Theorem msqge0 8889

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

Step	Hyp	Ref	Expression
1		remulcl 8254	. . . . 5 |- ( ( A e. RR /\ A e. RR ) -> ( A x. A ) e. RR )
2	1	anidms 397	. . . 4 |- ( A e. RR -> ( A x. A ) e. RR )
3		0re 8273	. . . 4 |- 0 e. RR
4		ltnsym2 8363	. . . 4 |- ( ( ( A x. A ) e. RR /\ 0 e. RR ) -> -. ( ( A x. A ) < 0 /\ 0 < ( A x. A ) ) )
5	2, 3, 4	sylancl 413	. . 3 |- ( A e. RR -> -. ( ( A x. A ) < 0 /\ 0 < ( A x. A ) ) )
6		orc 720	. . . . . 6 |- ( ( A x. A ) < 0 -> ( ( A x. A ) < 0 \/ 0 < ( A x. A ) ) )
7		reaplt 8861	. . . . . . 7 |- ( ( ( A x. A ) e. RR /\ 0 e. RR ) -> ( ( A x. A ) # 0 <-> ( ( A x. A ) < 0 \/ 0 < ( A x. A ) ) ) )
8	2, 3, 7	sylancl 413	. . . . . 6 |- ( A e. RR -> ( ( A x. A ) # 0 <-> ( ( A x. A ) < 0 \/ 0 < ( A x. A ) ) ) )
9	6, 8	imbitrrid 156	. . . . 5 |- ( A e. RR -> ( ( A x. A ) < 0 -> ( A x. A ) # 0 ) )
10		recn 8259	. . . . . . . . 9 |- ( A e. RR -> A e. CC )
11		mulap0r 8888	. . . . . . . . . 10 |- ( ( A e. CC /\ A e. CC /\ ( A x. A ) # 0 ) -> ( A # 0 /\ A # 0 ) )
12	10, 11	syl3an1 1307	. . . . . . . . 9 |- ( ( A e. RR /\ A e. CC /\ ( A x. A ) # 0 ) -> ( A # 0 /\ A # 0 ) )
13	10, 12	syl3an2 1308	. . . . . . . 8 |- ( ( A e. RR /\ A e. RR /\ ( A x. A ) # 0 ) -> ( A # 0 /\ A # 0 ) )
14	13	simpld 112	. . . . . . 7 |- ( ( A e. RR /\ A e. RR /\ ( A x. A ) # 0 ) -> A # 0 )
15	14	3expia 1232	. . . . . 6 |- ( ( A e. RR /\ A e. RR ) -> ( ( A x. A ) # 0 -> A # 0 ) )
16	15	anidms 397	. . . . 5 |- ( A e. RR -> ( ( A x. A ) # 0 -> A # 0 ) )
17		apsqgt0 8874	. . . . . 6 |- ( ( A e. RR /\ A # 0 ) -> 0 < ( A x. A ) )
18	17	ex 115	. . . . 5 |- ( A e. RR -> ( A # 0 -> 0 < ( A x. A ) ) )
19	9, 16, 18	3syld 57	. . . 4 |- ( A e. RR -> ( ( A x. A ) < 0 -> 0 < ( A x. A ) ) )
20	19	ancld 325	. . 3 |- ( A e. RR -> ( ( A x. A ) < 0 -> ( ( A x. A ) < 0 /\ 0 < ( A x. A ) ) ) )
21	5, 20	mtod 669	. 2 |- ( A e. RR -> -. ( A x. A ) < 0 )
22		lenlt 8348	. . 3 |- ( ( 0 e. RR /\ ( A x. A ) e. RR ) -> ( 0 <_ ( A x. A ) <-> -. ( A x. A ) < 0 ) )
23	3, 2, 22	sylancr 414	. 2 |- ( A e. RR -> ( 0 <_ ( A x. A ) <-> -. ( A x. A ) < 0 ) )
24	21, 23	mpbird 167	1 |- ( A e. RR -> 0 <_ ( A x. A ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 716   /\ w3a 1005   e. wcel 2203   class class class wbr 4108  (class class class)co 6049   CCcc 8124   RRcr 8125   0cc0 8126   x. cmul 8131   < clt 8307   <_ cle 8308   # cap 8854

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-cnex 8217  ax-resscn 8218  ax-1cn 8219  ax-1re 8220  ax-icn 8221  ax-addcl 8222  ax-addrcl 8223  ax-mulcl 8224  ax-mulrcl 8225  ax-addcom 8226  ax-mulcom 8227  ax-addass 8228  ax-mulass 8229  ax-distr 8230  ax-i2m1 8231  ax-0lt1 8232  ax-1rid 8233  ax-0id 8234  ax-rnegex 8235  ax-precex 8236  ax-cnre 8237  ax-pre-ltirr 8238  ax-pre-ltwlin 8239  ax-pre-lttrn 8240  ax-pre-apti 8241  ax-pre-ltadd 8242  ax-pre-mulgt0 8243  ax-pre-mulext 8244

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-br 4109  df-opab 4171  df-id 4413  df-po 4416  df-iso 4417  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-iota 5311  df-fun 5353  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-pnf 8309  df-mnf 8310  df-xr 8311  df-ltxr 8312  df-le 8313  df-sub 8445  df-neg 8446  df-reap 8848  df-ap 8855

This theorem is referenced by:  msqge0i  8890  msqge0d  8891  recexaplem2  8925  sqge0  10977  bernneq  11021