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Mirrors > Home > MPE Home > Th. List > sqeqd | Structured version Visualization version GIF version |
Description: A deduction for showing two numbers whose squares are equal are themselves equal. (Contributed by Mario Carneiro, 3-Apr-2015.) |
Ref | Expression |
---|---|
sqeqd.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
sqeqd.2 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
sqeqd.3 | ⊢ (𝜑 → (𝐴↑2) = (𝐵↑2)) |
sqeqd.4 | ⊢ (𝜑 → 0 ≤ (ℜ‘𝐴)) |
sqeqd.5 | ⊢ (𝜑 → 0 ≤ (ℜ‘𝐵)) |
sqeqd.6 | ⊢ ((𝜑 ∧ (ℜ‘𝐴) = 0 ∧ (ℜ‘𝐵) = 0) → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
sqeqd | ⊢ (𝜑 → 𝐴 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sqeqd.3 | . . . . 5 ⊢ (𝜑 → (𝐴↑2) = (𝐵↑2)) | |
2 | sqeqd.1 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
3 | sqeqd.2 | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
4 | sqeqor 13568 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴↑2) = (𝐵↑2) ↔ (𝐴 = 𝐵 ∨ 𝐴 = -𝐵))) | |
5 | 2, 3, 4 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → ((𝐴↑2) = (𝐵↑2) ↔ (𝐴 = 𝐵 ∨ 𝐴 = -𝐵))) |
6 | 1, 5 | mpbid 233 | . . . 4 ⊢ (𝜑 → (𝐴 = 𝐵 ∨ 𝐴 = -𝐵)) |
7 | 6 | ord 858 | . . 3 ⊢ (𝜑 → (¬ 𝐴 = 𝐵 → 𝐴 = -𝐵)) |
8 | simpl 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 = -𝐵) → 𝜑) | |
9 | fveq2 6664 | . . . . . . 7 ⊢ (𝐴 = -𝐵 → (ℜ‘𝐴) = (ℜ‘-𝐵)) | |
10 | reneg 14474 | . . . . . . . 8 ⊢ (𝐵 ∈ ℂ → (ℜ‘-𝐵) = -(ℜ‘𝐵)) | |
11 | 3, 10 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (ℜ‘-𝐵) = -(ℜ‘𝐵)) |
12 | 9, 11 | sylan9eqr 2878 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐴 = -𝐵) → (ℜ‘𝐴) = -(ℜ‘𝐵)) |
13 | sqeqd.4 | . . . . . . . . . . . 12 ⊢ (𝜑 → 0 ≤ (ℜ‘𝐴)) | |
14 | 13 | adantr 481 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐴 = -𝐵) → 0 ≤ (ℜ‘𝐴)) |
15 | 14, 12 | breqtrd 5084 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 = -𝐵) → 0 ≤ -(ℜ‘𝐵)) |
16 | 3 | adantr 481 | . . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 = -𝐵) → 𝐵 ∈ ℂ) |
17 | recl 14459 | . . . . . . . . . . . 12 ⊢ (𝐵 ∈ ℂ → (ℜ‘𝐵) ∈ ℝ) | |
18 | 16, 17 | syl 17 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐴 = -𝐵) → (ℜ‘𝐵) ∈ ℝ) |
19 | 18 | le0neg1d 11200 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 = -𝐵) → ((ℜ‘𝐵) ≤ 0 ↔ 0 ≤ -(ℜ‘𝐵))) |
20 | 15, 19 | mpbird 258 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 = -𝐵) → (ℜ‘𝐵) ≤ 0) |
21 | sqeqd.5 | . . . . . . . . . 10 ⊢ (𝜑 → 0 ≤ (ℜ‘𝐵)) | |
22 | 21 | adantr 481 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 = -𝐵) → 0 ≤ (ℜ‘𝐵)) |
23 | 0re 10632 | . . . . . . . . . 10 ⊢ 0 ∈ ℝ | |
24 | letri3 10715 | . . . . . . . . . 10 ⊢ (((ℜ‘𝐵) ∈ ℝ ∧ 0 ∈ ℝ) → ((ℜ‘𝐵) = 0 ↔ ((ℜ‘𝐵) ≤ 0 ∧ 0 ≤ (ℜ‘𝐵)))) | |
25 | 18, 23, 24 | sylancl 586 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 = -𝐵) → ((ℜ‘𝐵) = 0 ↔ ((ℜ‘𝐵) ≤ 0 ∧ 0 ≤ (ℜ‘𝐵)))) |
26 | 20, 22, 25 | mpbir2and 709 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 = -𝐵) → (ℜ‘𝐵) = 0) |
27 | 26 | negeqd 10869 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 = -𝐵) → -(ℜ‘𝐵) = -0) |
28 | neg0 10921 | . . . . . . 7 ⊢ -0 = 0 | |
29 | 27, 28 | syl6eq 2872 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐴 = -𝐵) → -(ℜ‘𝐵) = 0) |
30 | 12, 29 | eqtrd 2856 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 = -𝐵) → (ℜ‘𝐴) = 0) |
31 | sqeqd.6 | . . . . 5 ⊢ ((𝜑 ∧ (ℜ‘𝐴) = 0 ∧ (ℜ‘𝐵) = 0) → 𝐴 = 𝐵) | |
32 | 8, 30, 26, 31 | syl3anc 1363 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = -𝐵) → 𝐴 = 𝐵) |
33 | 32 | ex 413 | . . 3 ⊢ (𝜑 → (𝐴 = -𝐵 → 𝐴 = 𝐵)) |
34 | 7, 33 | syld 47 | . 2 ⊢ (𝜑 → (¬ 𝐴 = 𝐵 → 𝐴 = 𝐵)) |
35 | 34 | pm2.18d 127 | 1 ⊢ (𝜑 → 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 ∨ wo 841 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 class class class wbr 5058 ‘cfv 6349 (class class class)co 7145 ℂcc 10524 ℝcr 10525 0cc0 10526 ≤ cle 10665 -cneg 10860 2c2 11681 ↑cexp 13419 ℜcre 14446 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 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 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4833 df-iun 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7569 df-2nd 7681 df-wrecs 7938 df-recs 7999 df-rdg 8037 df-er 8279 df-en 8499 df-dom 8500 df-sdom 8501 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11628 df-2 11689 df-n0 11887 df-z 11971 df-uz 12233 df-seq 13360 df-exp 13420 df-cj 14448 df-re 14449 df-im 14450 |
This theorem is referenced by: (None) |
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