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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 14559 | . . 3 ⊢ (𝐵 ∈ ℂ → ∃!𝑥 ∈ ℂ ((𝑥↑2) = 𝐵 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ+)) | |
3 | reurmo 3393 | . . 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 3091 | . . . 4 ⊢ ((i · 𝐴) ∉ ℝ+ ↔ ¬ (i · 𝐴) ∈ ℝ+) | |
10 | 8, 9 | sylibr 235 | . . 3 ⊢ (𝜑 → (i · 𝐴) ∉ ℝ+) |
11 | 6, 7, 10 | 3jca 1121 | . 2 ⊢ (𝜑 → ((𝐴↑2) = 𝐵 ∧ 0 ≤ (ℜ‘𝐴) ∧ (i · 𝐴) ∉ ℝ+)) |
12 | sqrtcl 14560 | . . 3 ⊢ (𝐵 ∈ ℂ → (√‘𝐵) ∈ ℂ) | |
13 | 1, 12 | syl 17 | . 2 ⊢ (𝜑 → (√‘𝐵) ∈ ℂ) |
14 | sqrtthlem 14561 | . . 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 7028 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥↑2) = (𝐴↑2)) | |
17 | 16 | eqeq1d 2797 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥↑2) = 𝐵 ↔ (𝐴↑2) = 𝐵)) |
18 | fveq2 6543 | . . . . 5 ⊢ (𝑥 = 𝐴 → (ℜ‘𝑥) = (ℜ‘𝐴)) | |
19 | 18 | breq2d 4978 | . . . 4 ⊢ (𝑥 = 𝐴 → (0 ≤ (ℜ‘𝑥) ↔ 0 ≤ (ℜ‘𝐴))) |
20 | oveq2 7029 | . . . . 5 ⊢ (𝑥 = 𝐴 → (i · 𝑥) = (i · 𝐴)) | |
21 | neleq1 3095 | . . . . 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 1428 | . . 3 ⊢ (𝑥 = 𝐴 → (((𝑥↑2) = 𝐵 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ+) ↔ ((𝐴↑2) = 𝐵 ∧ 0 ≤ (ℜ‘𝐴) ∧ (i · 𝐴) ∉ ℝ+))) |
24 | oveq1 7028 | . . . . 5 ⊢ (𝑥 = (√‘𝐵) → (𝑥↑2) = ((√‘𝐵)↑2)) | |
25 | 24 | eqeq1d 2797 | . . . 4 ⊢ (𝑥 = (√‘𝐵) → ((𝑥↑2) = 𝐵 ↔ ((√‘𝐵)↑2) = 𝐵)) |
26 | fveq2 6543 | . . . . 5 ⊢ (𝑥 = (√‘𝐵) → (ℜ‘𝑥) = (ℜ‘(√‘𝐵))) | |
27 | 26 | breq2d 4978 | . . . 4 ⊢ (𝑥 = (√‘𝐵) → (0 ≤ (ℜ‘𝑥) ↔ 0 ≤ (ℜ‘(√‘𝐵)))) |
28 | oveq2 7029 | . . . . 5 ⊢ (𝑥 = (√‘𝐵) → (i · 𝑥) = (i · (√‘𝐵))) | |
29 | neleq1 3095 | . . . . 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 1428 | . . 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 3807 | . 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 1372 | 1 ⊢ (𝜑 → 𝐴 = (√‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ w3a 1080 = wceq 1522 ∈ wcel 2081 ∉ wnel 3090 ∃!wreu 3107 ∃*wrmo 3108 class class class wbr 4966 ‘cfv 6230 (class class class)co 7021 ℂcc 10386 0cc0 10388 ici 10390 · cmul 10393 ≤ cle 10527 2c2 11545 ℝ+crp 12244 ↑cexp 13284 ℜcre 14295 √csqrt 14431 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5099 ax-nul 5106 ax-pow 5162 ax-pr 5226 ax-un 7324 ax-cnex 10444 ax-resscn 10445 ax-1cn 10446 ax-icn 10447 ax-addcl 10448 ax-addrcl 10449 ax-mulcl 10450 ax-mulrcl 10451 ax-mulcom 10452 ax-addass 10453 ax-mulass 10454 ax-distr 10455 ax-i2m1 10456 ax-1ne0 10457 ax-1rid 10458 ax-rnegex 10459 ax-rrecex 10460 ax-cnre 10461 ax-pre-lttri 10462 ax-pre-lttrn 10463 ax-pre-ltadd 10464 ax-pre-mulgt0 10465 ax-pre-sup 10466 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3710 df-csb 3816 df-dif 3866 df-un 3868 df-in 3870 df-ss 3878 df-pss 3880 df-nul 4216 df-if 4386 df-pw 4459 df-sn 4477 df-pr 4479 df-tp 4481 df-op 4483 df-uni 4750 df-iun 4831 df-br 4967 df-opab 5029 df-mpt 5046 df-tr 5069 df-id 5353 df-eprel 5358 df-po 5367 df-so 5368 df-fr 5407 df-we 5409 df-xp 5454 df-rel 5455 df-cnv 5456 df-co 5457 df-dm 5458 df-rn 5459 df-res 5460 df-ima 5461 df-pred 6028 df-ord 6074 df-on 6075 df-lim 6076 df-suc 6077 df-iota 6194 df-fun 6232 df-fn 6233 df-f 6234 df-f1 6235 df-fo 6236 df-f1o 6237 df-fv 6238 df-riota 6982 df-ov 7024 df-oprab 7025 df-mpo 7026 df-om 7442 df-2nd 7551 df-wrecs 7803 df-recs 7865 df-rdg 7903 df-er 8144 df-en 8363 df-dom 8364 df-sdom 8365 df-sup 8757 df-pnf 10528 df-mnf 10529 df-xr 10530 df-ltxr 10531 df-le 10532 df-sub 10724 df-neg 10725 df-div 11151 df-nn 11492 df-2 11553 df-3 11554 df-n0 11751 df-z 11835 df-uz 12099 df-rp 12245 df-seq 13225 df-exp 13285 df-cj 14297 df-re 14298 df-im 14299 df-sqrt 14433 df-abs 14434 |
This theorem is referenced by: eqsqrt2d 14567 cphsqrtcl2 23478 |
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