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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 15396 | . . 3 ⊢ (𝐵 ∈ ℂ → ∃!𝑥 ∈ ℂ ((𝑥↑2) = 𝐵 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ+)) | |
3 | reurmo 3381 | . . 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 3045 | . . . 4 ⊢ ((i · 𝐴) ∉ ℝ+ ↔ ¬ (i · 𝐴) ∈ ℝ+) | |
10 | 8, 9 | sylibr 234 | . . 3 ⊢ (𝜑 → (i · 𝐴) ∉ ℝ+) |
11 | 6, 7, 10 | 3jca 1127 | . 2 ⊢ (𝜑 → ((𝐴↑2) = 𝐵 ∧ 0 ≤ (ℜ‘𝐴) ∧ (i · 𝐴) ∉ ℝ+)) |
12 | sqrtcl 15397 | . . 3 ⊢ (𝐵 ∈ ℂ → (√‘𝐵) ∈ ℂ) | |
13 | 1, 12 | syl 17 | . 2 ⊢ (𝜑 → (√‘𝐵) ∈ ℂ) |
14 | sqrtthlem 15398 | . . 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 7438 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥↑2) = (𝐴↑2)) | |
17 | 16 | eqeq1d 2737 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥↑2) = 𝐵 ↔ (𝐴↑2) = 𝐵)) |
18 | fveq2 6907 | . . . . 5 ⊢ (𝑥 = 𝐴 → (ℜ‘𝑥) = (ℜ‘𝐴)) | |
19 | 18 | breq2d 5160 | . . . 4 ⊢ (𝑥 = 𝐴 → (0 ≤ (ℜ‘𝑥) ↔ 0 ≤ (ℜ‘𝐴))) |
20 | oveq2 7439 | . . . . 5 ⊢ (𝑥 = 𝐴 → (i · 𝑥) = (i · 𝐴)) | |
21 | neleq1 3050 | . . . . 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 1435 | . . 3 ⊢ (𝑥 = 𝐴 → (((𝑥↑2) = 𝐵 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ+) ↔ ((𝐴↑2) = 𝐵 ∧ 0 ≤ (ℜ‘𝐴) ∧ (i · 𝐴) ∉ ℝ+))) |
24 | oveq1 7438 | . . . . 5 ⊢ (𝑥 = (√‘𝐵) → (𝑥↑2) = ((√‘𝐵)↑2)) | |
25 | 24 | eqeq1d 2737 | . . . 4 ⊢ (𝑥 = (√‘𝐵) → ((𝑥↑2) = 𝐵 ↔ ((√‘𝐵)↑2) = 𝐵)) |
26 | fveq2 6907 | . . . . 5 ⊢ (𝑥 = (√‘𝐵) → (ℜ‘𝑥) = (ℜ‘(√‘𝐵))) | |
27 | 26 | breq2d 5160 | . . . 4 ⊢ (𝑥 = (√‘𝐵) → (0 ≤ (ℜ‘𝑥) ↔ 0 ≤ (ℜ‘(√‘𝐵)))) |
28 | oveq2 7439 | . . . . 5 ⊢ (𝑥 = (√‘𝐵) → (i · 𝑥) = (i · (√‘𝐵))) | |
29 | neleq1 3050 | . . . . 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 1435 | . . 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 3900 | . 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 1378 | 1 ⊢ (𝜑 → 𝐴 = (√‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ∉ wnel 3044 ∃!wreu 3376 ∃*wrmo 3377 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 0cc0 11153 ici 11155 · cmul 11158 ≤ cle 11294 2c2 12319 ℝ+crp 13032 ↑cexp 14099 ℜcre 15133 √csqrt 15269 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-sup 9480 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12612 df-uz 12877 df-rp 13033 df-seq 14040 df-exp 14100 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 |
This theorem is referenced by: eqsqrt2d 15404 cphsqrtcl2 25234 sqrtcval 43631 |
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