Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > sqeqor | 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 Paul Chapman, 15-Mar-2008.) |
Ref | Expression |
---|---|
sqeqor | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴↑2) = (𝐵↑2) ↔ (𝐴 = 𝐵 ∨ 𝐴 = -𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 7166 | . . . 4 ⊢ (𝐴 = if(𝐴 ∈ ℂ, 𝐴, 0) → (𝐴↑2) = (if(𝐴 ∈ ℂ, 𝐴, 0)↑2)) | |
2 | 1 | eqeq1d 2826 | . . 3 ⊢ (𝐴 = if(𝐴 ∈ ℂ, 𝐴, 0) → ((𝐴↑2) = (𝐵↑2) ↔ (if(𝐴 ∈ ℂ, 𝐴, 0)↑2) = (𝐵↑2))) |
3 | eqeq1 2828 | . . . 4 ⊢ (𝐴 = if(𝐴 ∈ ℂ, 𝐴, 0) → (𝐴 = 𝐵 ↔ if(𝐴 ∈ ℂ, 𝐴, 0) = 𝐵)) | |
4 | eqeq1 2828 | . . . 4 ⊢ (𝐴 = if(𝐴 ∈ ℂ, 𝐴, 0) → (𝐴 = -𝐵 ↔ if(𝐴 ∈ ℂ, 𝐴, 0) = -𝐵)) | |
5 | 3, 4 | orbi12d 915 | . . 3 ⊢ (𝐴 = if(𝐴 ∈ ℂ, 𝐴, 0) → ((𝐴 = 𝐵 ∨ 𝐴 = -𝐵) ↔ (if(𝐴 ∈ ℂ, 𝐴, 0) = 𝐵 ∨ if(𝐴 ∈ ℂ, 𝐴, 0) = -𝐵))) |
6 | 2, 5 | bibi12d 348 | . 2 ⊢ (𝐴 = if(𝐴 ∈ ℂ, 𝐴, 0) → (((𝐴↑2) = (𝐵↑2) ↔ (𝐴 = 𝐵 ∨ 𝐴 = -𝐵)) ↔ ((if(𝐴 ∈ ℂ, 𝐴, 0)↑2) = (𝐵↑2) ↔ (if(𝐴 ∈ ℂ, 𝐴, 0) = 𝐵 ∨ if(𝐴 ∈ ℂ, 𝐴, 0) = -𝐵)))) |
7 | oveq1 7166 | . . . 4 ⊢ (𝐵 = if(𝐵 ∈ ℂ, 𝐵, 0) → (𝐵↑2) = (if(𝐵 ∈ ℂ, 𝐵, 0)↑2)) | |
8 | 7 | eqeq2d 2835 | . . 3 ⊢ (𝐵 = if(𝐵 ∈ ℂ, 𝐵, 0) → ((if(𝐴 ∈ ℂ, 𝐴, 0)↑2) = (𝐵↑2) ↔ (if(𝐴 ∈ ℂ, 𝐴, 0)↑2) = (if(𝐵 ∈ ℂ, 𝐵, 0)↑2))) |
9 | eqeq2 2836 | . . . 4 ⊢ (𝐵 = if(𝐵 ∈ ℂ, 𝐵, 0) → (if(𝐴 ∈ ℂ, 𝐴, 0) = 𝐵 ↔ if(𝐴 ∈ ℂ, 𝐴, 0) = if(𝐵 ∈ ℂ, 𝐵, 0))) | |
10 | negeq 10881 | . . . . 5 ⊢ (𝐵 = if(𝐵 ∈ ℂ, 𝐵, 0) → -𝐵 = -if(𝐵 ∈ ℂ, 𝐵, 0)) | |
11 | 10 | eqeq2d 2835 | . . . 4 ⊢ (𝐵 = if(𝐵 ∈ ℂ, 𝐵, 0) → (if(𝐴 ∈ ℂ, 𝐴, 0) = -𝐵 ↔ if(𝐴 ∈ ℂ, 𝐴, 0) = -if(𝐵 ∈ ℂ, 𝐵, 0))) |
12 | 9, 11 | orbi12d 915 | . . 3 ⊢ (𝐵 = if(𝐵 ∈ ℂ, 𝐵, 0) → ((if(𝐴 ∈ ℂ, 𝐴, 0) = 𝐵 ∨ if(𝐴 ∈ ℂ, 𝐴, 0) = -𝐵) ↔ (if(𝐴 ∈ ℂ, 𝐴, 0) = if(𝐵 ∈ ℂ, 𝐵, 0) ∨ if(𝐴 ∈ ℂ, 𝐴, 0) = -if(𝐵 ∈ ℂ, 𝐵, 0)))) |
13 | 8, 12 | bibi12d 348 | . 2 ⊢ (𝐵 = if(𝐵 ∈ ℂ, 𝐵, 0) → (((if(𝐴 ∈ ℂ, 𝐴, 0)↑2) = (𝐵↑2) ↔ (if(𝐴 ∈ ℂ, 𝐴, 0) = 𝐵 ∨ if(𝐴 ∈ ℂ, 𝐴, 0) = -𝐵)) ↔ ((if(𝐴 ∈ ℂ, 𝐴, 0)↑2) = (if(𝐵 ∈ ℂ, 𝐵, 0)↑2) ↔ (if(𝐴 ∈ ℂ, 𝐴, 0) = if(𝐵 ∈ ℂ, 𝐵, 0) ∨ if(𝐴 ∈ ℂ, 𝐴, 0) = -if(𝐵 ∈ ℂ, 𝐵, 0))))) |
14 | 0cn 10636 | . . . 4 ⊢ 0 ∈ ℂ | |
15 | 14 | elimel 4537 | . . 3 ⊢ if(𝐴 ∈ ℂ, 𝐴, 0) ∈ ℂ |
16 | 14 | elimel 4537 | . . 3 ⊢ if(𝐵 ∈ ℂ, 𝐵, 0) ∈ ℂ |
17 | 15, 16 | sqeqori 13579 | . 2 ⊢ ((if(𝐴 ∈ ℂ, 𝐴, 0)↑2) = (if(𝐵 ∈ ℂ, 𝐵, 0)↑2) ↔ (if(𝐴 ∈ ℂ, 𝐴, 0) = if(𝐵 ∈ ℂ, 𝐵, 0) ∨ if(𝐴 ∈ ℂ, 𝐴, 0) = -if(𝐵 ∈ ℂ, 𝐵, 0))) |
18 | 6, 13, 17 | dedth2h 4527 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴↑2) = (𝐵↑2) ↔ (𝐴 = 𝐵 ∨ 𝐴 = -𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 = wceq 1536 ∈ wcel 2113 ifcif 4470 (class class class)co 7159 ℂcc 10538 0cc0 10540 -cneg 10874 2c2 11695 ↑cexp 13432 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-2 11703 df-n0 11901 df-z 11985 df-uz 12247 df-seq 13373 df-exp 13433 |
This theorem is referenced by: sqeqd 14528 sqrmo 14614 eqsqrtor 14729 4sqlem10 16286 cxpsqrt 25289 quad2 25420 atandm3 25459 atans2 25512 dvasin 34982 dvacos 34983 itschlc0xyqsol1 44760 |
Copyright terms: Public domain | W3C validator |