Theorem normlem6 27972

Description: Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 2-Aug-1999.) (Revised by Mario Carneiro, 4-Jun-2014.) (New usage is discouraged.)

Hypotheses

Ref	Expression
normlem1.1	|- S e. CC
normlem1.2	|- F e. ~H
normlem1.3	|- G e. ~H
normlem2.4	|- B = -u ( ( ( * ` S ) x. ( F .ih G ) ) + ( S x. ( G .ih F ) ) )
normlem3.5	|- A = ( G .ih G )
normlem3.6	|- C = ( F .ih F )
normlem6.7	|- ( abs ` S ) = 1

Assertion

Ref	Expression
normlem6	|- ( abs ` B ) <_ ( 2 x. ( ( sqr ` A ) x. ( sqr ` C ) ) )

Proof of Theorem normlem6

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		normlem3.5	. . . . . . . . 9 |- A = ( G .ih G )
2		normlem1.3	. . . . . . . . . 10 |- G e. ~H
3		hiidrcl 27952	. . . . . . . . . 10 |- ( G e. ~H -> ( G .ih G ) e. RR )
4	2, 3	ax-mp 5	. . . . . . . . 9 |- ( G .ih G ) e. RR
5	1, 4	eqeltri 2697	. . . . . . . 8 |- A e. RR
6	5	a1i 11	. . . . . . 7 |- ( T. -> A e. RR )
7		normlem1.1	. . . . . . . . 9 |- S e. CC
8		normlem1.2	. . . . . . . . 9 |- F e. ~H
9		normlem2.4	. . . . . . . . 9 |- B = -u ( ( ( * ` S ) x. ( F .ih G ) ) + ( S x. ( G .ih F ) ) )
10	7, 8, 2, 9	normlem2 27968	. . . . . . . 8 |- B e. RR
11	10	a1i 11	. . . . . . 7 |- ( T. -> B e. RR )
12		normlem3.6	. . . . . . . . 9 |- C = ( F .ih F )
13		hiidrcl 27952	. . . . . . . . . 10 |- ( F e. ~H -> ( F .ih F ) e. RR )
14	8, 13	ax-mp 5	. . . . . . . . 9 |- ( F .ih F ) e. RR
15	12, 14	eqeltri 2697	. . . . . . . 8 |- C e. RR
16	15	a1i 11	. . . . . . 7 |- ( T. -> C e. RR )
17		oveq1 6657	. . . . . . . . . . . . 13 |- ( x = if ( x e. RR , x , 0 ) -> ( x ^ 2 ) = ( if ( x e. RR , x , 0 ) ^ 2 ) )
18	17	oveq2d 6666	. . . . . . . . . . . 12 |- ( x = if ( x e. RR , x , 0 ) -> ( A x. ( x ^ 2 ) ) = ( A x. ( if ( x e. RR , x , 0 ) ^ 2 ) ) )
19		oveq2 6658	. . . . . . . . . . . 12 |- ( x = if ( x e. RR , x , 0 ) -> ( B x. x ) = ( B x. if ( x e. RR , x , 0 ) ) )
20	18, 19	oveq12d 6668	. . . . . . . . . . 11 |- ( x = if ( x e. RR , x , 0 ) -> ( ( A x. ( x ^ 2 ) ) + ( B x. x ) ) = ( ( A x. ( if ( x e. RR , x , 0 ) ^ 2 ) ) + ( B x. if ( x e. RR , x , 0 ) ) ) )
21	20	oveq1d 6665	. . . . . . . . . 10 |- ( x = if ( x e. RR , x , 0 ) -> ( ( ( A x. ( x ^ 2 ) ) + ( B x. x ) ) + C ) = ( ( ( A x. ( if ( x e. RR , x , 0 ) ^ 2 ) ) + ( B x. if ( x e. RR , x , 0 ) ) ) + C ) )
22	21	breq2d 4665	. . . . . . . . 9 |- ( x = if ( x e. RR , x , 0 ) -> ( 0 <_ ( ( ( A x. ( x ^ 2 ) ) + ( B x. x ) ) + C ) <-> 0 <_ ( ( ( A x. ( if ( x e. RR , x , 0 ) ^ 2 ) ) + ( B x. if ( x e. RR , x , 0 ) ) ) + C ) ) )
23		0re 10040	. . . . . . . . . . 11 |- 0 e. RR
24	23	elimel 4150	. . . . . . . . . 10 |- if ( x e. RR , x , 0 ) e. RR
25		normlem6.7	. . . . . . . . . 10 |- ( abs ` S ) = 1
26	7, 8, 2, 9, 1, 12, 24, 25	normlem5 27971	. . . . . . . . 9 |- 0 <_ ( ( ( A x. ( if ( x e. RR , x , 0 ) ^ 2 ) ) + ( B x. if ( x e. RR , x , 0 ) ) ) + C )
27	22, 26	dedth 4139	. . . . . . . 8 |- ( x e. RR -> 0 <_ ( ( ( A x. ( x ^ 2 ) ) + ( B x. x ) ) + C ) )
28	27	adantl 482	. . . . . . 7 |- ( ( T. /\ x e. RR ) -> 0 <_ ( ( ( A x. ( x ^ 2 ) ) + ( B x. x ) ) + C ) )
29	6, 11, 16, 28	discr 13001	. . . . . 6 |- ( T. -> ( ( B ^ 2 ) - ( 4 x. ( A x. C ) ) ) <_ 0 )
30	29	trud 1493	. . . . 5 |- ( ( B ^ 2 ) - ( 4 x. ( A x. C ) ) ) <_ 0
31	10	resqcli 12949	. . . . . 6 |- ( B ^ 2 ) e. RR
32		4re 11097	. . . . . . 7 |- 4 e. RR
33	5, 15	remulcli 10054	. . . . . . 7 |- ( A x. C ) e. RR
34	32, 33	remulcli 10054	. . . . . 6 |- ( 4 x. ( A x. C ) ) e. RR
35	31, 34, 23	lesubadd2i 10588	. . . . 5 |- ( ( ( B ^ 2 ) - ( 4 x. ( A x. C ) ) ) <_ 0 <-> ( B ^ 2 ) <_ ( ( 4 x. ( A x. C ) ) + 0 ) )
36	30, 35	mpbi 220	. . . 4 |- ( B ^ 2 ) <_ ( ( 4 x. ( A x. C ) ) + 0 )
37	34	recni 10052	. . . . 5 |- ( 4 x. ( A x. C ) ) e. CC
38	37	addid1i 10223	. . . 4 |- ( ( 4 x. ( A x. C ) ) + 0 ) = ( 4 x. ( A x. C ) )
39	36, 38	breqtri 4678	. . 3 |- ( B ^ 2 ) <_ ( 4 x. ( A x. C ) )
40	10	sqge0i 12951	. . . 4 |- 0 <_ ( B ^ 2 )
41		4pos 11116	. . . . . 6 |- 0 < 4
42	23, 32, 41	ltleii 10160	. . . . 5 |- 0 <_ 4
43		hiidge0 27955	. . . . . . . 8 |- ( G e. ~H -> 0 <_ ( G .ih G ) )
44	2, 43	ax-mp 5	. . . . . . 7 |- 0 <_ ( G .ih G )
45	44, 1	breqtrri 4680	. . . . . 6 |- 0 <_ A
46		hiidge0 27955	. . . . . . . 8 |- ( F e. ~H -> 0 <_ ( F .ih F ) )
47	8, 46	ax-mp 5	. . . . . . 7 |- 0 <_ ( F .ih F )
48	47, 12	breqtrri 4680	. . . . . 6 |- 0 <_ C
49	5, 15	mulge0i 10575	. . . . . 6 |- ( ( 0 <_ A /\ 0 <_ C ) -> 0 <_ ( A x. C ) )
50	45, 48, 49	mp2an 708	. . . . 5 |- 0 <_ ( A x. C )
51	32, 33	mulge0i 10575	. . . . 5 |- ( ( 0 <_ 4 /\ 0 <_ ( A x. C ) ) -> 0 <_ ( 4 x. ( A x. C ) ) )
52	42, 50, 51	mp2an 708	. . . 4 |- 0 <_ ( 4 x. ( A x. C ) )
53	31, 34	sqrtlei 14128	. . . 4 |- ( ( 0 <_ ( B ^ 2 ) /\ 0 <_ ( 4 x. ( A x. C ) ) ) -> ( ( B ^ 2 ) <_ ( 4 x. ( A x. C ) ) <-> ( sqr ` ( B ^ 2 ) ) <_ ( sqr ` ( 4 x. ( A x. C ) ) ) ) )
54	40, 52, 53	mp2an 708	. . 3 |- ( ( B ^ 2 ) <_ ( 4 x. ( A x. C ) ) <-> ( sqr ` ( B ^ 2 ) ) <_ ( sqr ` ( 4 x. ( A x. C ) ) ) )
55	39, 54	mpbi 220	. 2 |- ( sqr ` ( B ^ 2 ) ) <_ ( sqr ` ( 4 x. ( A x. C ) ) )
56	10	absrei 14121	. 2 |- ( abs ` B ) = ( sqr ` ( B ^ 2 ) )
57	32, 33, 42, 50	sqrtmulii 14126	. . 3 |- ( sqr ` ( 4 x. ( A x. C ) ) ) = ( ( sqr ` 4 ) x. ( sqr ` ( A x. C ) ) )
58		sqrt4 14013	. . . 4 |- ( sqr ` 4 ) = 2
59	5, 15, 45, 48	sqrtmulii 14126	. . . 4 |- ( sqr ` ( A x. C ) ) = ( ( sqr ` A ) x. ( sqr ` C ) )
60	58, 59	oveq12i 6662	. . 3 |- ( ( sqr ` 4 ) x. ( sqr ` ( A x. C ) ) ) = ( 2 x. ( ( sqr ` A ) x. ( sqr ` C ) ) )
61	57, 60	eqtr2i 2645	. 2 |- ( 2 x. ( ( sqr ` A ) x. ( sqr ` C ) ) ) = ( sqr ` ( 4 x. ( A x. C ) ) )
62	55, 56, 61	3brtr4i 4683	1 |- ( abs ` B ) <_ ( 2 x. ( ( sqr ` A ) x. ( sqr ` C ) ) )

Syntax hints:   <-> wb 196   = wceq 1483   T. wtru 1484   e. wcel 1990   ifcif 4086   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   <_ cle 10075   - cmin 10266   -ucneg 10267   2c2 11070   4c4 11072   ^cexp 12860   *ccj 13836   sqrcsqrt 13973   abscabs 13974   ~Hchil 27776   .ih csp 27779

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-hfvadd 27857  ax-hv0cl 27860  ax-hfvmul 27862  ax-hvmulass 27864  ax-hvmul0 27867  ax-hfi 27936  ax-his1 27939  ax-his2 27940  ax-his3 27941  ax-his4 27942

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-hvsub 27828

This theorem is referenced by:  normlem7  27973