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Mirrors > Home > MPE Home > Th. List > sq01 | Structured version Visualization version GIF version |
Description: If a complex number equals its square, it must be 0 or 1. (Contributed by NM, 6-Jun-2006.) |
Ref | Expression |
---|---|
sq01 | ⊢ (𝐴 ∈ ℂ → ((𝐴↑2) = 𝐴 ↔ (𝐴 = 0 ∨ 𝐴 = 1))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ne 2940 | . . . . 5 ⊢ (𝐴 ≠ 0 ↔ ¬ 𝐴 = 0) | |
2 | sqval 14087 | . . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → (𝐴↑2) = (𝐴 · 𝐴)) | |
3 | mulrid 11219 | . . . . . . . . . . 11 ⊢ (𝐴 ∈ ℂ → (𝐴 · 1) = 𝐴) | |
4 | 3 | eqcomd 2737 | . . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → 𝐴 = (𝐴 · 1)) |
5 | 2, 4 | eqeq12d 2747 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → ((𝐴↑2) = 𝐴 ↔ (𝐴 · 𝐴) = (𝐴 · 1))) |
6 | 5 | adantr 480 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((𝐴↑2) = 𝐴 ↔ (𝐴 · 𝐴) = (𝐴 · 1))) |
7 | ax-1cn 11174 | . . . . . . . . . 10 ⊢ 1 ∈ ℂ | |
8 | mulcan 11858 | . . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ ∧ (𝐴 ∈ ℂ ∧ 𝐴 ≠ 0)) → ((𝐴 · 𝐴) = (𝐴 · 1) ↔ 𝐴 = 1)) | |
9 | 7, 8 | mp3an2 1448 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (𝐴 ∈ ℂ ∧ 𝐴 ≠ 0)) → ((𝐴 · 𝐴) = (𝐴 · 1) ↔ 𝐴 = 1)) |
10 | 9 | anabss5 665 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((𝐴 · 𝐴) = (𝐴 · 1) ↔ 𝐴 = 1)) |
11 | 6, 10 | bitrd 279 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((𝐴↑2) = 𝐴 ↔ 𝐴 = 1)) |
12 | 11 | biimpd 228 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((𝐴↑2) = 𝐴 → 𝐴 = 1)) |
13 | 12 | impancom 451 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (𝐴↑2) = 𝐴) → (𝐴 ≠ 0 → 𝐴 = 1)) |
14 | 1, 13 | biimtrrid 242 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (𝐴↑2) = 𝐴) → (¬ 𝐴 = 0 → 𝐴 = 1)) |
15 | 14 | orrd 860 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ (𝐴↑2) = 𝐴) → (𝐴 = 0 ∨ 𝐴 = 1)) |
16 | 15 | ex 412 | . 2 ⊢ (𝐴 ∈ ℂ → ((𝐴↑2) = 𝐴 → (𝐴 = 0 ∨ 𝐴 = 1))) |
17 | sq0 14163 | . . . 4 ⊢ (0↑2) = 0 | |
18 | oveq1 7419 | . . . 4 ⊢ (𝐴 = 0 → (𝐴↑2) = (0↑2)) | |
19 | id 22 | . . . 4 ⊢ (𝐴 = 0 → 𝐴 = 0) | |
20 | 17, 18, 19 | 3eqtr4a 2797 | . . 3 ⊢ (𝐴 = 0 → (𝐴↑2) = 𝐴) |
21 | sq1 14166 | . . . 4 ⊢ (1↑2) = 1 | |
22 | oveq1 7419 | . . . 4 ⊢ (𝐴 = 1 → (𝐴↑2) = (1↑2)) | |
23 | id 22 | . . . 4 ⊢ (𝐴 = 1 → 𝐴 = 1) | |
24 | 21, 22, 23 | 3eqtr4a 2797 | . . 3 ⊢ (𝐴 = 1 → (𝐴↑2) = 𝐴) |
25 | 20, 24 | jaoi 854 | . 2 ⊢ ((𝐴 = 0 ∨ 𝐴 = 1) → (𝐴↑2) = 𝐴) |
26 | 16, 25 | impbid1 224 | 1 ⊢ (𝐴 ∈ ℂ → ((𝐴↑2) = 𝐴 ↔ (𝐴 = 0 ∨ 𝐴 = 1))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∨ wo 844 = wceq 1540 ∈ wcel 2105 ≠ wne 2939 (class class class)co 7412 ℂcc 11114 0cc0 11116 1c1 11117 · cmul 11121 2c2 12274 ↑cexp 14034 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-nn 12220 df-2 12282 df-n0 12480 df-z 12566 df-uz 12830 df-seq 13974 df-exp 14035 |
This theorem is referenced by: cphsubrglem 25024 |
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