Theorem eigpos 9702

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
eigpos	|- ((((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 eigpos

Step	Hyp	Ref	Expression
1		eigpos.1	. . . . . . . . 9 |- A e. H~
2		eigpos.2	. . . . . . . . . 10 |- B e. CC
3	2, 1	hvmulcl 8823	. . . . . . . . 9 |- (B .h A) e. H~
4		hiret 8899	. . . . . . . . 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 696	. . . . . . . 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 3960	. . . . . . . 8 |- ((T` A) = (B .h A) -> (A .ih (T` A)) = (A .ih (B .h A)))
8	7	eleq1d 1537	. . . . . . 7 |- ((T` A) = (B .h A) -> ((A .ih (T` A)) e. RR <-> (A .ih (B .h A)) e. RR))
9		opreq1 3959	. . . . . . . 8 |- ((T` A) = (B .h A) -> ((T` A) .ih A) = ((B .h A) .ih A))
10	7, 9	eqeq12d 1486	. . . . . . 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 549	. . . . . 6 |- ((T` A) = (B .h A) -> ((A .ih (T` A)) e. RR <-> (A .ih (T` A)) = ((T` A) .ih A)))
12	11	adantr 389	. . . . 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	eigre 9700	. . . . 5 |- (((T` A) = (B .h A) /\ A =/= 0h) -> ((A .ih (T` A)) = ((T` A) .ih A) <-> B e. RR))
14	12, 13	bitrd 527	. . . 4 |- (((T` A) = (B .h A) /\ A =/= 0h) -> ((A .ih (T` A)) e. RR <-> B e. RR))
15	14	biimpac 418	. . 3 |- (((A .ih (T` A)) e. RR /\ ((T` A) = (B .h A) /\ A =/= 0h)) -> B e. RR)
16	15	adantlr 393	. 2 |- ((((A .ih (T` A)) e. RR /\ 0 <_ (A .ih (T` A))) /\ ((T` A) = (B .h A) /\ A =/= 0h)) -> B e. RR)
17		hiidrclt 8900	. . . . 5 |- (A e. H~ -> (A .ih A) e. RR)
18	1, 17	ax-mp 7	. . . 4 |- (A .ih A) e. RR
19		prodge02t 5793	. . . 4 |- (((B e. RR /\ (A .ih A) e. RR) /\ (0 < (A .ih A) /\ 0 <_ (B x. (A .ih A)))) -> 0 <_ B)
20	18, 19	mpanl2 706	. . 3 |- ((B e. RR /\ (0 < (A .ih A) /\ 0 <_ (B x. (A .ih A)))) -> 0 <_ B)
21		ax-his4 8891	. . . . . 6 |- ((A e. H~ /\ A =/= 0h) -> 0 < (A .ih A))
22	1, 21	mpan 694	. . . . 5 |- (A =/= 0h -> 0 < (A .ih A))
23	22	ad2antll 407	. . . 4 |- ((((A .ih (T` A)) e. RR /\ 0 <_ (A .ih (T` A))) /\ ((T` A) = (B .h A) /\ A =/= 0h)) -> 0 < (A .ih A))
24		simplr 413	. . . . 5 |- ((((A .ih (T` A)) e. RR /\ 0 <_ (A .ih (T` A))) /\ ((T` A) = (B .h A) /\ A =/= 0h)) -> 0 <_ (A .ih (T` A)))
25	7	ad2antrl 406	. . . . . 6 |- ((((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)))
26	2	cjreb 6724	. . . . . . . . 9 |- (B e. RR <-> (*` B) = B)
27	16, 26	sylib 198	. . . . . . . 8 |- ((((A .ih (T` A)) e. RR /\ 0 <_ (A .ih (T` A))) /\ ((T` A) = (B .h A) /\ A =/= 0h)) -> (*` B) = B)
28	27	opreq1d 3966	. . . . . . 7 |- ((((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)))
29		his5t 8892	. . . . . . . 8 |- ((B e. CC /\ A e. H~ /\ A e. H~) -> (A .ih (B .h A)) = ((*` B) x. (A .ih A)))
30	2, 1, 1, 29	mp3an 914	. . . . . . 7 |- (A .ih (B .h A)) = ((*` B) x. (A .ih A))
31	28, 30	syl5eq 1516	. . . . . 6 |- ((((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)))
32	25, 31	eqtrd 1504	. . . . 5 |- ((((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)))
33	24, 32	breqtrd 2634	. . . 4 |- ((((A .ih (T` A)) e. RR /\ 0 <_ (A .ih (T` A))) /\ ((T` A) = (B .h A) /\ A =/= 0h)) -> 0 <_ (B x. (A .ih A)))
34	23, 33	jca 288	. . 3 |- ((((A .ih (T` A)) e. RR /\ 0 <_ (A .ih (T` A))) /\ ((T` A) = (B .h A) /\ A =/= 0h)) -> (0 < (A .ih A) /\ 0 <_ (B x. (A .ih A))))
35	20, 16, 34	sylanc 471	. 2 |- ((((A .ih (T` A)) e. RR /\ 0 <_ (A .ih (T` A))) /\ ((T` A) = (B .h A) /\ A =/= 0h)) -> 0 <_ B)
36	16, 35	jca 288	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 146   /\ wa 223   = wceq 954   e. wcel 956   =/= wne 1582   class class class wbr 2614  ` cfv 3177  (class class class)co 3954  CCcc 5212  RRcr 5213  0cc0 5214   x. cmul 5219   <_ cle 5275   < clt 5466  *ccj 6688  H~chil 8727   .h csm 8729  0hc0v 8730   .ih csp 8732

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861  ax-inf2 4605  ax-hfvmul 8814  ax-hfi 8885  ax-his1 8888  ax-his3 8890  ax-his4 8891

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-nel 1585  df-ral 1646  df-rex 1647  df-reu 1648  df-rab 1649  df-v 1808  df-sbc 1938  df-csb 1998  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-pss 2051  df-nul 2277  df-if 2358  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-int 2529  df-iun 2563  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-id 2830  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-lim 2948  df-suc 2949  df-om 3127  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-f1 3190  df-fo 3191  df-f1o 3192  df-fv 3193  df-rdg 3923  df-opr 3956  df-oprab 3957  df-1st 4069  df-2nd 4070  df-1o 4123  df-oadd 4125  df-omul 4126  df-er 4251  df-ec 4253  df-qs 4256  df-en 4357  df-dom 4358  df-sdom 4359  df-ni 4980  df-pli 4981  df-mi 4982  df-lti 4983  df-plpq 5015  df-mpq 5016  df-enq 5017  df-nq 5018  df-plq 5019  df-mq 5020  df-rq 5021  df-ltq 5022  df-1q 5023  df-np 5066  df-1p 5067  df-plp 5068  df-mp 5069  df-ltp 5070  df-plpr 5144  df-mpr 5145  df-enr 5146  df-nr 5147  df-plr 5148  df-mr 5149  df-ltr 5150  df-0r 5151  df-1r 5152  df-m1r 5153  df-c 5220  df-0 5221  df-1 5222  df-i 5223  df-r 5224  df-plus 5225  df-mul 5226  df-lt 5227  df-sub 5336  df-neg 5338  df-pnf 5467  df-mnf 5468  df-xr 5469  df-ltxr 5470  df-le 5471  df-div 5680  df-re 6690  df-im 6691  df-cj 6692

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