Theorem msqge0 8643

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 8007	. . . . 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 8026	. . . 4 |- 0 e. RR
4		ltnsym2 8117	. . . 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 8615	. . . . . . 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 8012	. . . . . . . . 9 |- ( A e. RR -> A e. CC )
11		mulap0r 8642	. . . . . . . . . 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 8628	. . . . . 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 8102	. . 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 4033  (class class class)co 5922   CCcc 7877   RRcr 7878   0cc0 7879   x. cmul 7884   < clt 8061   <_ cle 8062   # cap 8608

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 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996  ax-pre-mulext 7997

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 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-id 4328  df-po 4331  df-iso 4332  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-iota 5219  df-fun 5260  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-reap 8602  df-ap 8609

This theorem is referenced by:  msqge0i  8644  msqge0d  8645  recexaplem2  8679  sqge0  10708  bernneq  10752