Theorem hmopidmchlem 10016

Description: Lemma for hmopidmch 10017.

Hypotheses

Ref	Expression
hmopidmch.1	|- H = {x e. H~ | (T` x) = x}
hmopidmch.2	|- (T e. HrmOp /\ (T o. T) = T)

Assertion

Ref	Expression
hmopidmchlem	|- (A e. H~ -> (normh` (T` A)) <_ (normh` A))

Distinct variable group:   x,T

Proof of Theorem hmopidmchlem

Step	Hyp	Ref	Expression
1		hmopidmch.2	. . . . . . 7 |- (T e. HrmOp /\ (T o. T) = T)
2	1	pm3.26i 320	. . . . . 6 |- T e. HrmOp
3		hmoplint 9805	. . . . . 6 |- (T e. HrmOp -> T e. LinOp)
4	2, 3	ax-mp 7	. . . . 5 |- T e. LinOp
5	4	lnopf 9832	. . . 4 |- T:H~-->H~
6	5	ffvelrni 3806	. . 3 |- (A e. H~ -> (T` A) e. H~)
7		normge0t 8931	. . 3 |- ((T` A) e. H~ -> 0 <_ (normh` (T` A)))
8	6, 7	syl 10	. 2 |- (A e. H~ -> 0 <_ (normh` (T` A)))
9		normclt 8930	. . . 4 |- ((T` A) e. H~ -> (normh` (T` A)) e. RR)
10		0re 5420	. . . . 5 |- 0 e. RR
11		leloet 5499	. . . . 5 |- ((0 e. RR /\ (normh` (T` A)) e. RR) -> (0 <_ (normh` (T` A)) <-> (0 < (normh` (T` A)) \/ 0 = (normh` (T` A)))))
12	10, 11	mpan 694	. . . 4 |- ((normh` (T` A)) e. RR -> (0 <_ (normh` (T` A)) <-> (0 < (normh` (T` A)) \/ 0 = (normh` (T` A)))))
13	6, 9, 12	3syl 20	. . 3 |- (A e. H~ -> (0 <_ (normh` (T` A)) <-> (0 < (normh` (T` A)) \/ 0 = (normh` (T` A)))))
14		normsqt 8940	. . . . . . . . . 10 |- ((T` A) e. H~ -> ((normh` (T` A))^2) = ((T` A) .ih (T` A)))
15	6, 14	syl 10	. . . . . . . . 9 |- (A e. H~ -> ((normh` (T` A))^2) = ((T` A) .ih (T` A)))
16	6, 9	syl 10	. . . . . . . . . . 11 |- (A e. H~ -> (normh` (T` A)) e. RR)
17	16	recnd 5295	. . . . . . . . . 10 |- (A e. H~ -> (normh` (T` A)) e. CC)
18		sqvalt 6548	. . . . . . . . . 10 |- ((normh` (T` A)) e. CC -> ((normh` (T` A))^2) = ((normh` (T` A)) x. (normh` (T` A))))
19	17, 18	syl 10	. . . . . . . . 9 |- (A e. H~ -> ((normh` (T` A))^2) = ((normh` (T` A)) x. (normh` (T` A))))
20		hmopt 9785	. . . . . . . . . . . . . 14 |- ((T e. HrmOp /\ (T` A) e. H~ /\ A e. H~) -> ((T` A) .ih (T` A)) = ((T` (T` A)) .ih A))
21	2, 20	mp3an1 901	. . . . . . . . . . . . 13 |- (((T` A) e. H~ /\ A e. H~) -> ((T` A) .ih (T` A)) = ((T` (T` A)) .ih A))
22	6, 21	mpancom 704	. . . . . . . . . . . 12 |- (A e. H~ -> ((T` A) .ih (T` A)) = ((T` (T` A)) .ih A))
23	5, 5	hoco 9630	. . . . . . . . . . . . . 14 |- (A e. H~ -> ((T o. T)` A) = (T` (T` A)))
24	1	pm3.27i 324	. . . . . . . . . . . . . . 15 |- (T o. T) = T
25	24	fveq1i 3716	. . . . . . . . . . . . . 14 |- ((T o. T)` A) = (T` A)
26	23, 25	syl5reqr 1519	. . . . . . . . . . . . 13 |- (A e. H~ -> (T` (T` A)) = (T` A))
27	26	opreq1d 3966	. . . . . . . . . . . 12 |- (A e. H~ -> ((T` (T` A)) .ih A) = ((T` A) .ih A))
28	22, 27	eqtr2d 1505	. . . . . . . . . . 11 |- (A e. H~ -> ((T` A) .ih A) = ((T` A) .ih (T` A)))
29	28	fveq2d 3719	. . . . . . . . . 10 |- (A e. H~ -> (abs` ((T` A) .ih A)) = (abs` ((T` A) .ih (T` A))))
30		absidt 6805	. . . . . . . . . . 11 |- ((((T` A) .ih (T` A)) e. RR /\ 0 <_ ((T` A) .ih (T` A))) -> (abs` ((T` A) .ih (T` A))) = ((T` A) .ih (T` A)))
31		hiidrclt 8900	. . . . . . . . . . . 12 |- ((T` A) e. H~ -> ((T` A) .ih (T` A)) e. RR)
32	6, 31	syl 10	. . . . . . . . . . 11 |- (A e. H~ -> ((T` A) .ih (T` A)) e. RR)
33		hiidge0t 8903	. . . . . . . . . . . 12 |- ((T` A) e. H~ -> 0 <_ ((T` A) .ih (T` A)))
34	6, 33	syl 10	. . . . . . . . . . 11 |- (A e. H~ -> 0 <_ ((T` A) .ih (T` A)))
35	30, 32, 34	sylanc 471	. . . . . . . . . 10 |- (A e. H~ -> (abs` ((T` A) .ih (T` A))) = ((T` A) .ih (T` A)))
36	29, 35	eqtr2d 1505	. . . . . . . . 9 |- (A e. H~ -> ((T` A) .ih (T` A)) = (abs` ((T` A) .ih A)))
37	15, 19, 36	3eqtr3d 1512	. . . . . . . 8 |- (A e. H~ -> ((normh` (T` A)) x. (normh` (T` A))) = (abs` ((T` A) .ih A)))
38		bcst 8987	. . . . . . . . 9 |- (((T` A) e. H~ /\ A e. H~) -> (abs` ((T` A) .ih A)) <_ ((normh` (T` A)) x. (normh` A)))
39	6, 38	mpancom 704	. . . . . . . 8 |- (A e. H~ -> (abs` ((T` A) .ih A)) <_ ((normh` (T` A)) x. (normh` A)))
40	37, 39	eqbrtrd 2630	. . . . . . 7 |- (A e. H~ -> ((normh` (T` A)) x. (normh` (T` A))) <_ ((normh` (T` A)) x. (normh` A)))
41	40	adantr 389	. . . . . 6 |- ((A e. H~ /\ 0 < (normh` (T` A))) -> ((normh` (T` A)) x. (normh` (T` A))) <_ ((normh` (T` A)) x. (normh` A)))
42		lemul2t 5797	. . . . . . 7 |- ((((normh` (T` A)) e. RR /\ (normh` A) e. RR /\ (normh` (T` A)) e. RR) /\ 0 < (normh` (T` A))) -> ((normh` (T` A)) <_ (normh` A) <-> ((normh` (T` A)) x. (normh` (T` A))) <_ ((normh` (T` A)) x. (normh` A))))
43		normclt 8930	. . . . . . . 8 |- (A e. H~ -> (normh` A) e. RR)
44	16, 43, 16	3jca 818	. . . . . . 7 |- (A e. H~ -> ((normh` (T` A)) e. RR /\ (normh` A) e. RR /\ (normh` (T` A)) e. RR))
45	42, 44	sylan 448	. . . . . 6 |- ((A e. H~ /\ 0 < (normh` (T` A))) -> ((normh` (T` A)) <_ (normh` A) <-> ((normh` (T` A)) x. (normh` (T` A))) <_ ((normh` (T` A)) x. (normh` A))))
46	41, 45	mpbird 196	. . . . 5 |- ((A e. H~ /\ 0 < (normh` (T` A))) -> (normh` (T` A)) <_ (normh` A))
47		pm3.27 323	. . . . . 6 |- ((A e. H~ /\ 0 = (normh` (T` A))) -> 0 = (normh` (T` A)))
48		normge0t 8931	. . . . . . 7 |- (A e. H~ -> 0 <_ (normh` A))
49	48	adantr 389	. . . . . 6 |- ((A e. H~ /\ 0 = (normh` (T` A))) -> 0 <_ (normh` A))
50	47, 49	eqbrtrrd 2632	. . . . 5 |- ((A e. H~ /\ 0 = (normh` (T` A))) -> (normh` (T` A)) <_ (normh` A))
51	46, 50	jaodan 426	. . . 4 |- ((A e. H~ /\ (0 < (normh` (T` A)) \/ 0 = (normh` (T` A)))) -> (normh` (T` A)) <_ (normh` A))
52	51	ex 373	. . 3 |- (A e. H~ -> ((0 < (normh` (T` A)) \/ 0 = (normh` (T` A))) -> (normh` (T` A)) <_ (normh` A)))
53	13, 52	sylbid 203	. 2 |- (A e. H~ -> (0 <_ (normh` (T` A)) -> (normh` (T` A)) <_ (normh` A)))
54	8, 53	mpd 26	1 |- (A e. H~ -> (normh` (T` A)) <_ (normh` A))

Syntax hints:   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223   /\ w3a 774   = wceq 954   e. wcel 956  {crab 1645   class class class wbr 2614   o. ccom 3169  ` cfv 3177  (class class class)co 3954  CCcc 5212  RRcr 5213  0cc0 5214   x. cmul 5219   <_ cle 5275   < clt 5466  2c2 5916  ^cexp 6508  abscabs 6689  H~chil 8727   .ih csp 8732  normhcno 8733  LinOpclo 8755  HrmOpcho 8758

This theorem is referenced by:  hmopidmch 10017

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-reg 4573  ax-inf2 4605  ax-ac 4724  ax-hilex 8808  ax-hfvadd 8809  ax-hvcom 8810  ax-hvass 8811  ax-hv0cl 8812  ax-hvaddid 8813  ax-hfvmul 8814  ax-hvmulid 8815  ax-hvmulass 8816  ax-hvdistr1 8817  ax-hvdistr2 8818  ax-hvmul0 8819  ax-hfi 8885  ax-his1 8888  ax-his2 8889  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