![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 13254 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (𝐴↑2) ∈ ℝ) | |
2 | sqge0 13264 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 0 ≤ (𝐴↑2)) | |
3 | absid 14450 | . . . . 5 ⊢ (((𝐴↑2) ∈ ℝ ∧ 0 ≤ (𝐴↑2)) → (abs‘(𝐴↑2)) = (𝐴↑2)) | |
4 | 1, 2, 3 | syl2anc 579 | . . . 4 ⊢ (𝐴 ∈ ℝ → (abs‘(𝐴↑2)) = (𝐴↑2)) |
5 | recn 10364 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
6 | 2nn0 11666 | . . . . 5 ⊢ 2 ∈ ℕ0 | |
7 | absexp 14458 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 2 ∈ ℕ0) → (abs‘(𝐴↑2)) = ((abs‘𝐴)↑2)) | |
8 | 5, 6, 7 | sylancl 580 | . . . 4 ⊢ (𝐴 ∈ ℝ → (abs‘(𝐴↑2)) = ((abs‘𝐴)↑2)) |
9 | 4, 8 | eqtr3d 2816 | . . 3 ⊢ (𝐴 ∈ ℝ → (𝐴↑2) = ((abs‘𝐴)↑2)) |
10 | resqcl 13254 | . . . . 5 ⊢ (𝐵 ∈ ℝ → (𝐵↑2) ∈ ℝ) | |
11 | sqge0 13264 | . . . . 5 ⊢ (𝐵 ∈ ℝ → 0 ≤ (𝐵↑2)) | |
12 | absid 14450 | . . . . 5 ⊢ (((𝐵↑2) ∈ ℝ ∧ 0 ≤ (𝐵↑2)) → (abs‘(𝐵↑2)) = (𝐵↑2)) | |
13 | 10, 11, 12 | syl2anc 579 | . . . 4 ⊢ (𝐵 ∈ ℝ → (abs‘(𝐵↑2)) = (𝐵↑2)) |
14 | recn 10364 | . . . . 5 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℂ) | |
15 | absexp 14458 | . . . . 5 ⊢ ((𝐵 ∈ ℂ ∧ 2 ∈ ℕ0) → (abs‘(𝐵↑2)) = ((abs‘𝐵)↑2)) | |
16 | 14, 6, 15 | sylancl 580 | . . . 4 ⊢ (𝐵 ∈ ℝ → (abs‘(𝐵↑2)) = ((abs‘𝐵)↑2)) |
17 | 13, 16 | eqtr3d 2816 | . . 3 ⊢ (𝐵 ∈ ℝ → (𝐵↑2) = ((abs‘𝐵)↑2)) |
18 | 9, 17 | eqeqan12d 2794 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴↑2) = (𝐵↑2) ↔ ((abs‘𝐴)↑2) = ((abs‘𝐵)↑2))) |
19 | abscl 14432 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) ∈ ℝ) | |
20 | absge0 14441 | . . . . 5 ⊢ (𝐴 ∈ ℂ → 0 ≤ (abs‘𝐴)) | |
21 | 19, 20 | jca 507 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((abs‘𝐴) ∈ ℝ ∧ 0 ≤ (abs‘𝐴))) |
22 | abscl 14432 | . . . . 5 ⊢ (𝐵 ∈ ℂ → (abs‘𝐵) ∈ ℝ) | |
23 | absge0 14441 | . . . . 5 ⊢ (𝐵 ∈ ℂ → 0 ≤ (abs‘𝐵)) | |
24 | 22, 23 | jca 507 | . . . 4 ⊢ (𝐵 ∈ ℂ → ((abs‘𝐵) ∈ ℝ ∧ 0 ≤ (abs‘𝐵))) |
25 | sq11 13260 | . . . 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 589 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (((abs‘𝐴)↑2) = ((abs‘𝐵)↑2) ↔ (abs‘𝐴) = (abs‘𝐵))) |
27 | 5, 14, 26 | syl2an 589 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (((abs‘𝐴)↑2) = ((abs‘𝐵)↑2) ↔ (abs‘𝐴) = (abs‘𝐵))) |
28 | 18, 27 | bitrd 271 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴↑2) = (𝐵↑2) ↔ (abs‘𝐴) = (abs‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1601 ∈ wcel 2107 class class class wbr 4888 ‘cfv 6137 (class class class)co 6924 ℂcc 10272 ℝcr 10273 0cc0 10274 ≤ cle 10414 2c2 11435 ℕ0cn0 11647 ↑cexp 13183 abscabs 14387 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 ax-pre-sup 10352 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-2nd 7448 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-er 8028 df-en 8244 df-dom 8245 df-sdom 8246 df-sup 8638 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-div 11036 df-nn 11380 df-2 11443 df-3 11444 df-n0 11648 df-z 11734 df-uz 11998 df-rp 12143 df-seq 13125 df-exp 13184 df-cj 14252 df-re 14253 df-im 14254 df-sqrt 14388 df-abs 14389 |
This theorem is referenced by: coskpi 24721 |
Copyright terms: Public domain | W3C validator |