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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 14076 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (𝐴↑2) ∈ ℝ) | |
2 | sqge0 14088 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 0 ≤ (𝐴↑2)) | |
3 | absid 15230 | . . . . 5 ⊢ (((𝐴↑2) ∈ ℝ ∧ 0 ≤ (𝐴↑2)) → (abs‘(𝐴↑2)) = (𝐴↑2)) | |
4 | 1, 2, 3 | syl2anc 585 | . . . 4 ⊢ (𝐴 ∈ ℝ → (abs‘(𝐴↑2)) = (𝐴↑2)) |
5 | recn 11187 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
6 | 2nn0 12476 | . . . . 5 ⊢ 2 ∈ ℕ0 | |
7 | absexp 15238 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 2 ∈ ℕ0) → (abs‘(𝐴↑2)) = ((abs‘𝐴)↑2)) | |
8 | 5, 6, 7 | sylancl 587 | . . . 4 ⊢ (𝐴 ∈ ℝ → (abs‘(𝐴↑2)) = ((abs‘𝐴)↑2)) |
9 | 4, 8 | eqtr3d 2775 | . . 3 ⊢ (𝐴 ∈ ℝ → (𝐴↑2) = ((abs‘𝐴)↑2)) |
10 | resqcl 14076 | . . . . 5 ⊢ (𝐵 ∈ ℝ → (𝐵↑2) ∈ ℝ) | |
11 | sqge0 14088 | . . . . 5 ⊢ (𝐵 ∈ ℝ → 0 ≤ (𝐵↑2)) | |
12 | absid 15230 | . . . . 5 ⊢ (((𝐵↑2) ∈ ℝ ∧ 0 ≤ (𝐵↑2)) → (abs‘(𝐵↑2)) = (𝐵↑2)) | |
13 | 10, 11, 12 | syl2anc 585 | . . . 4 ⊢ (𝐵 ∈ ℝ → (abs‘(𝐵↑2)) = (𝐵↑2)) |
14 | recn 11187 | . . . . 5 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℂ) | |
15 | absexp 15238 | . . . . 5 ⊢ ((𝐵 ∈ ℂ ∧ 2 ∈ ℕ0) → (abs‘(𝐵↑2)) = ((abs‘𝐵)↑2)) | |
16 | 14, 6, 15 | sylancl 587 | . . . 4 ⊢ (𝐵 ∈ ℝ → (abs‘(𝐵↑2)) = ((abs‘𝐵)↑2)) |
17 | 13, 16 | eqtr3d 2775 | . . 3 ⊢ (𝐵 ∈ ℝ → (𝐵↑2) = ((abs‘𝐵)↑2)) |
18 | 9, 17 | eqeqan12d 2747 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴↑2) = (𝐵↑2) ↔ ((abs‘𝐴)↑2) = ((abs‘𝐵)↑2))) |
19 | abscl 15212 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) ∈ ℝ) | |
20 | absge0 15221 | . . . . 5 ⊢ (𝐴 ∈ ℂ → 0 ≤ (abs‘𝐴)) | |
21 | 19, 20 | jca 513 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((abs‘𝐴) ∈ ℝ ∧ 0 ≤ (abs‘𝐴))) |
22 | abscl 15212 | . . . . 5 ⊢ (𝐵 ∈ ℂ → (abs‘𝐵) ∈ ℝ) | |
23 | absge0 15221 | . . . . 5 ⊢ (𝐵 ∈ ℂ → 0 ≤ (abs‘𝐵)) | |
24 | 22, 23 | jca 513 | . . . 4 ⊢ (𝐵 ∈ ℂ → ((abs‘𝐵) ∈ ℝ ∧ 0 ≤ (abs‘𝐵))) |
25 | sq11 14083 | . . . 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 279 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴↑2) = (𝐵↑2) ↔ (abs‘𝐴) = (abs‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 class class class wbr 5144 ‘cfv 6535 (class class class)co 7396 ℂcc 11095 ℝcr 11096 0cc0 11097 ≤ cle 11236 2c2 12254 ℕ0cn0 12459 ↑cexp 14014 abscabs 15168 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 ax-un 7712 ax-cnex 11153 ax-resscn 11154 ax-1cn 11155 ax-icn 11156 ax-addcl 11157 ax-addrcl 11158 ax-mulcl 11159 ax-mulrcl 11160 ax-mulcom 11161 ax-addass 11162 ax-mulass 11163 ax-distr 11164 ax-i2m1 11165 ax-1ne0 11166 ax-1rid 11167 ax-rnegex 11168 ax-rrecex 11169 ax-cnre 11170 ax-pre-lttri 11171 ax-pre-lttrn 11172 ax-pre-ltadd 11173 ax-pre-mulgt0 11174 ax-pre-sup 11175 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-iun 4995 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6292 df-ord 6359 df-on 6360 df-lim 6361 df-suc 6362 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-riota 7352 df-ov 7399 df-oprab 7400 df-mpo 7401 df-om 7843 df-2nd 7963 df-frecs 8253 df-wrecs 8284 df-recs 8358 df-rdg 8397 df-er 8691 df-en 8928 df-dom 8929 df-sdom 8930 df-sup 9424 df-pnf 11237 df-mnf 11238 df-xr 11239 df-ltxr 11240 df-le 11241 df-sub 11433 df-neg 11434 df-div 11859 df-nn 12200 df-2 12262 df-3 12263 df-n0 12460 df-z 12546 df-uz 12810 df-rp 12962 df-seq 13954 df-exp 14015 df-cj 15033 df-re 15034 df-im 15035 df-sqrt 15169 df-abs 15170 |
This theorem is referenced by: coskpi 26001 |
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