Theorem lnopeq0i 27352

Description: A condition implying that a linear Hilbert space operator is identically zero. Unlike ho01i 27173 for arbitrary operators, when the operator is linear we need to consider only the values of the quadratic form ( T ` x ) .ih x ). (Contributed by NM, 26-Jul-2006.) (New usage is discouraged.)

Hypothesis

Ref	Expression
lnopeq0.1	|- T e. LinOp

Assertion

Ref	Expression
lnopeq0i	|- ( A. x e. ~H ( ( T ` x ) .ih x ) = 0 <-> T = 0hop )

Distinct variable group:   x, T

Proof of Theorem lnopeq0i

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lnopeq0.1	. . . . . . 7 |- T e. LinOp
2	1	lnopeq0lem2 27351	. . . . . 6 |- ( ( y e. ~H /\ z e. ~H ) -> ( ( T ` y ) .ih z ) = ( ( ( ( ( T ` ( y +h z ) ) .ih ( y +h z ) ) - ( ( T ` ( y -h z ) ) .ih ( y -h z ) ) ) + ( _i x. ( ( ( T ` ( y +h ( _i .h z ) ) ) .ih ( y +h ( _i .h z ) ) ) - ( ( T ` ( y -h ( _i .h z ) ) ) .ih ( y -h ( _i .h z ) ) ) ) ) ) / 4 ) )
3	2	adantl 466	. . . . 5 |- ( ( A. x e. ~H ( ( T ` x ) .ih x ) = 0 /\ ( y e. ~H /\ z e. ~H ) ) -> ( ( T ` y ) .ih z ) = ( ( ( ( ( T ` ( y +h z ) ) .ih ( y +h z ) ) - ( ( T ` ( y -h z ) ) .ih ( y -h z ) ) ) + ( _i x. ( ( ( T ` ( y +h ( _i .h z ) ) ) .ih ( y +h ( _i .h z ) ) ) - ( ( T ` ( y -h ( _i .h z ) ) ) .ih ( y -h ( _i .h z ) ) ) ) ) ) / 4 ) )
4		hvaddcl 26356	. . . . . . . . . . . 12 |- ( ( y e. ~H /\ z e. ~H ) -> ( y +h z ) e. ~H )
5		fveq2 5851	. . . . . . . . . . . . . . 15 |- ( x = ( y +h z ) -> ( T ` x ) = ( T ` ( y +h z ) ) )
6		id 23	. . . . . . . . . . . . . . 15 |- ( x = ( y +h z ) -> x = ( y +h z ) )
7	5, 6	oveq12d 6298	. . . . . . . . . . . . . 14 |- ( x = ( y +h z ) -> ( ( T ` x ) .ih x ) = ( ( T ` ( y +h z ) ) .ih ( y +h z ) ) )
8	7	eqeq1d 2406	. . . . . . . . . . . . 13 |- ( x = ( y +h z ) -> ( ( ( T ` x ) .ih x ) = 0 <-> ( ( T ` ( y +h z ) ) .ih ( y +h z ) ) = 0 ) )
9	8	rspccva 3161	. . . . . . . . . . . 12 |- ( ( A. x e. ~H ( ( T ` x ) .ih x ) = 0 /\ ( y +h z ) e. ~H ) -> ( ( T ` ( y +h z ) ) .ih ( y +h z ) ) = 0 )
10	4, 9	sylan2 474	. . . . . . . . . . 11 |- ( ( A. x e. ~H ( ( T ` x ) .ih x ) = 0 /\ ( y e. ~H /\ z e. ~H ) ) -> ( ( T ` ( y +h z ) ) .ih ( y +h z ) ) = 0 )
11		hvsubcl 26361	. . . . . . . . . . . 12 |- ( ( y e. ~H /\ z e. ~H ) -> ( y -h z ) e. ~H )
12		fveq2 5851	. . . . . . . . . . . . . . 15 |- ( x = ( y -h z ) -> ( T ` x ) = ( T ` ( y -h z ) ) )
13		id 23	. . . . . . . . . . . . . . 15 |- ( x = ( y -h z ) -> x = ( y -h z ) )
14	12, 13	oveq12d 6298	. . . . . . . . . . . . . 14 |- ( x = ( y -h z ) -> ( ( T ` x ) .ih x ) = ( ( T ` ( y -h z ) ) .ih ( y -h z ) ) )
15	14	eqeq1d 2406	. . . . . . . . . . . . 13 |- ( x = ( y -h z ) -> ( ( ( T ` x ) .ih x ) = 0 <-> ( ( T ` ( y -h z ) ) .ih ( y -h z ) ) = 0 ) )
16	15	rspccva 3161	. . . . . . . . . . . 12 |- ( ( A. x e. ~H ( ( T ` x ) .ih x ) = 0 /\ ( y -h z ) e. ~H ) -> ( ( T ` ( y -h z ) ) .ih ( y -h z ) ) = 0 )
17	11, 16	sylan2 474	. . . . . . . . . . 11 |- ( ( A. x e. ~H ( ( T ` x ) .ih x ) = 0 /\ ( y e. ~H /\ z e. ~H ) ) -> ( ( T ` ( y -h z ) ) .ih ( y -h z ) ) = 0 )
18	10, 17	oveq12d 6298	. . . . . . . . . 10 |- ( ( A. x e. ~H ( ( T ` x ) .ih x ) = 0 /\ ( y e. ~H /\ z e. ~H ) ) -> ( ( ( T ` ( y +h z ) ) .ih ( y +h z ) ) - ( ( T ` ( y -h z ) ) .ih ( y -h z ) ) ) = ( 0 - 0 ) )
19		0m0e0 10688	. . . . . . . . . 10 |- ( 0 - 0 ) = 0
20	18, 19	syl6eq 2461	. . . . . . . . 9 |- ( ( A. x e. ~H ( ( T ` x ) .ih x ) = 0 /\ ( y e. ~H /\ z e. ~H ) ) -> ( ( ( T ` ( y +h z ) ) .ih ( y +h z ) ) - ( ( T ` ( y -h z ) ) .ih ( y -h z ) ) ) = 0 )
21		ax-icn 9583	. . . . . . . . . . . . . . . 16 |- _i e. CC
22		hvmulcl 26357	. . . . . . . . . . . . . . . 16 |- ( ( _i e. CC /\ z e. ~H ) -> ( _i .h z ) e. ~H )
23	21, 22	mpan 670	. . . . . . . . . . . . . . 15 |- ( z e. ~H -> ( _i .h z ) e. ~H )
24		hvaddcl 26356	. . . . . . . . . . . . . . 15 |- ( ( y e. ~H /\ ( _i .h z ) e. ~H ) -> ( y +h ( _i .h z ) ) e. ~H )
25	23, 24	sylan2 474	. . . . . . . . . . . . . 14 |- ( ( y e. ~H /\ z e. ~H ) -> ( y +h ( _i .h z ) ) e. ~H )
26		fveq2 5851	. . . . . . . . . . . . . . . . 17 |- ( x = ( y +h ( _i .h z ) ) -> ( T ` x ) = ( T ` ( y +h ( _i .h z ) ) ) )
27		id 23	. . . . . . . . . . . . . . . . 17 |- ( x = ( y +h ( _i .h z ) ) -> x = ( y +h ( _i .h z ) ) )
28	26, 27	oveq12d 6298	. . . . . . . . . . . . . . . 16 |- ( x = ( y +h ( _i .h z ) ) -> ( ( T ` x ) .ih x ) = ( ( T ` ( y +h ( _i .h z ) ) ) .ih ( y +h ( _i .h z ) ) ) )
29	28	eqeq1d 2406	. . . . . . . . . . . . . . 15 |- ( x = ( y +h ( _i .h z ) ) -> ( ( ( T ` x ) .ih x ) = 0 <-> ( ( T ` ( y +h ( _i .h z ) ) ) .ih ( y +h ( _i .h z ) ) ) = 0 ) )
30	29	rspccva 3161	. . . . . . . . . . . . . 14 |- ( ( A. x e. ~H ( ( T ` x ) .ih x ) = 0 /\ ( y +h ( _i .h z ) ) e. ~H ) -> ( ( T ` ( y +h ( _i .h z ) ) ) .ih ( y +h ( _i .h z ) ) ) = 0 )
31	25, 30	sylan2 474	. . . . . . . . . . . . 13 |- ( ( A. x e. ~H ( ( T ` x ) .ih x ) = 0 /\ ( y e. ~H /\ z e. ~H ) ) -> ( ( T ` ( y +h ( _i .h z ) ) ) .ih ( y +h ( _i .h z ) ) ) = 0 )
32		hvsubcl 26361	. . . . . . . . . . . . . . 15 |- ( ( y e. ~H /\ ( _i .h z ) e. ~H ) -> ( y -h ( _i .h z ) ) e. ~H )
33	23, 32	sylan2 474	. . . . . . . . . . . . . 14 |- ( ( y e. ~H /\ z e. ~H ) -> ( y -h ( _i .h z ) ) e. ~H )
34		fveq2 5851	. . . . . . . . . . . . . . . . 17 |- ( x = ( y -h ( _i .h z ) ) -> ( T ` x ) = ( T ` ( y -h ( _i .h z ) ) ) )
35		id 23	. . . . . . . . . . . . . . . . 17 |- ( x = ( y -h ( _i .h z ) ) -> x = ( y -h ( _i .h z ) ) )
36	34, 35	oveq12d 6298	. . . . . . . . . . . . . . . 16 |- ( x = ( y -h ( _i .h z ) ) -> ( ( T ` x ) .ih x ) = ( ( T ` ( y -h ( _i .h z ) ) ) .ih ( y -h ( _i .h z ) ) ) )
37	36	eqeq1d 2406	. . . . . . . . . . . . . . 15 |- ( x = ( y -h ( _i .h z ) ) -> ( ( ( T ` x ) .ih x ) = 0 <-> ( ( T ` ( y -h ( _i .h z ) ) ) .ih ( y -h ( _i .h z ) ) ) = 0 ) )
38	37	rspccva 3161	. . . . . . . . . . . . . 14 |- ( ( A. x e. ~H ( ( T ` x ) .ih x ) = 0 /\ ( y -h ( _i .h z ) ) e. ~H ) -> ( ( T ` ( y -h ( _i .h z ) ) ) .ih ( y -h ( _i .h z ) ) ) = 0 )
39	33, 38	sylan2 474	. . . . . . . . . . . . 13 |- ( ( A. x e. ~H ( ( T ` x ) .ih x ) = 0 /\ ( y e. ~H /\ z e. ~H ) ) -> ( ( T ` ( y -h ( _i .h z ) ) ) .ih ( y -h ( _i .h z ) ) ) = 0 )
40	31, 39	oveq12d 6298	. . . . . . . . . . . 12 |- ( ( A. x e. ~H ( ( T ` x ) .ih x ) = 0 /\ ( y e. ~H /\ z e. ~H ) ) -> ( ( ( T ` ( y +h ( _i .h z ) ) ) .ih ( y +h ( _i .h z ) ) ) - ( ( T ` ( y -h ( _i .h z ) ) ) .ih ( y -h ( _i .h z ) ) ) ) = ( 0 - 0 ) )
41	40, 19	syl6eq 2461	. . . . . . . . . . 11 |- ( ( A. x e. ~H ( ( T ` x ) .ih x ) = 0 /\ ( y e. ~H /\ z e. ~H ) ) -> ( ( ( T ` ( y +h ( _i .h z ) ) ) .ih ( y +h ( _i .h z ) ) ) - ( ( T ` ( y -h ( _i .h z ) ) ) .ih ( y -h ( _i .h z ) ) ) ) = 0 )
42	41	oveq2d 6296	. . . . . . . . . 10 |- ( ( A. x e. ~H ( ( T ` x ) .ih x ) = 0 /\ ( y e. ~H /\ z e. ~H ) ) -> ( _i x. ( ( ( T ` ( y +h ( _i .h z ) ) ) .ih ( y +h ( _i .h z ) ) ) - ( ( T ` ( y -h ( _i .h z ) ) ) .ih ( y -h ( _i .h z ) ) ) ) ) = ( _i x. 0 ) )
43		it0e0 10804	. . . . . . . . . 10 |- ( _i x. 0 ) = 0
44	42, 43	syl6eq 2461	. . . . . . . . 9 |- ( ( A. x e. ~H ( ( T ` x ) .ih x ) = 0 /\ ( y e. ~H /\ z e. ~H ) ) -> ( _i x. ( ( ( T ` ( y +h ( _i .h z ) ) ) .ih ( y +h ( _i .h z ) ) ) - ( ( T ` ( y -h ( _i .h z ) ) ) .ih ( y -h ( _i .h z ) ) ) ) ) = 0 )
45	20, 44	oveq12d 6298	. . . . . . . 8 |- ( ( A. x e. ~H ( ( T ` x ) .ih x ) = 0 /\ ( y e. ~H /\ z e. ~H ) ) -> ( ( ( ( T ` ( y +h z ) ) .ih ( y +h z ) ) - ( ( T ` ( y -h z ) ) .ih ( y -h z ) ) ) + ( _i x. ( ( ( T ` ( y +h ( _i .h z ) ) ) .ih ( y +h ( _i .h z ) ) ) - ( ( T ` ( y -h ( _i .h z ) ) ) .ih ( y -h ( _i .h z ) ) ) ) ) ) = ( 0 + 0 ) )
46		00id 9791	. . . . . . . 8 |- ( 0 + 0 ) = 0
47	45, 46	syl6eq 2461	. . . . . . 7 |- ( ( A. x e. ~H ( ( T ` x ) .ih x ) = 0 /\ ( y e. ~H /\ z e. ~H ) ) -> ( ( ( ( T ` ( y +h z ) ) .ih ( y +h z ) ) - ( ( T ` ( y -h z ) ) .ih ( y -h z ) ) ) + ( _i x. ( ( ( T ` ( y +h ( _i .h z ) ) ) .ih ( y +h ( _i .h z ) ) ) - ( ( T ` ( y -h ( _i .h z ) ) ) .ih ( y -h ( _i .h z ) ) ) ) ) ) = 0 )
48	47	oveq1d 6295	. . . . . 6 |- ( ( A. x e. ~H ( ( T ` x ) .ih x ) = 0 /\ ( y e. ~H /\ z e. ~H ) ) -> ( ( ( ( ( T ` ( y +h z ) ) .ih ( y +h z ) ) - ( ( T ` ( y -h z ) ) .ih ( y -h z ) ) ) + ( _i x. ( ( ( T ` ( y +h ( _i .h z ) ) ) .ih ( y +h ( _i .h z ) ) ) - ( ( T ` ( y -h ( _i .h z ) ) ) .ih ( y -h ( _i .h z ) ) ) ) ) ) / 4 ) = ( 0 / 4 ) )
49		4cn 10656	. . . . . . 7 |- 4 e. CC
50		4ne0 10675	. . . . . . 7 |- 4 =/= 0
51	49, 50	div0i 10321	. . . . . 6 |- ( 0 / 4 ) = 0
52	48, 51	syl6eq 2461	. . . . 5 |- ( ( A. x e. ~H ( ( T ` x ) .ih x ) = 0 /\ ( y e. ~H /\ z e. ~H ) ) -> ( ( ( ( ( T ` ( y +h z ) ) .ih ( y +h z ) ) - ( ( T ` ( y -h z ) ) .ih ( y -h z ) ) ) + ( _i x. ( ( ( T ` ( y +h ( _i .h z ) ) ) .ih ( y +h ( _i .h z ) ) ) - ( ( T ` ( y -h ( _i .h z ) ) ) .ih ( y -h ( _i .h z ) ) ) ) ) ) / 4 ) = 0 )
53	3, 52	eqtrd 2445	. . . 4 |- ( ( A. x e. ~H ( ( T ` x ) .ih x ) = 0 /\ ( y e. ~H /\ z e. ~H ) ) -> ( ( T ` y ) .ih z ) = 0 )
54	53	ralrimivva 2827	. . 3 |- ( A. x e. ~H ( ( T ` x ) .ih x ) = 0 -> A. y e. ~H A. z e. ~H ( ( T ` y ) .ih z ) = 0 )
55	1	lnopfi 27314	. . . 4 |- T : ~H --> ~H
56	55	ho01i 27173	. . 3 |- ( A. y e. ~H A. z e. ~H ( ( T ` y ) .ih z ) = 0 <-> T = 0hop )
57	54, 56	sylib 198	. 2 |- ( A. x e. ~H ( ( T ` x ) .ih x ) = 0 -> T = 0hop )
58		fveq1 5850	. . . . . 6 |- ( T = 0hop -> ( T ` x ) = ( 0hop ` x ) )
59		ho0val 27095	. . . . . 6 |- ( x e. ~H -> ( 0hop ` x ) = 0h )
60	58, 59	sylan9eq 2465	. . . . 5 |- ( ( T = 0hop /\ x e. ~H ) -> ( T ` x ) = 0h )
61	60	oveq1d 6295	. . . 4 |- ( ( T = 0hop /\ x e. ~H ) -> ( ( T ` x ) .ih x ) = ( 0h .ih x ) )
62		hi01 26440	. . . . 5 |- ( x e. ~H -> ( 0h .ih x ) = 0 )
63	62	adantl 466	. . . 4 |- ( ( T = 0hop /\ x e. ~H ) -> ( 0h .ih x ) = 0 )
64	61, 63	eqtrd 2445	. . 3 |- ( ( T = 0hop /\ x e. ~H ) -> ( ( T ` x ) .ih x ) = 0 )
65	64	ralrimiva 2820	. 2 |- ( T = 0hop -> A. x e. ~H ( ( T ` x ) .ih x ) = 0 )
66	57, 65	impbii 189	1 |- ( A. x e. ~H ( ( T ` x ) .ih x ) = 0 <-> T = 0hop )

Syntax hints:   <-> wb 186   /\ wa 369   = wceq 1407   e. wcel 1844   A.wral 2756   ` cfv 5571  (class class class)co 6280   CCcc 9522   0cc0 9524   _ici 9526   + caddc 9527   x. cmul 9529   - cmin 9843   / cdiv 10249   4c4 10630   ~Hchil 26263   +h cva 26264   .h csm 26265   .ih csp 26266   0hc0v 26268   -h cmv 26269   0hopch0o 26287   LinOpclo 26291

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-inf2 8093  ax-cc 8849  ax-cnex 9580  ax-resscn 9581  ax-1cn 9582  ax-icn 9583  ax-addcl 9584  ax-addrcl 9585  ax-mulcl 9586  ax-mulrcl 9587  ax-mulcom 9588  ax-addass 9589  ax-mulass 9590  ax-distr 9591  ax-i2m1 9592  ax-1ne0 9593  ax-1rid 9594  ax-rnegex 9595  ax-rrecex 9596  ax-cnre 9597  ax-pre-lttri 9598  ax-pre-lttrn 9599  ax-pre-ltadd 9600  ax-pre-mulgt0 9601  ax-pre-sup 9602  ax-addf 9603  ax-mulf 9604  ax-hilex 26343  ax-hfvadd 26344  ax-hvcom 26345  ax-hvass 26346  ax-hv0cl 26347  ax-hvaddid 26348  ax-hfvmul 26349  ax-hvmulid 26350  ax-hvmulass 26351  ax-hvdistr1 26352  ax-hvdistr2 26353  ax-hvmul0 26354  ax-hfi 26423  ax-his1 26426  ax-his2 26427  ax-his3 26428  ax-his4 26429  ax-hcompl 26546

This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-fal 1413  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-nel 2603  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-int 4230  df-iun 4275  df-iin 4276  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-se 4785  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-pred 5369  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-isom 5580  df-riota 6242  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-of 6523  df-om 6686  df-1st 6786  df-2nd 6787  df-supp 6905  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-1o 7169  df-2o 7170  df-oadd 7173  df-omul 7174  df-er 7350  df-map 7461  df-pm 7462  df-ixp 7510  df-en 7557  df-dom 7558  df-sdom 7559  df-fin 7560  df-fsupp 7866  df-fi 7907  df-sup 7937  df-oi 7971  df-card 8354  df-acn 8357  df-cda 8582  df-pnf 9662  df-mnf 9663  df-xr 9664  df-ltxr 9665  df-le 9666  df-sub 9845  df-neg 9846  df-div 10250  df-nn 10579  df-2 10637  df-3 10638  df-4 10639  df-5 10640  df-6 10641  df-7 10642  df-8 10643  df-9 10644  df-10 10645  df-n0 10839  df-z 10908  df-dec 11022  df-uz 11130  df-q 11230  df-rp 11268  df-xneg 11373  df-xadd 11374  df-xmul 11375  df-ioo 11588  df-ico 11590  df-icc 11591  df-fz 11729  df-fzo 11857  df-fl 11968  df-seq 12154  df-exp 12213  df-hash 12455  df-cj 13083  df-re 13084  df-im 13085  df-sqrt 13219  df-abs 13220  df-clim 13462  df-rlim 13463  df-sum 13660  df-struct 14845  df-ndx 14846  df-slot 14847  df-base 14848  df-sets 14849  df-ress 14850  df-plusg 14924  df-mulr 14925  df-starv 14926  df-sca 14927  df-vsca 14928  df-ip 14929  df-tset 14930  df-ple 14931  df-ds 14933  df-unif 14934  df-hom 14935  df-cco 14936  df-rest 15039  df-topn 15040  df-0g 15058  df-gsum 15059  df-topgen 15060  df-pt 15061  df-prds 15064  df-xrs 15118  df-qtop 15123  df-imas 15124  df-xps 15126  df-mre 15202  df-mrc 15203  df-acs 15205  df-mgm 16198  df-sgrp 16237  df-mnd 16247  df-submnd 16293  df-mulg 16386  df-cntz 16681  df-cmn 17126  df-psmet 18733  df-xmet 18734  df-met 18735  df-bl 18736  df-mopn 18737  df-fbas 18738  df-fg 18739  df-cnfld 18743  df-top 19693  df-bases 19695  df-topon 19696  df-topsp 19697  df-cld 19814  df-ntr 19815  df-cls 19816  df-nei 19894  df-cn 20023  df-cnp 20024  df-lm 20025  df-haus 20111  df-tx 20357  df-hmeo 20550  df-fil 20641  df-fm 20733  df-flim 20734  df-flf 20735  df-xms 21117  df-ms 21118  df-tms 21119  df-cfil 21988  df-cau 21989  df-cmet 21990  df-grpo 25620  df-gid 25621  df-ginv 25622  df-gdiv 25623  df-ablo 25711  df-subgo 25731  df-vc 25866  df-nv 25912  df-va 25915  df-ba 25916  df-sm 25917  df-0v 25918  df-vs 25919  df-nmcv 25920  df-ims 25921  df-dip 26038  df-ssp 26062  df-ph 26155  df-cbn 26206  df-hnorm 26312  df-hba 26313  df-hvsub 26315  df-hlim 26316  df-hcau 26317  df-sh 26551  df-ch 26566  df-oc 26597  df-ch0 26598  df-shs 26653  df-pjh 26740  df-h0op 27093  df-lnop 27186

This theorem is referenced by:  lnopeqi  27353