Theorem hmopidmchlem 10078

Description: Lemma for hmopidmch 10079.

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 9866	. . . . . 6 |- (T e. HrmOp -> T e. LinOp)
4	2, 3	ax-mp 7	. . . . 5 |- T e. LinOp
5	4	lnopf 9893	. . . 4 |- T:H~-->H~
6	5	ffvelrni 3815	. . 3 |- (A e. H~ -> (T` A) e. H~)
7		normge0t 8992	. . 3 |- ((T` A) e. H~ -> 0 <_ (normh` (T` A)))
8	6, 7	syl 10	. 2 |- (A e. H~ -> 0 <_ (normh` (T` A)))
9		normclt 8991	. . . 4 |- ((T` A) e. H~ -> (normh` (T` A)) e. RR)
10		0re 5440	. . . . 5 |- 0 e. RR
11		leloet 5518	. . . . 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 695	. . . 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 9001	. . . . . . . . . 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 5315	. . . . . . . . . 10 |- (A e. H~ -> (normh` (T` A)) e. CC)
18		sqvalt 6609	. . . . . . . . . 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 9846	. . . . . . . . . . . . . 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 903	. . . . . . . . . . . . 13 |- (((T` A) e. H~ /\ A e. H~) -> ((T` A) .ih (T` A)) = ((T` (T` A)) .ih A))
22	6, 21	mpancom 705	. . . . . . . . . . . 12 |- (A e. H~ -> ((T` A) .ih (T` A)) = ((T` (T` A)) .ih A))
23	5, 5	hoco 9690	. . . . . . . . . . . . . 14 |- (A e. H~ -> ((T o. T)` A) = (T` (T` A)))
24	1	pm3.27i 324	. . . . . . . . . . . . . . 15 |- (T o. T) = T
25	24	fveq1i 3725	. . . . . . . . . . . . . 14 |- ((T o. T)` A) = (T` A)
26	23, 25	syl5reqr 1522	. . . . . . . . . . . . 13 |- (A e. H~ -> (T` (T` A)) = (T` A))
27	26	opreq1d 3975	. . . . . . . . . . . 12 |- (A e. H~ -> ((T` (T` A)) .ih A) = ((T` A) .ih A))
28	22, 27	eqtr2d 1508	. . . . . . . . . . 11 |- (A e. H~ -> ((T` A) .ih A) = ((T` A) .ih (T` A)))
29	28	fveq2d 3728	. . . . . . . . . 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 8961	. . . . . . . . . . . 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 8964	. . . . . . . . . . . 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 1508	. . . . . . . . 9 |- (A e. H~ -> ((T` A) .ih (T` A)) = (abs` ((T` A) .ih A)))
37	15, 19, 36	3eqtr3d 1515	. . . . . . . 8 |- (A e. H~ -> ((normh` (T` A)) x. (normh` (T` A))) = (abs` ((T` A) .ih A)))
38		bcst 9048	. . . . . . . . 9 |- (((T` A) e. H~ /\ A e. H~) -> (abs` ((T` A) .ih A)) <_ ((normh` (T` A)) x. (normh` A)))
39	6, 38	mpancom 705	. . . . . . . 8 |- (A e. H~ -> (abs` ((T` A) .ih A)) <_ ((normh` (T` A)) x. (normh` A)))
40	37, 39	eqbrtrd 2635	. . . . . . 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 5833	. . . . . . 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 8991	. . . . . . . 8 |- (A e. H~ -> (normh` A) e. RR)
44	16, 43, 16	3jca 819	. . . . . . 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 8992	. . . . . . 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 2637	. . . . 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 775   = wceq 956   e. wcel 958  {crab 1648   class class class wbr 2619   o. ccom 3174  ` cfv 3182  (class class class)co 3963  CCcc 5232  RRcr 5233  0cc0 5234   x. cmul 5239   <_ cle 5295   < clt 5486  2c2 5961  ^cexp 6568  abscabs 6750  H~chil 8788   .ih csp 8793  normhcno 8794  LinOpclo 8816  HrmOpcho 8819

This theorem is referenced by:  hmopidmch 10079

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-reg 4593  ax-inf2 4625  ax-ac 4744  ax-hilex 8869  ax-hfvadd 8870  ax-hvcom 8871  ax-hvass 8872  ax-hv0cl 8873  ax-hvaddid 8874  ax-hfvmul 8875  ax-hvmulid 8876  ax-hvmulass 8877  ax-hvdistr1 8878  ax-hvdistr2 8879  ax-hvmul0 8880  ax-hfi 8946  ax-his1 8949  ax-his2 8950  ax-his3 8951  ax-his4 8952

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-nel 1588  df-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  df-v 1812  df-sbc 1942