 | |
11 | fveq2 6891 | . . . . . . . . 9 ⊢ (𝑥 = 0 → (ℜ‘𝑥) = (ℜ‘0)) | |
12 | re0 15125 | . . . . . . . . 9 ⊢ (ℜ‘0) = 0 | |
13 | 11, 12 | eqtrdi 2784 | . . . . . . . 8 ⊢ (𝑥 = 0 → (ℜ‘𝑥) = 0) |
14 | 10, 13 | breqtrrid 5180 | . . . . . . 7 ⊢ (𝑥 = 0 → 0 ≤ (ℜ‘𝑥)) |
15 | 0re 11240 | . . . . . . . . 9 ⊢ 0 ∈ ℝ | |
16 | eleq1 2817 | . . . . . . . . 9 ⊢ (𝑥 = 0 → (𝑥 ∈ ℝ ↔ 0 ∈ ℝ)) | |
17 | 15, 16 | mpbiri 258 | . . . . . . . 8 ⊢ (𝑥 = 0 → 𝑥 ∈ ℝ) |
18 | rennim 15212 | . . . . . . . 8 ⊢ (𝑥 ∈ ℝ → (i · 𝑥) ∉ ℝ+) | |
19 | 17, 18 | syl 17 | . . . . . . 7 ⊢ (𝑥 = 0 → (i · 𝑥) ∉ ℝ+) |
20 | 9, 14, 19 | 3jca 1126 | . . . . . 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 7400 | . . 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 2756 | 1 ⊢ (√‘0) = 0 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ∉ wnel 3042 class class class wbr 5142 ‘cfv 6542 ℩crio 7369 (class class class)co 7414 ℂcc 11130 ℝcr 11131 0cc0 11132 ici 11134 · cmul 11137 ≤ cle 11273 2c2 12291 ℝ+crp 13000 ↑cexp 14052 ℜcre 15070 √csqrt 15206 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-div 11896 df-nn 12237 df-2 12299 df-n0 12497 df-z 12583 df-uz 12847 df-rp 13001 df-seq 13993 df-exp 14053 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 |
This theorem is referenced by: sqrt00 15236 abs0 15258 cnsqrt00 15365 cphsqrtcl2 25107 cxpsqrt 26630 cxpsqrtth 26657 loglesqrt 26686 asin1 26819 normgt0 30930 norm0 30931 ftc1anclem3 37162 areacirc 37180 rrncmslem 37299 sqrtcval 43065 |
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