Theorem cphsqrtcl2 22986

Description: The scalar field of a subcomplex pre-Hilbert space is closed under square roots of all numbers except possibly the negative reals. (Contributed by Mario Carneiro, 8-Oct-2015.)

Hypotheses

Ref	Expression
cphsca.f	|- F = (Scalar ` W )
cphsca.k	|- K = ( Base ` F )

Assertion

Ref	Expression
cphsqrtcl2	|- ( ( W e. CPreHil /\ A e. K /\ -. -u A e. RR+ ) -> ( sqr ` A ) e. K )

Proof of Theorem cphsqrtcl2

Step	Hyp	Ref	Expression
1		sqrt0 13982	. . . . 5 |- ( sqr ` 0 ) = 0
2		fveq2 6191	. . . . 5 |- ( A = 0 -> ( sqr ` A ) = ( sqr ` 0 ) )
3		id 22	. . . . 5 |- ( A = 0 -> A = 0 )
4	1, 2, 3	3eqtr4a 2682	. . . 4 |- ( A = 0 -> ( sqr ` A ) = A )
5	4	adantl 482	. . 3 |- ( ( ( W e. CPreHil /\ A e. K /\ -. -u A e. RR+ ) /\ A = 0 ) -> ( sqr ` A ) = A )
6		simpl2 1065	. . 3 |- ( ( ( W e. CPreHil /\ A e. K /\ -. -u A e. RR+ ) /\ A = 0 ) -> A e. K )
7	5, 6	eqeltrd 2701	. 2 |- ( ( ( W e. CPreHil /\ A e. K /\ -. -u A e. RR+ ) /\ A = 0 ) -> ( sqr ` A ) e. K )
8		simpl1 1064	. . . . . . 7 |- ( ( ( W e. CPreHil /\ A e. K /\ -. -u A e. RR+ ) /\ A =/= 0 ) -> W e. CPreHil )
9		cphsca.f	. . . . . . . 8 |- F = (Scalar ` W )
10		cphsca.k	. . . . . . . 8 |- K = ( Base ` F )
11	9, 10	cphsubrg 22980	. . . . . . 7 |- ( W e. CPreHil -> K e. (SubRing ` ℂfld ) )
12	8, 11	syl 17	. . . . . 6 |- ( ( ( W e. CPreHil /\ A e. K /\ -. -u A e. RR+ ) /\ A =/= 0 ) -> K e. (SubRing ` ℂfld ) )
13		cnfldbas 19750	. . . . . . 7 |- CC = ( Base ` ℂfld )
14	13	subrgss 18781	. . . . . 6 |- ( K e. (SubRing ` ℂfld ) -> K C_ CC )
15	12, 14	syl 17	. . . . 5 |- ( ( ( W e. CPreHil /\ A e. K /\ -. -u A e. RR+ ) /\ A =/= 0 ) -> K C_ CC )
16		simpl2 1065	. . . . . . . 8 |- ( ( ( W e. CPreHil /\ A e. K /\ -. -u A e. RR+ ) /\ A =/= 0 ) -> A e. K )
17	9, 10	cphabscl 22985	. . . . . . . 8 |- ( ( W e. CPreHil /\ A e. K ) -> ( abs ` A ) e. K )
18	8, 16, 17	syl2anc 693	. . . . . . 7 |- ( ( ( W e. CPreHil /\ A e. K /\ -. -u A e. RR+ ) /\ A =/= 0 ) -> ( abs ` A ) e. K )
19	15, 16	sseldd 3604	. . . . . . . 8 |- ( ( ( W e. CPreHil /\ A e. K /\ -. -u A e. RR+ ) /\ A =/= 0 ) -> A e. CC )
20	19	abscld 14175	. . . . . . 7 |- ( ( ( W e. CPreHil /\ A e. K /\ -. -u A e. RR+ ) /\ A =/= 0 ) -> ( abs ` A ) e. RR )
21	19	absge0d 14183	. . . . . . 7 |- ( ( ( W e. CPreHil /\ A e. K /\ -. -u A e. RR+ ) /\ A =/= 0 ) -> 0 <_ ( abs ` A ) )
22	9, 10	cphsqrtcl 22984	. . . . . . 7 |- ( ( W e. CPreHil /\ ( ( abs ` A ) e. K /\ ( abs ` A ) e. RR /\ 0 <_ ( abs ` A ) ) ) -> ( sqr ` ( abs ` A ) ) e. K )
23	8, 18, 20, 21, 22	syl13anc 1328	. . . . . 6 |- ( ( ( W e. CPreHil /\ A e. K /\ -. -u A e. RR+ ) /\ A =/= 0 ) -> ( sqr ` ( abs ` A ) ) e. K )
24		cnfldadd 19751	. . . . . . . . 9 |- + = ( +g ` ℂfld )
25	24	subrgacl 18791	. . . . . . . 8 |- ( ( K e. (SubRing ` ℂfld ) /\ ( abs ` A ) e. K /\ A e. K ) -> ( ( abs ` A ) + A ) e. K )
26	12, 18, 16, 25	syl3anc 1326	. . . . . . 7 |- ( ( ( W e. CPreHil /\ A e. K /\ -. -u A e. RR+ ) /\ A =/= 0 ) -> ( ( abs ` A ) + A ) e. K )
27	9, 10	cphabscl 22985	. . . . . . . 8 |- ( ( W e. CPreHil /\ ( ( abs ` A ) + A ) e. K ) -> ( abs ` ( ( abs ` A ) + A ) ) e. K )
28	8, 26, 27	syl2anc 693	. . . . . . 7 |- ( ( ( W e. CPreHil /\ A e. K /\ -. -u A e. RR+ ) /\ A =/= 0 ) -> ( abs ` ( ( abs ` A ) + A ) ) e. K )
29	15, 26	sseldd 3604	. . . . . . . 8 |- ( ( ( W e. CPreHil /\ A e. K /\ -. -u A e. RR+ ) /\ A =/= 0 ) -> ( ( abs ` A ) + A ) e. CC )
30		simpl3 1066	. . . . . . . . 9 |- ( ( ( W e. CPreHil /\ A e. K /\ -. -u A e. RR+ ) /\ A =/= 0 ) -> -. -u A e. RR+ )
31	20	recnd 10068	. . . . . . . . . . . . . 14 |- ( ( ( W e. CPreHil /\ A e. K /\ -. -u A e. RR+ ) /\ A =/= 0 ) -> ( abs ` A ) e. CC )
32	31, 19	subnegd 10399	. . . . . . . . . . . . 13 |- ( ( ( W e. CPreHil /\ A e. K /\ -. -u A e. RR+ ) /\ A =/= 0 ) -> ( ( abs ` A ) - -u A ) = ( ( abs ` A ) + A ) )
33	32	eqeq1d 2624	. . . . . . . . . . . 12 |- ( ( ( W e. CPreHil /\ A e. K /\ -. -u A e. RR+ ) /\ A =/= 0 ) -> ( ( ( abs ` A ) - -u A ) = 0 <-> ( ( abs ` A ) + A ) = 0 ) )
34	19	negcld 10379	. . . . . . . . . . . . 13 |- ( ( ( W e. CPreHil /\ A e. K /\ -. -u A e. RR+ ) /\ A =/= 0 ) -> -u A e. CC )
35	31, 34	subeq0ad 10402	. . . . . . . . . . . 12 |- ( ( ( W e. CPreHil /\ A e. K /\ -. -u A e. RR+ ) /\ A =/= 0 ) -> ( ( ( abs ` A ) - -u A ) = 0 <-> ( abs ` A ) = -u A ) )
36	33, 35	bitr3d 270	. . . . . . . . . . 11 |- ( ( ( W e. CPreHil /\ A e. K /\ -. -u A e. RR+ ) /\ A =/= 0 ) -> ( ( ( abs ` A ) + A ) = 0 <-> ( abs ` A ) = -u A ) )
37		absrpcl 14028	. . . . . . . . . . . . 13 |- ( ( A e. CC /\ A =/= 0 ) -> ( abs ` A ) e. RR+ )
38	19, 37	sylancom 701	. . . . . . . . . . . 12 |- ( ( ( W e. CPreHil /\ A e. K /\ -. -u A e. RR+ ) /\ A =/= 0 ) -> ( abs ` A ) e. RR+ )
39		eleq1 2689	. . . . . . . . . . . 12 |- ( ( abs ` A ) = -u A -> ( ( abs ` A ) e. RR+ <-> -u A e. RR+ ) )
40	38, 39	syl5ibcom 235	. . . . . . . . . . 11 |- ( ( ( W e. CPreHil /\ A e. K /\ -. -u A e. RR+ ) /\ A =/= 0 ) -> ( ( abs ` A ) = -u A -> -u A e. RR+ ) )
41	36, 40	sylbid 230	. . . . . . . . . 10 |- ( ( ( W e. CPreHil /\ A e. K /\ -. -u A e. RR+ ) /\ A =/= 0 ) -> ( ( ( abs ` A ) + A ) = 0 -> -u A e. RR+ ) )
42	41	necon3bd 2808	. . . . . . . . 9 |- ( ( ( W e. CPreHil /\ A e. K /\ -. -u A e. RR+ ) /\ A =/= 0 ) -> ( -. -u A e. RR+ -> ( ( abs ` A ) + A ) =/= 0 ) )
43	30, 42	mpd 15	. . . . . . . 8 |- ( ( ( W e. CPreHil /\ A e. K /\ -. -u A e. RR+ ) /\ A =/= 0 ) -> ( ( abs ` A ) + A ) =/= 0 )
44	29, 43	absne0d 14186	. . . . . . 7 |- ( ( ( W e. CPreHil /\ A e. K /\ -. -u A e. RR+ ) /\ A =/= 0 ) -> ( abs ` ( ( abs ` A ) + A ) ) =/= 0 )
45	9, 10	cphdivcl 22982	. . . . . . 7 |- ( ( W e. CPreHil /\ ( ( ( abs ` A ) + A ) e. K /\ ( abs ` ( ( abs ` A ) + A ) ) e. K /\ ( abs ` ( ( abs ` A ) + A ) ) =/= 0 ) ) -> ( ( ( abs ` A ) + A ) / ( abs ` ( ( abs ` A ) + A ) ) ) e. K )
46	8, 26, 28, 44, 45	syl13anc 1328	. . . . . 6 |- ( ( ( W e. CPreHil /\ A e. K /\ -. -u A e. RR+ ) /\ A =/= 0 ) -> ( ( ( abs ` A ) + A ) / ( abs ` ( ( abs ` A ) + A ) ) ) e. K )
47		cnfldmul 19752	. . . . . . 7 |- x. = ( .r ` ℂfld )
48	47	subrgmcl 18792	. . . . . 6 |- ( ( K e. (SubRing ` ℂfld ) /\ ( sqr ` ( abs ` A ) ) e. K /\ ( ( ( abs ` A ) + A ) / ( abs ` ( ( abs ` A ) + A ) ) ) e. K ) -> ( ( sqr ` ( abs ` A ) ) x. ( ( ( abs ` A ) + A ) / ( abs ` ( ( abs ` A ) + A ) ) ) ) e. K )
49	12, 23, 46, 48	syl3anc 1326	. . . . 5 |- ( ( ( W e. CPreHil /\ A e. K /\ -. -u A e. RR+ ) /\ A =/= 0 ) -> ( ( sqr ` ( abs ` A ) ) x. ( ( ( abs ` A ) + A ) / ( abs ` ( ( abs ` A ) + A ) ) ) ) e. K )
50	15, 49	sseldd 3604	. . . 4 |- ( ( ( W e. CPreHil /\ A e. K /\ -. -u A e. RR+ ) /\ A =/= 0 ) -> ( ( sqr ` ( abs ` A ) ) x. ( ( ( abs ` A ) + A ) / ( abs ` ( ( abs ` A ) + A ) ) ) ) e. CC )
51		eqid 2622	. . . . . . 7 |- ( ( sqr ` ( abs ` A ) ) x. ( ( ( abs ` A ) + A ) / ( abs ` ( ( abs ` A ) + A ) ) ) ) = ( ( sqr ` ( abs ` A ) ) x. ( ( ( abs ` A ) + A ) / ( abs ` ( ( abs ` A ) + A ) ) ) )
52	51	sqreulem 14099	. . . . . 6 |- ( ( A e. CC /\ ( ( abs ` A ) + A ) =/= 0 ) -> ( ( ( ( sqr ` ( abs ` A ) ) x. ( ( ( abs ` A ) + A ) / ( abs ` ( ( abs ` A ) + A ) ) ) ) ^ 2 ) = A /\ 0 <_ ( Re ` ( ( sqr ` ( abs ` A ) ) x. ( ( ( abs ` A ) + A ) / ( abs ` ( ( abs ` A ) + A ) ) ) ) ) /\ ( _i x. ( ( sqr ` ( abs ` A ) ) x. ( ( ( abs ` A ) + A ) / ( abs ` ( ( abs ` A ) + A ) ) ) ) ) e/ RR+ ) )
53	19, 43, 52	syl2anc 693	. . . . 5 |- ( ( ( W e. CPreHil /\ A e. K /\ -. -u A e. RR+ ) /\ A =/= 0 ) -> ( ( ( ( sqr ` ( abs ` A ) ) x. ( ( ( abs ` A ) + A ) / ( abs ` ( ( abs ` A ) + A ) ) ) ) ^ 2 ) = A /\ 0 <_ ( Re ` ( ( sqr ` ( abs ` A ) ) x. ( ( ( abs ` A ) + A ) / ( abs ` ( ( abs ` A ) + A ) ) ) ) ) /\ ( _i x. ( ( sqr ` ( abs ` A ) ) x. ( ( ( abs ` A ) + A ) / ( abs ` ( ( abs ` A ) + A ) ) ) ) ) e/ RR+ ) )
54	53	simp1d 1073	. . . 4 |- ( ( ( W e. CPreHil /\ A e. K /\ -. -u A e. RR+ ) /\ A =/= 0 ) -> ( ( ( sqr ` ( abs ` A ) ) x. ( ( ( abs ` A ) + A ) / ( abs ` ( ( abs ` A ) + A ) ) ) ) ^ 2 ) = A )
55	53	simp2d 1074	. . . 4 |- ( ( ( W e. CPreHil /\ A e. K /\ -. -u A e. RR+ ) /\ A =/= 0 ) -> 0 <_ ( Re ` ( ( sqr ` ( abs ` A ) ) x. ( ( ( abs ` A ) + A ) / ( abs ` ( ( abs ` A ) + A ) ) ) ) ) )
56	53	simp3d 1075	. . . . 5 |- ( ( ( W e. CPreHil /\ A e. K /\ -. -u A e. RR+ ) /\ A =/= 0 ) -> ( _i x. ( ( sqr ` ( abs ` A ) ) x. ( ( ( abs ` A ) + A ) / ( abs ` ( ( abs ` A ) + A ) ) ) ) ) e/ RR+ )
57		df-nel 2898	. . . . 5 |- ( ( _i x. ( ( sqr ` ( abs ` A ) ) x. ( ( ( abs ` A ) + A ) / ( abs ` ( ( abs ` A ) + A ) ) ) ) ) e/ RR+ <-> -. ( _i x. ( ( sqr ` ( abs ` A ) ) x. ( ( ( abs ` A ) + A ) / ( abs ` ( ( abs ` A ) + A ) ) ) ) ) e. RR+ )
58	56, 57	sylib 208	. . . 4 |- ( ( ( W e. CPreHil /\ A e. K /\ -. -u A e. RR+ ) /\ A =/= 0 ) -> -. ( _i x. ( ( sqr ` ( abs ` A ) ) x. ( ( ( abs ` A ) + A ) / ( abs ` ( ( abs ` A ) + A ) ) ) ) ) e. RR+ )
59	50, 19, 54, 55, 58	eqsqrtd 14107	. . 3 |- ( ( ( W e. CPreHil /\ A e. K /\ -. -u A e. RR+ ) /\ A =/= 0 ) -> ( ( sqr ` ( abs ` A ) ) x. ( ( ( abs ` A ) + A ) / ( abs ` ( ( abs ` A ) + A ) ) ) ) = ( sqr ` A ) )
60	59, 49	eqeltrrd 2702	. 2 |- ( ( ( W e. CPreHil /\ A e. K /\ -. -u A e. RR+ ) /\ A =/= 0 ) -> ( sqr ` A ) e. K )
61	7, 60	pm2.61dane 2881	1 |- ( ( W e. CPreHil /\ A e. K /\ -. -u A e. RR+ ) -> ( sqr ` A ) e. K )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   e/ wnel 2897   C_ wss 3574   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   _ici 9938   + caddc 9939   x. cmul 9941   <_ cle 10075   - cmin 10266   -ucneg 10267   / cdiv 10684   2c2 11070   RR+crp 11832   ^cexp 12860   Recre 13837   sqrcsqrt 13973   abscabs 13974   Basecbs 15857  Scalarcsca 15944  SubRingcsubrg 18776  ℂ_fldccnfld 19746   CPreHilccph 22966

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-rep 4771  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-addf 10015  ax-mulf 10016

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-int 4476  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-1st 7168  df-2nd 7169  df-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  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-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-rp 11833  df-ico 12181  df-fz 12327  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-grp 17425  df-minusg 17426  df-subg 17591  df-ghm 17658  df-cmn 18195  df-mgp 18490  df-ur 18502  df-ring 18549  df-cring 18550  df-oppr 18623  df-dvdsr 18641  df-unit 18642  df-invr 18672  df-dvr 18683  df-rnghom 18715  df-drng 18749  df-subrg 18778  df-staf 18845  df-srng 18846  df-lvec 19103  df-cnfld 19747  df-phl 19971  df-cph 22968

This theorem is referenced by:  cphsqrtcl3  22987