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Mirrors > Home > ILE Home > Th. List > sqabs | 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 10486 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (𝐴↑2) ∈ ℝ) | |
2 | sqge0 10495 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 0 ≤ (𝐴↑2)) | |
3 | absid 10971 | . . . . 5 ⊢ (((𝐴↑2) ∈ ℝ ∧ 0 ≤ (𝐴↑2)) → (abs‘(𝐴↑2)) = (𝐴↑2)) | |
4 | 1, 2, 3 | syl2anc 409 | . . . 4 ⊢ (𝐴 ∈ ℝ → (abs‘(𝐴↑2)) = (𝐴↑2)) |
5 | recn 7865 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
6 | 2nn0 9107 | . . . . 5 ⊢ 2 ∈ ℕ0 | |
7 | absexp 10979 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 2 ∈ ℕ0) → (abs‘(𝐴↑2)) = ((abs‘𝐴)↑2)) | |
8 | 5, 6, 7 | sylancl 410 | . . . 4 ⊢ (𝐴 ∈ ℝ → (abs‘(𝐴↑2)) = ((abs‘𝐴)↑2)) |
9 | 4, 8 | eqtr3d 2192 | . . 3 ⊢ (𝐴 ∈ ℝ → (𝐴↑2) = ((abs‘𝐴)↑2)) |
10 | resqcl 10486 | . . . . 5 ⊢ (𝐵 ∈ ℝ → (𝐵↑2) ∈ ℝ) | |
11 | sqge0 10495 | . . . . 5 ⊢ (𝐵 ∈ ℝ → 0 ≤ (𝐵↑2)) | |
12 | absid 10971 | . . . . 5 ⊢ (((𝐵↑2) ∈ ℝ ∧ 0 ≤ (𝐵↑2)) → (abs‘(𝐵↑2)) = (𝐵↑2)) | |
13 | 10, 11, 12 | syl2anc 409 | . . . 4 ⊢ (𝐵 ∈ ℝ → (abs‘(𝐵↑2)) = (𝐵↑2)) |
14 | recn 7865 | . . . . 5 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℂ) | |
15 | absexp 10979 | . . . . 5 ⊢ ((𝐵 ∈ ℂ ∧ 2 ∈ ℕ0) → (abs‘(𝐵↑2)) = ((abs‘𝐵)↑2)) | |
16 | 14, 6, 15 | sylancl 410 | . . . 4 ⊢ (𝐵 ∈ ℝ → (abs‘(𝐵↑2)) = ((abs‘𝐵)↑2)) |
17 | 13, 16 | eqtr3d 2192 | . . 3 ⊢ (𝐵 ∈ ℝ → (𝐵↑2) = ((abs‘𝐵)↑2)) |
18 | 9, 17 | eqeqan12d 2173 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴↑2) = (𝐵↑2) ↔ ((abs‘𝐴)↑2) = ((abs‘𝐵)↑2))) |
19 | abscl 10951 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) ∈ ℝ) | |
20 | absge0 10960 | . . . . 5 ⊢ (𝐴 ∈ ℂ → 0 ≤ (abs‘𝐴)) | |
21 | 19, 20 | jca 304 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((abs‘𝐴) ∈ ℝ ∧ 0 ≤ (abs‘𝐴))) |
22 | abscl 10951 | . . . . 5 ⊢ (𝐵 ∈ ℂ → (abs‘𝐵) ∈ ℝ) | |
23 | absge0 10960 | . . . . 5 ⊢ (𝐵 ∈ ℂ → 0 ≤ (abs‘𝐵)) | |
24 | 22, 23 | jca 304 | . . . 4 ⊢ (𝐵 ∈ ℂ → ((abs‘𝐵) ∈ ℝ ∧ 0 ≤ (abs‘𝐵))) |
25 | sq11 10491 | . . . 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 287 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (((abs‘𝐴)↑2) = ((abs‘𝐵)↑2) ↔ (abs‘𝐴) = (abs‘𝐵))) |
27 | 5, 14, 26 | syl2an 287 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (((abs‘𝐴)↑2) = ((abs‘𝐵)↑2) ↔ (abs‘𝐴) = (abs‘𝐵))) |
28 | 18, 27 | bitrd 187 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴↑2) = (𝐵↑2) ↔ (abs‘𝐴) = (abs‘𝐵))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1335 ∈ wcel 2128 class class class wbr 3965 ‘cfv 5170 (class class class)co 5824 ℂcc 7730 ℝcr 7731 0cc0 7732 ≤ cle 7913 2c2 8884 ℕ0cn0 9090 ↑cexp 10418 abscabs 10897 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4079 ax-sep 4082 ax-nul 4090 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-iinf 4547 ax-cnex 7823 ax-resscn 7824 ax-1cn 7825 ax-1re 7826 ax-icn 7827 ax-addcl 7828 ax-addrcl 7829 ax-mulcl 7830 ax-mulrcl 7831 ax-addcom 7832 ax-mulcom 7833 ax-addass 7834 ax-mulass 7835 ax-distr 7836 ax-i2m1 7837 ax-0lt1 7838 ax-1rid 7839 ax-0id 7840 ax-rnegex 7841 ax-precex 7842 ax-cnre 7843 ax-pre-ltirr 7844 ax-pre-ltwlin 7845 ax-pre-lttrn 7846 ax-pre-apti 7847 ax-pre-ltadd 7848 ax-pre-mulgt0 7849 ax-pre-mulext 7850 ax-arch 7851 ax-caucvg 7852 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rmo 2443 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-if 3506 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-iun 3851 df-br 3966 df-opab 4026 df-mpt 4027 df-tr 4063 df-id 4253 df-po 4256 df-iso 4257 df-iord 4326 df-on 4328 df-ilim 4329 df-suc 4331 df-iom 4550 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-res 4598 df-ima 4599 df-iota 5135 df-fun 5172 df-fn 5173 df-f 5174 df-f1 5175 df-fo 5176 df-f1o 5177 df-fv 5178 df-riota 5780 df-ov 5827 df-oprab 5828 df-mpo 5829 df-1st 6088 df-2nd 6089 df-recs 6252 df-frec 6338 df-pnf 7914 df-mnf 7915 df-xr 7916 df-ltxr 7917 df-le 7918 df-sub 8048 df-neg 8049 df-reap 8450 df-ap 8457 df-div 8546 df-inn 8834 df-2 8892 df-3 8893 df-4 8894 df-n0 9091 df-z 9168 df-uz 9440 df-rp 9561 df-seqfrec 10345 df-exp 10419 df-cj 10742 df-re 10743 df-im 10744 df-rsqrt 10898 df-abs 10899 |
This theorem is referenced by: coskpi 13180 |
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