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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 14714 | . . 3 ⊢ (𝐵 ∈ ℂ → ∃!𝑥 ∈ ℂ ((𝑥↑2) = 𝐵 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ+)) | |
3 | reurmo 3434 | . . 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 3124 | . . . 4 ⊢ ((i · 𝐴) ∉ ℝ+ ↔ ¬ (i · 𝐴) ∈ ℝ+) | |
10 | 8, 9 | sylibr 236 | . . 3 ⊢ (𝜑 → (i · 𝐴) ∉ ℝ+) |
11 | 6, 7, 10 | 3jca 1124 | . 2 ⊢ (𝜑 → ((𝐴↑2) = 𝐵 ∧ 0 ≤ (ℜ‘𝐴) ∧ (i · 𝐴) ∉ ℝ+)) |
12 | sqrtcl 14715 | . . 3 ⊢ (𝐵 ∈ ℂ → (√‘𝐵) ∈ ℂ) | |
13 | 1, 12 | syl 17 | . 2 ⊢ (𝜑 → (√‘𝐵) ∈ ℂ) |
14 | sqrtthlem 14716 | . . 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 7157 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥↑2) = (𝐴↑2)) | |
17 | 16 | eqeq1d 2823 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥↑2) = 𝐵 ↔ (𝐴↑2) = 𝐵)) |
18 | fveq2 6665 | . . . . 5 ⊢ (𝑥 = 𝐴 → (ℜ‘𝑥) = (ℜ‘𝐴)) | |
19 | 18 | breq2d 5071 | . . . 4 ⊢ (𝑥 = 𝐴 → (0 ≤ (ℜ‘𝑥) ↔ 0 ≤ (ℜ‘𝐴))) |
20 | oveq2 7158 | . . . . 5 ⊢ (𝑥 = 𝐴 → (i · 𝑥) = (i · 𝐴)) | |
21 | neleq1 3128 | . . . . 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 1432 | . . 3 ⊢ (𝑥 = 𝐴 → (((𝑥↑2) = 𝐵 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ+) ↔ ((𝐴↑2) = 𝐵 ∧ 0 ≤ (ℜ‘𝐴) ∧ (i · 𝐴) ∉ ℝ+))) |
24 | oveq1 7157 | . . . . 5 ⊢ (𝑥 = (√‘𝐵) → (𝑥↑2) = ((√‘𝐵)↑2)) | |
25 | 24 | eqeq1d 2823 | . . . 4 ⊢ (𝑥 = (√‘𝐵) → ((𝑥↑2) = 𝐵 ↔ ((√‘𝐵)↑2) = 𝐵)) |
26 | fveq2 6665 | . . . . 5 ⊢ (𝑥 = (√‘𝐵) → (ℜ‘𝑥) = (ℜ‘(√‘𝐵))) | |
27 | 26 | breq2d 5071 | . . . 4 ⊢ (𝑥 = (√‘𝐵) → (0 ≤ (ℜ‘𝑥) ↔ 0 ≤ (ℜ‘(√‘𝐵)))) |
28 | oveq2 7158 | . . . . 5 ⊢ (𝑥 = (√‘𝐵) → (i · 𝑥) = (i · (√‘𝐵))) | |
29 | neleq1 3128 | . . . . 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 1432 | . . 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 3875 | . 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 1375 | 1 ⊢ (𝜑 → 𝐴 = (√‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ∉ wnel 3123 ∃!wreu 3140 ∃*wrmo 3141 class class class wbr 5059 ‘cfv 6350 (class class class)co 7150 ℂcc 10529 0cc0 10531 ici 10533 · cmul 10536 ≤ cle 10670 2c2 11686 ℝ+crp 12383 ↑cexp 13423 ℜcre 14450 √csqrt 14586 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-sup 8900 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-n0 11892 df-z 11976 df-uz 12238 df-rp 12384 df-seq 13364 df-exp 13424 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 |
This theorem is referenced by: eqsqrt2d 14722 cphsqrtcl2 23784 |
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