Theorem msqge0 8907

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 8271	. . . . 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 8290	. . . 4 |- 0 e. RR
4		ltnsym2 8380	. . . 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 8879	. . . . . . 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 8276	. . . . . . . . 9 |- ( A e. RR -> A e. CC )
11		mulap0r 8906	. . . . . . . . . 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 8892	. . . . . 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 8365	. . 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 2205   class class class wbr 4114  (class class class)co 6058   CCcc 8141   RRcr 8142   0cc0 8143   x. cmul 8148   < clt 8324   <_ cle 8325   # cap 8872

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 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261

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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-id 4419  df-po 4422  df-iso 4423  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-iota 5317  df-fun 5359  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873

This theorem is referenced by:  msqge0i  8908  msqge0d  8909  recexaplem2  8943  sqge0  11002  bernneq  11047