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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 12840 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴↑2) = (𝐵↑2) ↔ (𝐴 = 𝐵 ∨ 𝐴 = -𝐵))) | |
5 | 2, 3, 4 | syl2anc 691 | . . . . 5 ⊢ (𝜑 → ((𝐴↑2) = (𝐵↑2) ↔ (𝐴 = 𝐵 ∨ 𝐴 = -𝐵))) |
6 | 1, 5 | mpbid 221 | . . . 4 ⊢ (𝜑 → (𝐴 = 𝐵 ∨ 𝐴 = -𝐵)) |
7 | 6 | ord 391 | . . 3 ⊢ (𝜑 → (¬ 𝐴 = 𝐵 → 𝐴 = -𝐵)) |
8 | simpl 472 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 = -𝐵) → 𝜑) | |
9 | fveq2 6103 | . . . . . . 7 ⊢ (𝐴 = -𝐵 → (ℜ‘𝐴) = (ℜ‘-𝐵)) | |
10 | reneg 13713 | . . . . . . . 8 ⊢ (𝐵 ∈ ℂ → (ℜ‘-𝐵) = -(ℜ‘𝐵)) | |
11 | 3, 10 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (ℜ‘-𝐵) = -(ℜ‘𝐵)) |
12 | 9, 11 | sylan9eqr 2666 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐴 = -𝐵) → (ℜ‘𝐴) = -(ℜ‘𝐵)) |
13 | sqeqd.4 | . . . . . . . . . . . 12 ⊢ (𝜑 → 0 ≤ (ℜ‘𝐴)) | |
14 | 13 | adantr 480 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐴 = -𝐵) → 0 ≤ (ℜ‘𝐴)) |
15 | 14, 12 | breqtrd 4609 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 = -𝐵) → 0 ≤ -(ℜ‘𝐵)) |
16 | 3 | adantr 480 | . . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 = -𝐵) → 𝐵 ∈ ℂ) |
17 | recl 13698 | . . . . . . . . . . . 12 ⊢ (𝐵 ∈ ℂ → (ℜ‘𝐵) ∈ ℝ) | |
18 | 16, 17 | syl 17 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐴 = -𝐵) → (ℜ‘𝐵) ∈ ℝ) |
19 | 18 | le0neg1d 10478 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 = -𝐵) → ((ℜ‘𝐵) ≤ 0 ↔ 0 ≤ -(ℜ‘𝐵))) |
20 | 15, 19 | mpbird 246 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 = -𝐵) → (ℜ‘𝐵) ≤ 0) |
21 | sqeqd.5 | . . . . . . . . . 10 ⊢ (𝜑 → 0 ≤ (ℜ‘𝐵)) | |
22 | 21 | adantr 480 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 = -𝐵) → 0 ≤ (ℜ‘𝐵)) |
23 | 0re 9919 | . . . . . . . . . 10 ⊢ 0 ∈ ℝ | |
24 | letri3 10002 | . . . . . . . . . 10 ⊢ (((ℜ‘𝐵) ∈ ℝ ∧ 0 ∈ ℝ) → ((ℜ‘𝐵) = 0 ↔ ((ℜ‘𝐵) ≤ 0 ∧ 0 ≤ (ℜ‘𝐵)))) | |
25 | 18, 23, 24 | sylancl 693 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 = -𝐵) → ((ℜ‘𝐵) = 0 ↔ ((ℜ‘𝐵) ≤ 0 ∧ 0 ≤ (ℜ‘𝐵)))) |
26 | 20, 22, 25 | mpbir2and 959 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 = -𝐵) → (ℜ‘𝐵) = 0) |
27 | 26 | negeqd 10154 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 = -𝐵) → -(ℜ‘𝐵) = -0) |
28 | neg0 10206 | . . . . . . 7 ⊢ -0 = 0 | |
29 | 27, 28 | syl6eq 2660 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐴 = -𝐵) → -(ℜ‘𝐵) = 0) |
30 | 12, 29 | eqtrd 2644 | . . . . 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 1318 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = -𝐵) → 𝐴 = 𝐵) |
33 | 32 | ex 449 | . . 3 ⊢ (𝜑 → (𝐴 = -𝐵 → 𝐴 = 𝐵)) |
34 | 7, 33 | syld 46 | . 2 ⊢ (𝜑 → (¬ 𝐴 = 𝐵 → 𝐴 = 𝐵)) |
35 | 34 | pm2.18d 123 | 1 ⊢ (𝜑 → 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 ℝcr 9814 0cc0 9815 ≤ cle 9954 -cneg 10146 2c2 10947 ↑cexp 12722 ℜcre 13685 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-n0 11170 df-z 11255 df-uz 11564 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 |
This theorem is referenced by: (None) |
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