Theorem eigposi 11399

Description: A sufficient condition (first conjunct pair, that holds when T is a positive operator) for an eigenvalue B (second conjunct pair) to be nonnegative. Remark (ii) in [Hughes] p. 137.

Hypotheses

Ref	Expression
eigpos.1	|- A e. ~H
eigpos.2	|- B e. CC

Assertion

Ref	Expression
eigposi	|- ((((A .ih (T` A)) e. RR /\ 0 <_ (A .ih (T` A))) /\ ((T` A) = (B .h A) /\ A =/= 0h)) -> (B e. RR /\ 0 <_ B))

Proof of Theorem eigposi

Step	Hyp	Ref	Expression
1		eigpos.1	. . . . . . . . 9 |- A e. ~H
2		eigpos.2	. . . . . . . . . 10 |- B e. CC
3	2, 1	hvmulcli 10516	. . . . . . . . 9 |- (B .h A) e. ~H
4		hire 10593	. . . . . . . . 9 |- ((A e. ~H /\ (B .h A) e. ~H) -> ((A .ih (B .h A)) e. RR <-> (A .ih (B .h A)) = ((B .h A) .ih A)))
5	1, 3, 4	mp2an 761	. . . . . . . 8 |- ((A .ih (B .h A)) e. RR <-> (A .ih (B .h A)) = ((B .h A) .ih A))
6	5	a1i 8	. . . . . . 7 |- ((T` A) = (B .h A) -> ((A .ih (B .h A)) e. RR <-> (A .ih (B .h A)) = ((B .h A) .ih A)))
7		opreq2 4890	. . . . . . . 8 |- ((T` A) = (B .h A) -> (A .ih (T` A)) = (A .ih (B .h A)))
8	7	eleq1d 1963	. . . . . . 7 |- ((T` A) = (B .h A) -> ((A .ih (T` A)) e. RR <-> (A .ih (B .h A)) e. RR))
9		opreq1 4889	. . . . . . . 8 |- ((T` A) = (B .h A) -> ((T` A) .ih A) = ((B .h A) .ih A))
10	7, 9	eqeq12d 1899	. . . . . . 7 |- ((T` A) = (B .h A) -> ((A .ih (T` A)) = ((T` A) .ih A) <-> (A .ih (B .h A)) = ((B .h A) .ih A)))
11	6, 8, 10	3bitr4d 609	. . . . . 6 |- ((T` A) = (B .h A) -> ((A .ih (T` A)) e. RR <-> (A .ih (T` A)) = ((T` A) .ih A)))
12	11	adantr 425	. . . . 5 |- (((T` A) = (B .h A) /\ A =/= 0h) -> ((A .ih (T` A)) e. RR <-> (A .ih (T` A)) = ((T` A) .ih A)))
13	1, 2	eigrei 11397	. . . . 5 |- (((T` A) = (B .h A) /\ A =/= 0h) -> ((A .ih (T` A)) = ((T` A) .ih A) <-> B e. RR))
14	12, 13	bitrd 587	. . . 4 |- (((T` A) = (B .h A) /\ A =/= 0h) -> ((A .ih (T` A)) e. RR <-> B e. RR))
15	14	biimpac 462	. . 3 |- (((A .ih (T` A)) e. RR /\ ((T` A) = (B .h A) /\ A =/= 0h)) -> B e. RR)
16	15	adantlr 429	. 2 |- ((((A .ih (T` A)) e. RR /\ 0 <_ (A .ih (T` A))) /\ ((T` A) = (B .h A) /\ A =/= 0h)) -> B e. RR)
17		ax-his4 10585	. . . . 5 |- ((A e. ~H /\ A =/= 0h) -> 0 < (A .ih A))
18	1, 17	mpan 759	. . . 4 |- (A =/= 0h -> 0 < (A .ih A))
19	18	ad2antll 443	. . 3 |- ((((A .ih (T` A)) e. RR /\ 0 <_ (A .ih (T` A))) /\ ((T` A) = (B .h A) /\ A =/= 0h)) -> 0 < (A .ih A))
20		simplr 449	. . . 4 |- ((((A .ih (T` A)) e. RR /\ 0 <_ (A .ih (T` A))) /\ ((T` A) = (B .h A) /\ A =/= 0h)) -> 0 <_ (A .ih (T` A)))
21	7	ad2antrl 442	. . . . 5 |- ((((A .ih (T` A)) e. RR /\ 0 <_ (A .ih (T` A))) /\ ((T` A) = (B .h A) /\ A =/= 0h)) -> (A .ih (T` A)) = (A .ih (B .h A)))
22	2	cjrebi 8031	. . . . . . . 8 |- (B e. RR <-> (*` B) = B)
23	16, 22	sylib 215	. . . . . . 7 |- ((((A .ih (T` A)) e. RR /\ 0 <_ (A .ih (T` A))) /\ ((T` A) = (B .h A) /\ A =/= 0h)) -> (*` B) = B)
24	23	opreq1d 4897	. . . . . 6 |- ((((A .ih (T` A)) e. RR /\ 0 <_ (A .ih (T` A))) /\ ((T` A) = (B .h A) /\ A =/= 0h)) -> ((*` B) x. (A .ih A)) = (B x. (A .ih A)))
25		his5 10586	. . . . . . 7 |- ((B e. CC /\ A e. ~H /\ A e. ~H) -> (A .ih (B .h A)) = ((*` B) x. (A .ih A)))
26	2, 1, 1, 25	mp3an 1191	. . . . . 6 |- (A .ih (B .h A)) = ((*` B) x. (A .ih A))
27	24, 26	syl5eq 1940	. . . . 5 |- ((((A .ih (T` A)) e. RR /\ 0 <_ (A .ih (T` A))) /\ ((T` A) = (B .h A) /\ A =/= 0h)) -> (A .ih (B .h A)) = (B x. (A .ih A)))
28	21, 27	eqtrd 1925	. . . 4 |- ((((A .ih (T` A)) e. RR /\ 0 <_ (A .ih (T` A))) /\ ((T` A) = (B .h A) /\ A =/= 0h)) -> (A .ih (T` A)) = (B x. (A .ih A)))
29	20, 28	breqtrd 3361	. . 3 |- ((((A .ih (T` A)) e. RR /\ 0 <_ (A .ih (T` A))) /\ ((T` A) = (B .h A) /\ A =/= 0h)) -> 0 <_ (B x. (A .ih A)))
30		hiidrcl 10594	. . . . 5 |- (A e. ~H -> (A .ih A) e. RR)
31	1, 30	ax-mp 7	. . . 4 |- (A .ih A) e. RR
32		prodge02 7007	. . . 4 |- (((B e. RR /\ (A .ih A) e. RR) /\ (0 < (A .ih A) /\ 0 <_ (B x. (A .ih A)))) -> 0 <_ B)
33	31, 32	mpanl2 771	. . 3 |- ((B e. RR /\ (0 < (A .ih A) /\ 0 <_ (B x. (A .ih A)))) -> 0 <_ B)
34	16, 19, 29, 33	syl12anc 1098	. 2 |- ((((A .ih (T` A)) e. RR /\ 0 <_ (A .ih (T` A))) /\ ((T` A) = (B .h A) /\ A =/= 0h)) -> 0 <_ B)
35	16, 34	jca 310	1 |- ((((A .ih (T` A)) e. RR /\ 0 <_ (A .ih (T` A))) /\ ((T` A) = (B .h A) /\ A =/= 0h)) -> (B e. RR /\ 0 <_ B))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300   =/= wne 2017   class class class wbr 3338  ` cfv 3998  (class class class)co 4884  CCcc 6384  RRcr 6385  0cc0 6386   x. cmul 6391   <_ cle 6448   < clt 6653  *ccj 7999  ~Hchil 10420   .h csm 10422  0hc0v 10423   .ih csp 10425

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731  ax-hfvmul 10507  ax-hfi 10579  ax-his1 10582  ax-his3 10584  ax-his4 10585

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-div 6892  df-re 8001  df-im 8002  df-cj 8003

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