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Mirrors > Home > MPE Home > Th. List > sqabs | Structured version Visualization version GIF version |
Description: The squares of two reals are equal iff their absolute values are equal. (Contributed by NM, 6-Mar-2009.) |
Ref | Expression |
---|---|
sqabs | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴↑2) = (𝐵↑2) ↔ (abs‘𝐴) = (abs‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resqcl 13489 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (𝐴↑2) ∈ ℝ) | |
2 | sqge0 13500 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 0 ≤ (𝐴↑2)) | |
3 | absid 14655 | . . . . 5 ⊢ (((𝐴↑2) ∈ ℝ ∧ 0 ≤ (𝐴↑2)) → (abs‘(𝐴↑2)) = (𝐴↑2)) | |
4 | 1, 2, 3 | syl2anc 586 | . . . 4 ⊢ (𝐴 ∈ ℝ → (abs‘(𝐴↑2)) = (𝐴↑2)) |
5 | recn 10626 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
6 | 2nn0 11913 | . . . . 5 ⊢ 2 ∈ ℕ0 | |
7 | absexp 14663 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 2 ∈ ℕ0) → (abs‘(𝐴↑2)) = ((abs‘𝐴)↑2)) | |
8 | 5, 6, 7 | sylancl 588 | . . . 4 ⊢ (𝐴 ∈ ℝ → (abs‘(𝐴↑2)) = ((abs‘𝐴)↑2)) |
9 | 4, 8 | eqtr3d 2858 | . . 3 ⊢ (𝐴 ∈ ℝ → (𝐴↑2) = ((abs‘𝐴)↑2)) |
10 | resqcl 13489 | . . . . 5 ⊢ (𝐵 ∈ ℝ → (𝐵↑2) ∈ ℝ) | |
11 | sqge0 13500 | . . . . 5 ⊢ (𝐵 ∈ ℝ → 0 ≤ (𝐵↑2)) | |
12 | absid 14655 | . . . . 5 ⊢ (((𝐵↑2) ∈ ℝ ∧ 0 ≤ (𝐵↑2)) → (abs‘(𝐵↑2)) = (𝐵↑2)) | |
13 | 10, 11, 12 | syl2anc 586 | . . . 4 ⊢ (𝐵 ∈ ℝ → (abs‘(𝐵↑2)) = (𝐵↑2)) |
14 | recn 10626 | . . . . 5 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℂ) | |
15 | absexp 14663 | . . . . 5 ⊢ ((𝐵 ∈ ℂ ∧ 2 ∈ ℕ0) → (abs‘(𝐵↑2)) = ((abs‘𝐵)↑2)) | |
16 | 14, 6, 15 | sylancl 588 | . . . 4 ⊢ (𝐵 ∈ ℝ → (abs‘(𝐵↑2)) = ((abs‘𝐵)↑2)) |
17 | 13, 16 | eqtr3d 2858 | . . 3 ⊢ (𝐵 ∈ ℝ → (𝐵↑2) = ((abs‘𝐵)↑2)) |
18 | 9, 17 | eqeqan12d 2838 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴↑2) = (𝐵↑2) ↔ ((abs‘𝐴)↑2) = ((abs‘𝐵)↑2))) |
19 | abscl 14637 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) ∈ ℝ) | |
20 | absge0 14646 | . . . . 5 ⊢ (𝐴 ∈ ℂ → 0 ≤ (abs‘𝐴)) | |
21 | 19, 20 | jca 514 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((abs‘𝐴) ∈ ℝ ∧ 0 ≤ (abs‘𝐴))) |
22 | abscl 14637 | . . . . 5 ⊢ (𝐵 ∈ ℂ → (abs‘𝐵) ∈ ℝ) | |
23 | absge0 14646 | . . . . 5 ⊢ (𝐵 ∈ ℂ → 0 ≤ (abs‘𝐵)) | |
24 | 22, 23 | jca 514 | . . . 4 ⊢ (𝐵 ∈ ℂ → ((abs‘𝐵) ∈ ℝ ∧ 0 ≤ (abs‘𝐵))) |
25 | sq11 13495 | . . . 4 ⊢ ((((abs‘𝐴) ∈ ℝ ∧ 0 ≤ (abs‘𝐴)) ∧ ((abs‘𝐵) ∈ ℝ ∧ 0 ≤ (abs‘𝐵))) → (((abs‘𝐴)↑2) = ((abs‘𝐵)↑2) ↔ (abs‘𝐴) = (abs‘𝐵))) | |
26 | 21, 24, 25 | syl2an 597 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (((abs‘𝐴)↑2) = ((abs‘𝐵)↑2) ↔ (abs‘𝐴) = (abs‘𝐵))) |
27 | 5, 14, 26 | syl2an 597 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (((abs‘𝐴)↑2) = ((abs‘𝐵)↑2) ↔ (abs‘𝐴) = (abs‘𝐵))) |
28 | 18, 27 | bitrd 281 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴↑2) = (𝐵↑2) ↔ (abs‘𝐴) = (abs‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 class class class wbr 5065 ‘cfv 6354 (class class class)co 7155 ℂcc 10534 ℝcr 10535 0cc0 10536 ≤ cle 10675 2c2 11691 ℕ0cn0 11896 ↑cexp 13428 abscabs 14592 |
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 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-pre-sup 10614 |
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 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-sup 8905 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-nn 11638 df-2 11699 df-3 11700 df-n0 11897 df-z 11981 df-uz 12243 df-rp 12389 df-seq 13369 df-exp 13429 df-cj 14457 df-re 14458 df-im 14459 df-sqrt 14593 df-abs 14594 |
This theorem is referenced by: coskpi 25107 |
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