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Theorem msqge0 8535
Description: A square is nonnegative. Lemma 2.35 of [Geuvers], p. 9. (Contributed by NM, 23-May-2007.) (Revised by Mario Carneiro, 27-May-2016.)
Assertion
Ref Expression
msqge0  |-  ( A  e.  RR  ->  0  <_  ( A  x.  A
) )

Proof of Theorem msqge0
StepHypRef Expression
1 remulcl 7902 . . . . 5  |-  ( ( A  e.  RR  /\  A  e.  RR )  ->  ( A  x.  A
)  e.  RR )
21anidms 395 . . . 4  |-  ( A  e.  RR  ->  ( A  x.  A )  e.  RR )
3 0re 7920 . . . 4  |-  0  e.  RR
4 ltnsym2 8010 . . . 4  |-  ( ( ( A  x.  A
)  e.  RR  /\  0  e.  RR )  ->  -.  ( ( A  x.  A )  <  0  /\  0  < 
( A  x.  A
) ) )
52, 3, 4sylancl 411 . . 3  |-  ( A  e.  RR  ->  -.  ( ( A  x.  A )  <  0  /\  0  <  ( A  x.  A ) ) )
6 orc 707 . . . . . 6  |-  ( ( A  x.  A )  <  0  ->  (
( A  x.  A
)  <  0  \/  0  <  ( A  x.  A ) ) )
7 reaplt 8507 . . . . . . 7  |-  ( ( ( A  x.  A
)  e.  RR  /\  0  e.  RR )  ->  ( ( A  x.  A ) #  0  <->  ( ( A  x.  A )  <  0  \/  0  < 
( A  x.  A
) ) ) )
82, 3, 7sylancl 411 . . . . . 6  |-  ( A  e.  RR  ->  (
( A  x.  A
) #  0  <->  ( ( A  x.  A )  <  0  \/  0  < 
( A  x.  A
) ) ) )
96, 8syl5ibr 155 . . . . 5  |-  ( A  e.  RR  ->  (
( A  x.  A
)  <  0  ->  ( A  x.  A ) #  0 ) )
10 recn 7907 . . . . . . . . 9  |-  ( A  e.  RR  ->  A  e.  CC )
11 mulap0r 8534 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A  e.  CC  /\  ( A  x.  A ) #  0 )  ->  ( A #  0  /\  A #  0 ) )
1210, 11syl3an1 1266 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  A  e.  CC  /\  ( A  x.  A ) #  0 )  ->  ( A #  0  /\  A #  0 ) )
1310, 12syl3an2 1267 . . . . . . . 8  |-  ( ( A  e.  RR  /\  A  e.  RR  /\  ( A  x.  A ) #  0 )  ->  ( A #  0  /\  A #  0 ) )
1413simpld 111 . . . . . . 7  |-  ( ( A  e.  RR  /\  A  e.  RR  /\  ( A  x.  A ) #  0 )  ->  A #  0 )
15143expia 1200 . . . . . 6  |-  ( ( A  e.  RR  /\  A  e.  RR )  ->  ( ( A  x.  A ) #  0  ->  A #  0 ) )
1615anidms 395 . . . . 5  |-  ( A  e.  RR  ->  (
( A  x.  A
) #  0  ->  A #  0 ) )
17 apsqgt0 8520 . . . . . 6  |-  ( ( A  e.  RR  /\  A #  0 )  ->  0  <  ( A  x.  A
) )
1817ex 114 . . . . 5  |-  ( A  e.  RR  ->  ( A #  0  ->  0  < 
( A  x.  A
) ) )
199, 16, 183syld 57 . . . 4  |-  ( A  e.  RR  ->  (
( A  x.  A
)  <  0  ->  0  <  ( A  x.  A ) ) )
2019ancld 323 . . 3  |-  ( A  e.  RR  ->  (
( A  x.  A
)  <  0  ->  ( ( A  x.  A
)  <  0  /\  0  <  ( A  x.  A ) ) ) )
215, 20mtod 658 . 2  |-  ( A  e.  RR  ->  -.  ( A  x.  A
)  <  0 )
22 lenlt 7995 . . 3  |-  ( ( 0  e.  RR  /\  ( A  x.  A
)  e.  RR )  ->  ( 0  <_ 
( A  x.  A
)  <->  -.  ( A  x.  A )  <  0
) )
233, 2, 22sylancr 412 . 2  |-  ( A  e.  RR  ->  (
0  <_  ( A  x.  A )  <->  -.  ( A  x.  A )  <  0 ) )
2421, 23mpbird 166 1  |-  ( A  e.  RR  ->  0  <_  ( A  x.  A
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 703    /\ w3a 973    e. wcel 2141   class class class wbr 3989  (class class class)co 5853   CCcc 7772   RRcr 7773   0cc0 7774    x. cmul 7779    < clt 7954    <_ cle 7955   # cap 8500
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-po 4281  df-iso 4282  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501
This theorem is referenced by:  msqge0i  8536  msqge0d  8537  recexaplem2  8570  sqge0  10552  bernneq  10596
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