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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 14255 | . . . . . 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 232 | . . . 4 ⊢ (𝜑 → (𝐴 = 𝐵 ∨ 𝐴 = -𝐵)) |
| 7 | 6 | ord 865 | . . 3 ⊢ (𝜑 → (¬ 𝐴 = 𝐵 → 𝐴 = -𝐵)) |
| 8 | simpl 482 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 = -𝐵) → 𝜑) | |
| 9 | fveq2 6906 | . . . . . . 7 ⊢ (𝐴 = -𝐵 → (ℜ‘𝐴) = (ℜ‘-𝐵)) | |
| 10 | reneg 15164 | . . . . . . . 8 ⊢ (𝐵 ∈ ℂ → (ℜ‘-𝐵) = -(ℜ‘𝐵)) | |
| 11 | 3, 10 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (ℜ‘-𝐵) = -(ℜ‘𝐵)) |
| 12 | 9, 11 | sylan9eqr 2799 | . . . . . 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 5169 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 = -𝐵) → 0 ≤ -(ℜ‘𝐵)) |
| 16 | 3 | adantr 480 | . . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 = -𝐵) → 𝐵 ∈ ℂ) |
| 17 | recl 15149 | . . . . . . . . . . . 12 ⊢ (𝐵 ∈ ℂ → (ℜ‘𝐵) ∈ ℝ) | |
| 18 | 16, 17 | syl 17 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐴 = -𝐵) → (ℜ‘𝐵) ∈ ℝ) |
| 19 | 18 | le0neg1d 11834 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 = -𝐵) → ((ℜ‘𝐵) ≤ 0 ↔ 0 ≤ -(ℜ‘𝐵))) |
| 20 | 15, 19 | mpbird 257 | . . . . . . . . 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 11263 | . . . . . . . . . 10 ⊢ 0 ∈ ℝ | |
| 24 | letri3 11346 | . . . . . . . . . 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 713 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 = -𝐵) → (ℜ‘𝐵) = 0) |
| 27 | 26 | negeqd 11502 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 = -𝐵) → -(ℜ‘𝐵) = -0) |
| 28 | neg0 11555 | . . . . . . 7 ⊢ -0 = 0 | |
| 29 | 27, 28 | eqtrdi 2793 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐴 = -𝐵) → -(ℜ‘𝐵) = 0) |
| 30 | 12, 29 | eqtrd 2777 | . . . . 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 1373 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = -𝐵) → 𝐴 = 𝐵) |
| 33 | 32 | ex 412 | . . 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 206 ∧ wa 395 ∨ wo 848 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 ℝcr 11154 0cc0 11155 ≤ cle 11296 -cneg 11493 2c2 12321 ↑cexp 14102 ℜcre 15136 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-n0 12527 df-z 12614 df-uz 12879 df-seq 14043 df-exp 14103 df-cj 15138 df-re 15139 df-im 15140 |
| This theorem is referenced by: (None) |
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