Theorem hmopidmchlem 10073

Description: Lemma for hmopidmch 10074.

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 9861	. . . . . 6 |- (T e. HrmOp -> T e. LinOp)
4	2, 3	ax-mp 7	. . . . 5 |- T e. LinOp
5	4	lnopf 9888	. . . 4 |- T:H~-->H~
6	5	ffvelrni 3821	. . 3 |- (A e. H~ -> (T` A) e. H~)
7		normge0t 8987	. . 3 |- ((T` A) e. H~ -> 0 <_ (normh` (T` A)))
8	6, 7	syl 10	. 2 |- (A e. H~ -> 0 <_ (normh` (T` A)))
9		normclt 8986	. . . 4 |- ((T` A) e. H~ -> (normh` (T` A)) e. RR)
10		0re 5452	. . . . 5 |- 0 e. RR
11		leloet 5530	. . . . 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 697	. . . 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 8996	. . . . . . . . . 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 5327	. . . . . . . . . 10 |- (A e. H~ -> (normh` (T` A)) e. CC)
18		sqvalt 6610	. . . . . . . . . 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 9841	. . . . . . . . . . . . . 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 905	. . . . . . . . . . . . 13 |- (((T` A) e. H~ /\ A e. H~) -> ((T` A) .ih (T` A)) = ((T` (T` A)) .ih A))
22	6, 21	mpancom 707	. . . . . . . . . . . 12 |- (A e. H~ -> ((T` A) .ih (T` A)) = ((T` (T` A)) .ih A))
23	5, 5	hoco 9685	. . . . . . . . . . . . . 14 |- (A e. H~ -> ((T o. T)` A) = (T` (T` A)))
24	1	pm3.27i 324	. . . . . . . . . . . . . . 15 |- (T o. T) = T
25	24	fveq1i 3731	. . . . . . . . . . . . . 14 |- ((T o. T)` A) = (T` A)
26	23, 25	syl5reqr 1525	. . . . . . . . . . . . 13 |- (A e. H~ -> (T` (T` A)) = (T` A))
27	26	opreq1d 3981	. . . . . . . . . . . 12 |- (A e. H~ -> ((T` (T` A)) .ih A) = ((T` A) .ih A))
28	22, 27	eqtr2d 1511	. . . . . . . . . . 11 |- (A e. H~ -> ((T` A) .ih A) = ((T` A) .ih (T` A)))
29	28	fveq2d 3734	. . . . . . . . . 10 |- (A e. H~ -> (abs` ((T` A) .ih A)) = (abs` ((T` A) .ih (T` A))))
30		absidt 6862	. . . . . . . . . . 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 8956	. . . . . . . . . . . 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 8959	. . . . . . . . . . . 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 473	. . . . . . . . . 10 |- (A e. H~ -> (abs` ((T` A) .ih (T` A))) = ((T` A) .ih (T` A)))
36	29, 35	eqtr2d 1511	. . . . . . . . 9 |- (A e. H~ -> ((T` A) .ih (T` A)) = (abs` ((T` A) .ih A)))
37	15, 19, 36	3eqtr3d 1518	. . . . . . . 8 |- (A e. H~ -> ((normh` (T` A)) x. (normh` (T` A))) = (abs` ((T` A) .ih A)))
38		bcst 9043	. . . . . . . . 9 |- (((T` A) e. H~ /\ A e. H~) -> (abs` ((T` A) .ih A)) <_ ((normh` (T` A)) x. (normh` A)))
39	6, 38	mpancom 707	. . . . . . . 8 |- (A e. H~ -> (abs` ((T` A) .ih A)) <_ ((normh` (T` A)) x. (normh` A)))
40	37, 39	eqbrtrd 2640	. . . . . . 7 |- (A e. H~ -> ((normh` (T` A)) x. (normh` (T` A))) <_ ((normh` (T` A)) x. (normh` A)))
41	40	adantr 391	. . . . . 6 |- ((A e. H~ /\ 0 < (normh` (T` A))) -> ((normh` (T` A)) x. (normh` (T` A))) <_ ((normh` (T` A)) x. (normh` A)))
42		lemul2t 5835	. . . . . . 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 8986	. . . . . . . 8 |- (A e. H~ -> (normh` A) e. RR)
44	16, 43, 16	3jca 821	. . . . . . 7 |- (A e. H~ -> ((normh` (T` A)) e. RR /\ (normh` A) e. RR /\ (normh` (T` A)) e. RR))
45	42, 44	sylan 450	. . . . . 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 8987	. . . . . . 7 |- (A e. H~ -> 0 <_ (normh` A))
49	48	adantr 391	. . . . . 6 |- ((A e. H~ /\ 0 = (normh` (T` A))) -> 0 <_ (normh` A))
50	47, 49	eqbrtrrd 2642	. . . . 5 |- ((A e. H~ /\ 0 = (normh` (T` A))) -> (normh` (T` A)) <_ (normh` A))
51	46, 50	jaodan 428	. . . 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 777   = wceq 958   e. wcel 960  {crab 1651   class class class wbr 2624   o. ccom 3180  ` cfv 3188  (class class class)co 3969  CCcc 5244  RRcr 5245  0cc0 5246   x. cmul 5251   <_ cle 5307   < clt 5498  2c2 5963  ^cexp 6569  abscabs 6751  H~chil 8783   .ih csp 8788  normhcno 8789  LinOpclo 8811  HrmOpcho 8814

This theorem is referenced by:  hmopidmch 10074

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-reg 4602  ax-inf2 4634  ax-ac 4754  ax-hilex 8864  ax-hfvadd 8865  ax-hvcom 8866  ax-hvass 8867  ax-hv0cl 8868  ax-hvaddid 8869  ax-hfvmul 8870  ax-hvmulid 8871  ax-hvmulass 8872  ax-hvdistr1 8873  ax-hvdistr2 8874  ax-hvmul0 8875  ax-hfi 8941  ax-his1 8944  ax-his2 8945  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