Theorem msqge0 8648

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 8012	. . . . 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 8031	. . . 4 |- 0 e. RR
4		ltnsym2 8122	. . . 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 8620	. . . . . . 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 8017	. . . . . . . . 9 |- ( A e. RR -> A e. CC )
11		mulap0r 8647	. . . . . . . . . 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 8633	. . . . . 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 8107	. . 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 2167   class class class wbr 4034  (class class class)co 5925   CCcc 7882   RRcr 7883   0cc0 7884   x. cmul 7889   < clt 8066   <_ cle 8067   # cap 8613

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7975  ax-resscn 7976  ax-1cn 7977  ax-1re 7978  ax-icn 7979  ax-addcl 7980  ax-addrcl 7981  ax-mulcl 7982  ax-mulrcl 7983  ax-addcom 7984  ax-mulcom 7985  ax-addass 7986  ax-mulass 7987  ax-distr 7988  ax-i2m1 7989  ax-0lt1 7990  ax-1rid 7991  ax-0id 7992  ax-rnegex 7993  ax-precex 7994  ax-cnre 7995  ax-pre-ltirr 7996  ax-pre-ltwlin 7997  ax-pre-lttrn 7998  ax-pre-apti 7999  ax-pre-ltadd 8000  ax-pre-mulgt0 8001  ax-pre-mulext 8002

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-opab 4096  df-id 4329  df-po 4332  df-iso 4333  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-iota 5220  df-fun 5261  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-pnf 8068  df-mnf 8069  df-xr 8070  df-ltxr 8071  df-le 8072  df-sub 8204  df-neg 8205  df-reap 8607  df-ap 8614

This theorem is referenced by:  msqge0i  8649  msqge0d  8650  recexaplem2  8684  sqge0  10713  bernneq  10757