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| Description: The squares of two complex numbers are equal iff one number equals the other or its negative. Lemma 15-4.7 of [Gleason] p. 311 and its converse. (Contributed by NM, 15-Jan-2006.) | 
| Ref | Expression | 
|---|---|
| binom2.1 | ⊢ 𝐴 ∈ ℂ | 
| binom2.2 | ⊢ 𝐵 ∈ ℂ | 
| Ref | Expression | 
|---|---|
| sqeqori | ⊢ ((𝐴↑2) = (𝐵↑2) ↔ (𝐴 = 𝐵 ∨ 𝐴 = -𝐵)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | binom2.1 | . . . . 5 ⊢ 𝐴 ∈ ℂ | |
| 2 | binom2.2 | . . . . 5 ⊢ 𝐵 ∈ ℂ | |
| 3 | 1, 2 | subsqi 14253 | . . . 4 ⊢ ((𝐴↑2) − (𝐵↑2)) = ((𝐴 + 𝐵) · (𝐴 − 𝐵)) | 
| 4 | 3 | eqeq1i 2741 | . . 3 ⊢ (((𝐴↑2) − (𝐵↑2)) = 0 ↔ ((𝐴 + 𝐵) · (𝐴 − 𝐵)) = 0) | 
| 5 | 1 | sqcli 14221 | . . . 4 ⊢ (𝐴↑2) ∈ ℂ | 
| 6 | 2 | sqcli 14221 | . . . 4 ⊢ (𝐵↑2) ∈ ℂ | 
| 7 | 5, 6 | subeq0i 11590 | . . 3 ⊢ (((𝐴↑2) − (𝐵↑2)) = 0 ↔ (𝐴↑2) = (𝐵↑2)) | 
| 8 | 1, 2 | addcli 11268 | . . . 4 ⊢ (𝐴 + 𝐵) ∈ ℂ | 
| 9 | 1, 2 | subcli 11586 | . . . 4 ⊢ (𝐴 − 𝐵) ∈ ℂ | 
| 10 | 8, 9 | mul0ori 11912 | . . 3 ⊢ (((𝐴 + 𝐵) · (𝐴 − 𝐵)) = 0 ↔ ((𝐴 + 𝐵) = 0 ∨ (𝐴 − 𝐵) = 0)) | 
| 11 | 4, 7, 10 | 3bitr3i 301 | . 2 ⊢ ((𝐴↑2) = (𝐵↑2) ↔ ((𝐴 + 𝐵) = 0 ∨ (𝐴 − 𝐵) = 0)) | 
| 12 | orcom 870 | . 2 ⊢ (((𝐴 + 𝐵) = 0 ∨ (𝐴 − 𝐵) = 0) ↔ ((𝐴 − 𝐵) = 0 ∨ (𝐴 + 𝐵) = 0)) | |
| 13 | 1, 2 | subeq0i 11590 | . . 3 ⊢ ((𝐴 − 𝐵) = 0 ↔ 𝐴 = 𝐵) | 
| 14 | 1, 2 | subnegi 11589 | . . . . 5 ⊢ (𝐴 − -𝐵) = (𝐴 + 𝐵) | 
| 15 | 14 | eqeq1i 2741 | . . . 4 ⊢ ((𝐴 − -𝐵) = 0 ↔ (𝐴 + 𝐵) = 0) | 
| 16 | 2 | negcli 11578 | . . . . 5 ⊢ -𝐵 ∈ ℂ | 
| 17 | 1, 16 | subeq0i 11590 | . . . 4 ⊢ ((𝐴 − -𝐵) = 0 ↔ 𝐴 = -𝐵) | 
| 18 | 15, 17 | bitr3i 277 | . . 3 ⊢ ((𝐴 + 𝐵) = 0 ↔ 𝐴 = -𝐵) | 
| 19 | 13, 18 | orbi12i 914 | . 2 ⊢ (((𝐴 − 𝐵) = 0 ∨ (𝐴 + 𝐵) = 0) ↔ (𝐴 = 𝐵 ∨ 𝐴 = -𝐵)) | 
| 20 | 11, 12, 19 | 3bitri 297 | 1 ⊢ ((𝐴↑2) = (𝐵↑2) ↔ (𝐴 = 𝐵 ∨ 𝐴 = -𝐵)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∨ wo 847 = wceq 1539 ∈ wcel 2107 (class class class)co 7432 ℂcc 11154 0cc0 11156 + caddc 11159 · cmul 11161 − cmin 11493 -cneg 11494 2c2 12322 ↑cexp 14103 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-2 12330 df-n0 12529 df-z 12616 df-uz 12880 df-seq 14044 df-exp 14104 | 
| This theorem is referenced by: subsq0i 14255 sqeqor 14256 sinhalfpilem 26506 | 
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