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Theorem lt2msq 8816
Description: Two nonnegative numbers compare the same as their squares. (Contributed by Roy F. Longton, 8-Aug-2005.) (Revised by Mario Carneiro, 27-May-2016.)
Assertion
Ref Expression
lt2msq  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  ( A  <  B  <->  ( A  x.  A )  <  ( B  x.  B )
) )

Proof of Theorem lt2msq
StepHypRef Expression
1 lt2msq1 8815 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  B  e.  RR  /\  A  <  B )  ->  ( A  x.  A )  <  ( B  x.  B )
)
213expia 1205 . . 3  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  B  e.  RR )  ->  ( A  < 
B  ->  ( A  x.  A )  <  ( B  x.  B )
) )
32adantrr 479 . 2  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  ( A  <  B  ->  ( A  x.  A )  <  ( B  x.  B )
) )
4 simpr 110 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  ( B  e.  RR  /\  0  <_  B ) )
5 simpll 527 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  A  e.  RR )
6 lt2msq1 8815 . . . . . . . 8  |-  ( ( ( B  e.  RR  /\  0  <_  B )  /\  A  e.  RR  /\  B  <  A )  ->  ( B  x.  B )  <  ( A  x.  A )
)
763expia 1205 . . . . . . 7  |-  ( ( ( B  e.  RR  /\  0  <_  B )  /\  A  e.  RR )  ->  ( B  < 
A  ->  ( B  x.  B )  <  ( A  x.  A )
) )
84, 5, 7syl2anc 411 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  ( B  <  A  ->  ( B  x.  B )  <  ( A  x.  A )
) )
98con3d 631 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  ( -.  ( B  x.  B
)  <  ( A  x.  A )  ->  -.  B  <  A ) )
105, 5remulcld 7962 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  ( A  x.  A )  e.  RR )
11 simprl 529 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  B  e.  RR )
1211, 11remulcld 7962 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  ( B  x.  B )  e.  RR )
1310, 12lenltd 8049 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  ( ( A  x.  A )  <_  ( B  x.  B
)  <->  -.  ( B  x.  B )  <  ( A  x.  A )
) )
145, 11lenltd 8049 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  ( A  <_  B  <->  -.  B  <  A ) )
159, 13, 143imtr4d 203 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  ( ( A  x.  A )  <_  ( B  x.  B
)  ->  A  <_  B ) )
165recnd 7960 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  A  e.  CC )
1711recnd 7960 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  B  e.  CC )
18 mulext 8545 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  e.  CC )  /\  ( B  e.  CC  /\  B  e.  CC ) )  -> 
( ( A  x.  A ) #  ( B  x.  B )  ->  ( A #  B  \/  A #  B ) ) )
1916, 16, 17, 17, 18syl22anc 1239 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  ( ( A  x.  A ) #  ( B  x.  B
)  ->  ( A #  B  \/  A #  B
) ) )
20 oridm 757 . . . . 5  |-  ( ( A #  B  \/  A #  B )  <->  A #  B
)
2119, 20syl6ib 161 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  ( ( A  x.  A ) #  ( B  x.  B
)  ->  A #  B
) )
2215, 21anim12d 335 . . 3  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  ( (
( A  x.  A
)  <_  ( B  x.  B )  /\  ( A  x.  A ) #  ( B  x.  B
) )  ->  ( A  <_  B  /\  A #  B ) ) )
23 ltleap 8563 . . . 4  |-  ( ( ( A  x.  A
)  e.  RR  /\  ( B  x.  B
)  e.  RR )  ->  ( ( A  x.  A )  < 
( B  x.  B
)  <->  ( ( A  x.  A )  <_ 
( B  x.  B
)  /\  ( A  x.  A ) #  ( B  x.  B ) ) ) )
2410, 12, 23syl2anc 411 . . 3  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  ( ( A  x.  A )  <  ( B  x.  B
)  <->  ( ( A  x.  A )  <_ 
( B  x.  B
)  /\  ( A  x.  A ) #  ( B  x.  B ) ) ) )
25 ltleap 8563 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  ( A  <_  B  /\  A #  B ) ) )
265, 11, 25syl2anc 411 . . 3  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  ( A  <  B  <->  ( A  <_  B  /\  A #  B ) ) )
2722, 24, 263imtr4d 203 . 2  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  ( ( A  x.  A )  <  ( B  x.  B
)  ->  A  <  B ) )
283, 27impbid 129 1  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  ( A  <  B  <->  ( A  x.  A )  <  ( B  x.  B )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    <-> wb 105    \/ wo 708    e. wcel 2146   class class class wbr 3998  (class class class)co 5865   CCcc 7784   RRcr 7785   0cc0 7786    x. cmul 7791    < clt 7966    <_ cle 7967   # cap 8512
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-1re 7880  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-mulrcl 7885  ax-addcom 7886  ax-mulcom 7887  ax-addass 7888  ax-mulass 7889  ax-distr 7890  ax-i2m1 7891  ax-0lt1 7892  ax-1rid 7893  ax-0id 7894  ax-rnegex 7895  ax-precex 7896  ax-cnre 7897  ax-pre-ltirr 7898  ax-pre-ltwlin 7899  ax-pre-lttrn 7900  ax-pre-apti 7901  ax-pre-ltadd 7902  ax-pre-mulgt0 7903  ax-pre-mulext 7904
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-reu 2460  df-rab 2462  df-v 2737  df-sbc 2961  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-br 3999  df-opab 4060  df-id 4287  df-po 4290  df-iso 4291  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-iota 5170  df-fun 5210  df-fv 5216  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-pnf 7968  df-mnf 7969  df-xr 7970  df-ltxr 7971  df-le 7972  df-sub 8104  df-neg 8105  df-reap 8506  df-ap 8513
This theorem is referenced by:  le2msq  8831  lt2msqi  8844  lt2sq  10563
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