Theorem msqge0 8603

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 7969	. . . . 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 7987	. . . 4 |- 0 e. RR
4		ltnsym2 8078	. . . 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 713	. . . . . 6 |- ( ( A x. A ) < 0 -> ( ( A x. A ) < 0 \/ 0 < ( A x. A ) ) )
7		reaplt 8575	. . . . . . 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 7974	. . . . . . . . 9 |- ( A e. RR -> A e. CC )
11		mulap0r 8602	. . . . . . . . . 10 |- ( ( A e. CC /\ A e. CC /\ ( A x. A ) # 0 ) -> ( A # 0 /\ A # 0 ) )
12	10, 11	syl3an1 1282	. . . . . . . . 9 |- ( ( A e. RR /\ A e. CC /\ ( A x. A ) # 0 ) -> ( A # 0 /\ A # 0 ) )
13	10, 12	syl3an2 1283	. . . . . . . 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 1207	. . . . . 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 8588	. . . . . 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 664	. 2 |- ( A e. RR -> -. ( A x. A ) < 0 )
22		lenlt 8063	. . 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 709   /\ w3a 980   e. wcel 2160   class class class wbr 4018  (class class class)co 5896   CCcc 7839   RRcr 7840   0cc0 7841   x. cmul 7846   < clt 8022   <_ cle 8023   # cap 8568

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7932  ax-resscn 7933  ax-1cn 7934  ax-1re 7935  ax-icn 7936  ax-addcl 7937  ax-addrcl 7938  ax-mulcl 7939  ax-mulrcl 7940  ax-addcom 7941  ax-mulcom 7942  ax-addass 7943  ax-mulass 7944  ax-distr 7945  ax-i2m1 7946  ax-0lt1 7947  ax-1rid 7948  ax-0id 7949  ax-rnegex 7950  ax-precex 7951  ax-cnre 7952  ax-pre-ltirr 7953  ax-pre-ltwlin 7954  ax-pre-lttrn 7955  ax-pre-apti 7956  ax-pre-ltadd 7957  ax-pre-mulgt0 7958  ax-pre-mulext 7959

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-id 4311  df-po 4314  df-iso 4315  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-iota 5196  df-fun 5237  df-fv 5243  df-riota 5852  df-ov 5899  df-oprab 5900  df-mpo 5901  df-pnf 8024  df-mnf 8025  df-xr 8026  df-ltxr 8027  df-le 8028  df-sub 8160  df-neg 8161  df-reap 8562  df-ap 8569

This theorem is referenced by:  msqge0i  8604  msqge0d  8605  recexaplem2  8639  sqge0  10628  bernneq  10672