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Theorem lt2msq 8832
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 8831 . . . 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 8831 . . . . . . . 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 7978 . . . . . 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 7978 . . . . . 6 |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> ( B x. B ) e. RR )
1310, 12lenltd 8065 . . . . 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 8065 . . . . 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 7976 . . . . . 6 |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> A e. CC )
1711recnd 7976 . . . . . 6 |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> B e. CC )
18 mulext 8561 . . . . . 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 8579 . . . 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 8579 . . . 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 2148   class class class wbr 4000  (class class class)co 5869   CCcc 7800   RRcr 7801   0cc0 7802   x. cmul 7807   < clt 7982   <_ cle 7983   # cap 8528
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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-id 4290  df-po 4293  df-iso 4294  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-iota 5174  df-fun 5214  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529
This theorem is referenced by:  le2msq  8847  lt2msqi  8860  lt2sq  10579
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