Theorem eigpos 9757

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 8879	. . . . . . . . 9 |- (B .h A) e. H~
4		hiret 8955	. . . . . . . . 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 699	. . . . . . . 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 3975	. . . . . . . 8 |- ((T` A) = (B .h A) -> (A .ih (T` A)) = (A .ih (B .h A)))
8	7	eleq1d 1543	. . . . . . 7 |- ((T` A) = (B .h A) -> ((A .ih (T` A)) e. RR <-> (A .ih (B .h A)) e. RR))
9		opreq1 3974	. . . . . . . 8 |- ((T` A) = (B .h A) -> ((T` A) .ih A) = ((B .h A) .ih A))
10	7, 9	eqeq12d 1492	. . . . . . 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 552	. . . . . 6 |- ((T` A) = (B .h A) -> ((A .ih (T` A)) e. RR <-> (A .ih (T` A)) = ((T` A) .ih A)))
12	11	adantr 391	. . . . 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 9755	. . . . 5 |- (((T` A) = (B .h A) /\ A =/= 0h) -> ((A .ih (T` A)) = ((T` A) .ih A) <-> B e. RR))
14	12, 13	bitrd 530	. . . 4 |- (((T` A) = (B .h A) /\ A =/= 0h) -> ((A .ih (T` A)) e. RR <-> B e. RR))
15	14	biimpac 420	. . 3 |- (((A .ih (T` A)) e. RR /\ ((T` A) = (B .h A) /\ A =/= 0h)) -> B e. RR)
16	15	adantlr 395	. 2 |- ((((A .ih (T` A)) e. RR /\ 0 <_ (A .ih (T` A))) /\ ((T` A) = (B .h A) /\ A =/= 0h)) -> B e. RR)
17		hiidrclt 8956	. . . . 5 |- (A e. H~ -> (A .ih A) e. RR)
18	1, 17	ax-mp 7	. . . 4 |- (A .ih A) e. RR
19		prodge02t 5831	. . . 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 709	. . 3 |- ((B e. RR /\ (0 < (A .ih A) /\ 0 <_ (B x. (A .ih A)))) -> 0 <_ B)
21		ax-his4 8947	. . . . . 6 |- ((A e. H~ /\ A =/= 0h) -> 0 < (A .ih A))
22	1, 21	mpan 697	. . . . 5 |- (A =/= 0h -> 0 < (A .ih A))
23	22	ad2antll 409	. . . 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 415	. . . . 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 408	. . . . . 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 6781	. . . . . . . . 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 3981	. . . . . . 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 8948	. . . . . . . 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 918	. . . . . . 7 |- (A .ih (B .h A)) = ((*` B) x. (A .ih A))
31	28, 30	syl5eq 1522	. . . . . 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 1510	. . . . 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 2644	. . . 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 473	. 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 958   e. wcel 960   =/= wne 1588   class class class wbr 2624  ` cfv 3188  (class class class)co 3969  CCcc 5244  RRcr 5245  0cc0 5246   x. cmul 5251   <_ cle 5307   < clt 5498  *ccj 6750  H~chil 8783   .h csm 8785  0hc0v 8786   .ih csp 8788

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-inf2 4634  ax-hfvmul 8870  ax-hfi 8941  ax-his1 8944  ax-his3 8946  ax-his4 8947

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-nel 1591  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-pss 2058  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203  df-fv 3204  df-rdg 3938  df-opr 3971  df-oprab 3972  df-1st 4085  df-2nd 4086  df-1o 4139  df-oadd 4141  df-omul 4142  df-er 4267  df-ec 4269  df-qs 4272  df-en 4374  df-dom 4375  df-sdom 4376  df-ni 5012  df-pli 5013  df-mi 5014  df-lti 5015  df-plpq 5047  df-mpq 5048  df-enq 5049  df-nq 5050  df-plq 5051  df-mq 5052  df-rq 5053  df-ltq 5054  df-1q 5055  df-np 5098  df-1p 5099  df-plp 5100  df-mp 5101  df-ltp 5102  df-plpr 5176  df-mpr 5177  df-enr 5178  df-nr 5179  df-plr 5180  df-mr 5181  df-ltr 5182  df-0r 5183  df-1r 5184  df-m1r 5185  df-c 5252  df-0 5253  df-1 5254  df-i 5255  df-r 5256  df-plus 5257  df-mul 5258  df-lt 5259  df-sub 5368  df-neg 5370  df-pnf 5499  df-mnf 5500  df-xr 5501  df-ltxr 5502  df-le 5503  df-div 5715  df-re 6752  df-im 6753  df-cj 6754

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