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Mirrors > Home > MPE Home > Th. List > eqsqrtd | Structured version Visualization version GIF version |
Description: A deduction for showing that a number equals the square root of another. (Contributed by Mario Carneiro, 3-Apr-2015.) |
Ref | Expression |
---|---|
eqsqrtd.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
eqsqrtd.2 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
eqsqrtd.3 | ⊢ (𝜑 → (𝐴↑2) = 𝐵) |
eqsqrtd.4 | ⊢ (𝜑 → 0 ≤ (ℜ‘𝐴)) |
eqsqrtd.5 | ⊢ (𝜑 → ¬ (i · 𝐴) ∈ ℝ+) |
Ref | Expression |
---|---|
eqsqrtd | ⊢ (𝜑 → 𝐴 = (√‘𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqsqrtd.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
2 | sqreu 14319 | . . 3 ⊢ (𝐵 ∈ ℂ → ∃!𝑥 ∈ ℂ ((𝑥↑2) = 𝐵 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ+)) | |
3 | reurmo 3300 | . . 3 ⊢ (∃!𝑥 ∈ ℂ ((𝑥↑2) = 𝐵 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ+) → ∃*𝑥 ∈ ℂ ((𝑥↑2) = 𝐵 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ+)) | |
4 | 1, 2, 3 | 3syl 18 | . 2 ⊢ (𝜑 → ∃*𝑥 ∈ ℂ ((𝑥↑2) = 𝐵 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ+)) |
5 | eqsqrtd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
6 | eqsqrtd.3 | . . 3 ⊢ (𝜑 → (𝐴↑2) = 𝐵) | |
7 | eqsqrtd.4 | . . 3 ⊢ (𝜑 → 0 ≤ (ℜ‘𝐴)) | |
8 | eqsqrtd.5 | . . . 4 ⊢ (𝜑 → ¬ (i · 𝐴) ∈ ℝ+) | |
9 | df-nel 3036 | . . . 4 ⊢ ((i · 𝐴) ∉ ℝ+ ↔ ¬ (i · 𝐴) ∈ ℝ+) | |
10 | 8, 9 | sylibr 224 | . . 3 ⊢ (𝜑 → (i · 𝐴) ∉ ℝ+) |
11 | 6, 7, 10 | 3jca 1123 | . 2 ⊢ (𝜑 → ((𝐴↑2) = 𝐵 ∧ 0 ≤ (ℜ‘𝐴) ∧ (i · 𝐴) ∉ ℝ+)) |
12 | sqrtcl 14320 | . . 3 ⊢ (𝐵 ∈ ℂ → (√‘𝐵) ∈ ℂ) | |
13 | 1, 12 | syl 17 | . 2 ⊢ (𝜑 → (√‘𝐵) ∈ ℂ) |
14 | sqrtthlem 14321 | . . 3 ⊢ (𝐵 ∈ ℂ → (((√‘𝐵)↑2) = 𝐵 ∧ 0 ≤ (ℜ‘(√‘𝐵)) ∧ (i · (√‘𝐵)) ∉ ℝ+)) | |
15 | 1, 14 | syl 17 | . 2 ⊢ (𝜑 → (((√‘𝐵)↑2) = 𝐵 ∧ 0 ≤ (ℜ‘(√‘𝐵)) ∧ (i · (√‘𝐵)) ∉ ℝ+)) |
16 | oveq1 6821 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥↑2) = (𝐴↑2)) | |
17 | 16 | eqeq1d 2762 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥↑2) = 𝐵 ↔ (𝐴↑2) = 𝐵)) |
18 | fveq2 6353 | . . . . 5 ⊢ (𝑥 = 𝐴 → (ℜ‘𝑥) = (ℜ‘𝐴)) | |
19 | 18 | breq2d 4816 | . . . 4 ⊢ (𝑥 = 𝐴 → (0 ≤ (ℜ‘𝑥) ↔ 0 ≤ (ℜ‘𝐴))) |
20 | oveq2 6822 | . . . . 5 ⊢ (𝑥 = 𝐴 → (i · 𝑥) = (i · 𝐴)) | |
21 | neleq1 3040 | . . . . 5 ⊢ ((i · 𝑥) = (i · 𝐴) → ((i · 𝑥) ∉ ℝ+ ↔ (i · 𝐴) ∉ ℝ+)) | |
22 | 20, 21 | syl 17 | . . . 4 ⊢ (𝑥 = 𝐴 → ((i · 𝑥) ∉ ℝ+ ↔ (i · 𝐴) ∉ ℝ+)) |
23 | 17, 19, 22 | 3anbi123d 1548 | . . 3 ⊢ (𝑥 = 𝐴 → (((𝑥↑2) = 𝐵 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ+) ↔ ((𝐴↑2) = 𝐵 ∧ 0 ≤ (ℜ‘𝐴) ∧ (i · 𝐴) ∉ ℝ+))) |
24 | oveq1 6821 | . . . . 5 ⊢ (𝑥 = (√‘𝐵) → (𝑥↑2) = ((√‘𝐵)↑2)) | |
25 | 24 | eqeq1d 2762 | . . . 4 ⊢ (𝑥 = (√‘𝐵) → ((𝑥↑2) = 𝐵 ↔ ((√‘𝐵)↑2) = 𝐵)) |
26 | fveq2 6353 | . . . . 5 ⊢ (𝑥 = (√‘𝐵) → (ℜ‘𝑥) = (ℜ‘(√‘𝐵))) | |
27 | 26 | breq2d 4816 | . . . 4 ⊢ (𝑥 = (√‘𝐵) → (0 ≤ (ℜ‘𝑥) ↔ 0 ≤ (ℜ‘(√‘𝐵)))) |
28 | oveq2 6822 | . . . . 5 ⊢ (𝑥 = (√‘𝐵) → (i · 𝑥) = (i · (√‘𝐵))) | |
29 | neleq1 3040 | . . . . 5 ⊢ ((i · 𝑥) = (i · (√‘𝐵)) → ((i · 𝑥) ∉ ℝ+ ↔ (i · (√‘𝐵)) ∉ ℝ+)) | |
30 | 28, 29 | syl 17 | . . . 4 ⊢ (𝑥 = (√‘𝐵) → ((i · 𝑥) ∉ ℝ+ ↔ (i · (√‘𝐵)) ∉ ℝ+)) |
31 | 25, 27, 30 | 3anbi123d 1548 | . . 3 ⊢ (𝑥 = (√‘𝐵) → (((𝑥↑2) = 𝐵 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ+) ↔ (((√‘𝐵)↑2) = 𝐵 ∧ 0 ≤ (ℜ‘(√‘𝐵)) ∧ (i · (√‘𝐵)) ∉ ℝ+))) |
32 | 23, 31 | rmoi 3671 | . 2 ⊢ ((∃*𝑥 ∈ ℂ ((𝑥↑2) = 𝐵 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ+) ∧ (𝐴 ∈ ℂ ∧ ((𝐴↑2) = 𝐵 ∧ 0 ≤ (ℜ‘𝐴) ∧ (i · 𝐴) ∉ ℝ+)) ∧ ((√‘𝐵) ∈ ℂ ∧ (((√‘𝐵)↑2) = 𝐵 ∧ 0 ≤ (ℜ‘(√‘𝐵)) ∧ (i · (√‘𝐵)) ∉ ℝ+))) → 𝐴 = (√‘𝐵)) |
33 | 4, 5, 11, 13, 15, 32 | syl122anc 1486 | 1 ⊢ (𝜑 → 𝐴 = (√‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ w3a 1072 = wceq 1632 ∈ wcel 2139 ∉ wnel 3035 ∃!wreu 3052 ∃*wrmo 3053 class class class wbr 4804 ‘cfv 6049 (class class class)co 6814 ℂcc 10146 0cc0 10148 ici 10150 · cmul 10153 ≤ cle 10287 2c2 11282 ℝ+crp 12045 ↑cexp 13074 ℜcre 14056 √csqrt 14192 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7115 ax-cnex 10204 ax-resscn 10205 ax-1cn 10206 ax-icn 10207 ax-addcl 10208 ax-addrcl 10209 ax-mulcl 10210 ax-mulrcl 10211 ax-mulcom 10212 ax-addass 10213 ax-mulass 10214 ax-distr 10215 ax-i2m1 10216 ax-1ne0 10217 ax-1rid 10218 ax-rnegex 10219 ax-rrecex 10220 ax-cnre 10221 ax-pre-lttri 10222 ax-pre-lttrn 10223 ax-pre-ltadd 10224 ax-pre-mulgt0 10225 ax-pre-sup 10226 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6775 df-ov 6817 df-oprab 6818 df-mpt2 6819 df-om 7232 df-2nd 7335 df-wrecs 7577 df-recs 7638 df-rdg 7676 df-er 7913 df-en 8124 df-dom 8125 df-sdom 8126 df-sup 8515 df-pnf 10288 df-mnf 10289 df-xr 10290 df-ltxr 10291 df-le 10292 df-sub 10480 df-neg 10481 df-div 10897 df-nn 11233 df-2 11291 df-3 11292 df-n0 11505 df-z 11590 df-uz 11900 df-rp 12046 df-seq 13016 df-exp 13075 df-cj 14058 df-re 14059 df-im 14060 df-sqrt 14194 df-abs 14195 |
This theorem is referenced by: eqsqrt2d 14327 cphsqrtcl2 23206 |
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