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Mirrors > Home > MPE Home > Th. List > sqeqori | Structured version Visualization version GIF version |
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 13571 | . . . 4 ⊢ ((𝐴↑2) − (𝐵↑2)) = ((𝐴 + 𝐵) · (𝐴 − 𝐵)) |
4 | 3 | eqeq1i 2803 | . . 3 ⊢ (((𝐴↑2) − (𝐵↑2)) = 0 ↔ ((𝐴 + 𝐵) · (𝐴 − 𝐵)) = 0) |
5 | 1 | sqcli 13540 | . . . 4 ⊢ (𝐴↑2) ∈ ℂ |
6 | 2 | sqcli 13540 | . . . 4 ⊢ (𝐵↑2) ∈ ℂ |
7 | 5, 6 | subeq0i 10955 | . . 3 ⊢ (((𝐴↑2) − (𝐵↑2)) = 0 ↔ (𝐴↑2) = (𝐵↑2)) |
8 | 1, 2 | addcli 10636 | . . . 4 ⊢ (𝐴 + 𝐵) ∈ ℂ |
9 | 1, 2 | subcli 10951 | . . . 4 ⊢ (𝐴 − 𝐵) ∈ ℂ |
10 | 8, 9 | mul0ori 11277 | . . 3 ⊢ (((𝐴 + 𝐵) · (𝐴 − 𝐵)) = 0 ↔ ((𝐴 + 𝐵) = 0 ∨ (𝐴 − 𝐵) = 0)) |
11 | 4, 7, 10 | 3bitr3i 304 | . 2 ⊢ ((𝐴↑2) = (𝐵↑2) ↔ ((𝐴 + 𝐵) = 0 ∨ (𝐴 − 𝐵) = 0)) |
12 | orcom 867 | . 2 ⊢ (((𝐴 + 𝐵) = 0 ∨ (𝐴 − 𝐵) = 0) ↔ ((𝐴 − 𝐵) = 0 ∨ (𝐴 + 𝐵) = 0)) | |
13 | 1, 2 | subeq0i 10955 | . . 3 ⊢ ((𝐴 − 𝐵) = 0 ↔ 𝐴 = 𝐵) |
14 | 1, 2 | subnegi 10954 | . . . . 5 ⊢ (𝐴 − -𝐵) = (𝐴 + 𝐵) |
15 | 14 | eqeq1i 2803 | . . . 4 ⊢ ((𝐴 − -𝐵) = 0 ↔ (𝐴 + 𝐵) = 0) |
16 | 2 | negcli 10943 | . . . . 5 ⊢ -𝐵 ∈ ℂ |
17 | 1, 16 | subeq0i 10955 | . . . 4 ⊢ ((𝐴 − -𝐵) = 0 ↔ 𝐴 = -𝐵) |
18 | 15, 17 | bitr3i 280 | . . 3 ⊢ ((𝐴 + 𝐵) = 0 ↔ 𝐴 = -𝐵) |
19 | 13, 18 | orbi12i 912 | . 2 ⊢ (((𝐴 − 𝐵) = 0 ∨ (𝐴 + 𝐵) = 0) ↔ (𝐴 = 𝐵 ∨ 𝐴 = -𝐵)) |
20 | 11, 12, 19 | 3bitri 300 | 1 ⊢ ((𝐴↑2) = (𝐵↑2) ↔ (𝐴 = 𝐵 ∨ 𝐴 = -𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∨ wo 844 = wceq 1538 ∈ wcel 2111 (class class class)co 7135 ℂcc 10524 0cc0 10526 + caddc 10529 · cmul 10531 − cmin 10859 -cneg 10860 2c2 11680 ↑cexp 13425 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 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 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-n0 11886 df-z 11970 df-uz 12232 df-seq 13365 df-exp 13426 |
This theorem is referenced by: subsq0i 13573 sqeqor 13574 sinhalfpilem 25056 |
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