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Theorem sqrm1 12887
Description: The imaginary unit is the square root of negative 1. A lot of people like to call this the "definition" of  _i, but the definition of  sqr df-sqr 12846 has already been crafted with  _i being mentioned explicitly, and in any case it doesn't make too much sense to define a value based on a function evaluated outside its domain. A more appropriate view is to take ax-i2m1 9465 or i2 12087 as the "definition", and simply postulate the existence of a number satisfying this property. This is the approach we take here. (Contributed by Mario Carneiro, 10-Jul-2013.)
Assertion
Ref Expression
sqrm1  |-  _i  =  ( sqr `  -u 1
)

Proof of Theorem sqrm1
StepHypRef Expression
1 1re 9500 . . 3  |-  1  e.  RR
2 0le1 9978 . . 3  |-  0  <_  1
3 sqrneg 12879 . . 3  |-  ( ( 1  e.  RR  /\  0  <_  1 )  -> 
( sqr `  -u 1
)  =  ( _i  x.  ( sqr `  1
) ) )
41, 2, 3mp2an 672 . 2  |-  ( sqr `  -u 1 )  =  ( _i  x.  ( sqr `  1 ) )
5 sqr1 12883 . . 3  |-  ( sqr `  1 )  =  1
65oveq2i 6214 . 2  |-  ( _i  x.  ( sqr `  1
) )  =  ( _i  x.  1 )
7 ax-icn 9456 . . 3  |-  _i  e.  CC
87mulid1i 9503 . 2  |-  ( _i  x.  1 )  =  _i
94, 6, 83eqtrri 2488 1  |-  _i  =  ( sqr `  -u 1
)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1370    e. wcel 1758   class class class wbr 4403   ` cfv 5529  (class class class)co 6203   RRcr 9396   0cc0 9397   1c1 9398   _ici 9399    x. cmul 9402    <_ cle 9534   -ucneg 9711   sqrcsqr 12844
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474  ax-pre-sup 9475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-2nd 6691  df-recs 6945  df-rdg 6979  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-sup 7806  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-div 10109  df-nn 10438  df-2 10495  df-3 10496  df-n0 10695  df-z 10762  df-uz 10977  df-rp 11107  df-seq 11928  df-exp 11987  df-cj 12710  df-re 12711  df-im 12712  df-sqr 12846
This theorem is referenced by: (None)
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