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Theorem blocni 27660

Description: A linear operator is continuous iff it is bounded. Theorem 2.7-9(a) of [Kreyszig] p. 97. (Contributed by NM, 18-Dec-2007.) (Revised by Mario Carneiro, 10-Jan-2014.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
blocni.8
blocni.d
blocni.j
blocni.k
blocni.4
blocni.5
blocni.u
blocni.w
blocni.l

**Assertion**
Ref	Expression
blocni

Proof of Theorem blocni Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		blocni.u	. . . 4
2		eqid 2622	. . . . 5
3		eqid 2622	. . . . 5
4	2, 3	nvzcl 27489	. . . 4
5	1, 4	ax-mp 5	. . 3
6		blocni.8	. . . . . . . . . 10
7	2, 6	imsmet 27546	. . . . . . . . 9
8	1, 7	ax-mp 5	. . . . . . . 8
9		metxmet 22139	. . . . . . . 8
10	8, 9	ax-mp 5	. . . . . . 7
11		blocni.j	. . . . . . . 8
12	11	mopntopon 22244	. . . . . . 7 TopOn
13	10, 12	ax-mp 5	. . . . . 6 TopOn
14	13	toponunii 20721	. . . . 5
15	14	cncnpi 21082	. . . 4 $/\$
16	5, 15	mpan2 707	. . 3
17		blocni.d	. . . 4
18		blocni.k	. . . 4
19		blocni.4	. . . 4
20		blocni.5	. . . 4
21		blocni.w	. . . 4
22		blocni.l	. . . 4
23	6, 17, 11, 18, 19, 20, 1, 21, 22, 2	blocnilem 27659	. . 3 $/\$
24	5, 16, 23	sylancr 695	. 2
25		eleq1 2689	. . 3
26		simprr 796	. . . . . . . 8 $/\$ $/\$ $/\$
27		eqid 2622	. . . . . . . . . . . . 13
28		eqid 2622	. . . . . . . . . . . . 13
29	2, 27, 28, 20	nmblore 27641	. . . . . . . . . . . 12 $/\$ $/\$
30	1, 21, 29	mp3an12 1414	. . . . . . . . . . 11
31		eqid 2622	. . . . . . . . . . . . . 14
32	28, 31, 19	nmlnogt0 27652	. . . . . . . . . . . . 13 $/\$ $/\$
33	1, 21, 22, 32	mp3an 1424	. . . . . . . . . . . 12
34	33	biimpi 206	. . . . . . . . . . 11
35	30, 34	anim12i 590	. . . . . . . . . 10 $/\$ $/\$
36		elrp 11834	. . . . . . . . . 10 $/\$
37	35, 36	sylibr 224	. . . . . . . . 9 $/\$
38	37	adantr 481	. . . . . . . 8 $/\$ $/\$ $/\$
39	26, 38	rpdivcld 11889	. . . . . . 7 $/\$ $/\$ $/\$
40		simprl 794	. . . . . . . . . . 11 $/\$ $/\$ $/\$
41		metcl 22137	. . . . . . . . . . . 12 $/\$ $/\$
42	8, 41	mp3an1 1411	. . . . . . . . . . 11 $/\$
43	40, 42	sylan 488	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
44		simplrr 801	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
45	44	rpred 11872	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
46	35	ad2antrr 762	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
47		ltmuldiv2 10897	. . . . . . . . . 10 $/\$ $/\$ $/\$
48	43, 45, 46, 47	syl3anc 1326	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
49		id 22	. . . . . . . . . . . 12 $/\$ $/\$
50	49	ad2ant2r 783	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
51	2, 27, 6, 17, 28, 20, 1, 21	blometi 27658	. . . . . . . . . . . 12 $/\$ $/\$
52	51	3expa 1265	. . . . . . . . . . 11 $/\$ $/\$
53	50, 52	sylan 488	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
54	2, 27, 19	lnof 27610	. . . . . . . . . . . . . . 15 $/\$ $/\$
55	1, 21, 22, 54	mp3an 1424	. . . . . . . . . . . . . 14
56	55	ffvelrni 6358	. . . . . . . . . . . . 13
57	55	ffvelrni 6358	. . . . . . . . . . . . 13
58	27, 17	imsmet 27546	. . . . . . . . . . . . . . 15
59	21, 58	ax-mp 5	. . . . . . . . . . . . . 14
60		metcl 22137	. . . . . . . . . . . . . 14 $/\$ $/\$
61	59, 60	mp3an1 1411	. . . . . . . . . . . . 13 $/\$
62	56, 57, 61	syl2an 494	. . . . . . . . . . . 12 $/\$
63	40, 62	sylan 488	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
64		remulcl 10021	. . . . . . . . . . . . . . 15 $/\$
65	30, 42, 64	syl2an 494	. . . . . . . . . . . . . 14 $/\$ $/\$
66	65	anassrs 680	. . . . . . . . . . . . 13 $/\$ $/\$
67	66	adantllr 755	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
68	67	adantlrr 757	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
69		lelttr 10128	. . . . . . . . . . 11 $/\$ $/\$ $/\$
70	63, 68, 45, 69	syl3anc 1326	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
71	53, 70	mpand 711	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
72	48, 71	sylbird 250	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
73	72	ralrimiva 2966	. . . . . . 7 $/\$ $/\$ $/\$
74		breq2 4657	. . . . . . . . . 10
75	74	imbi1d 331	. . . . . . . . 9
76	75	ralbidv 2986	. . . . . . . 8
77	76	rspcev 3309	. . . . . . 7 $/\$
78	39, 73, 77	syl2anc 693	. . . . . 6 $/\$ $/\$ $/\$
79	78	ralrimivva 2971	. . . . 5 $/\$
80	79, 55	jctil 560	. . . 4 $/\$ $/\$
81		metxmet 22139	. . . . . 6
82	59, 81	ax-mp 5	. . . . 5
83	11, 18	metcn 22348	. . . . 5 $/\$ $/\$
84	10, 82, 83	mp2an 708	. . . 4 $/\$
85	80, 84	sylibr 224	. . 3 $/\$
86		eqid 2622	. . . . . . 7
87	2, 86, 31	0ofval 27642	. . . . . 6 $/\$
88	1, 21, 87	mp2an 708	. . . . 5
89	18	mopntopon 22244	. . . . . . 7 TopOn
90	82, 89	ax-mp 5	. . . . . 6 TopOn
91	27, 86	nvzcl 27489	. . . . . . 7
92	21, 91	ax-mp 5	. . . . . 6
93		cnconst2 21087	. . . . . 6 TopOn $/\$ TopOn $/\$
94	13, 90, 92, 93	mp3an 1424	. . . . 5
95	88, 94	eqeltri 2697	. . . 4
96	95	a1i 11	. . 3
97	25, 85, 96	pm2.61ne 2879	. 2
98	24, 97	impbii 199	1

Colors of variables: wff setvar class

Syntax hints:

wi 4

wb 196 $/\$ wa 384

wceq 1483

wcel 1990

wne 2794

wral 2912

wrex 2913

csn 4177 class class class wbr 4653

cxp 5112

wf 5884

cfv 5888 (class class class)co 6650

cr 9935

cc0 9936

cmul 9941

clt 10074

cle 10075

cdiv 10684

crp 11832

cxmt 19731

cme 19732

cmopn 19736 TopOnctopon 20715

ccn 21028

ccnp 21029

cnv 27439

cba 27441

cn0v 27443

cims 27446

clno 27595

cnmoo 27596

cblo 27597

c0o 27598

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-topgen 16104 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-top 20699 df-topon 20716 df-bases 20750 df-cn 21031 df-cnp 21032 df-grpo 27347 df-gid 27348 df-ginv 27349 df-gdiv 27350 df-ablo 27399 df-vc 27414 df-nv 27447 df-va 27450 df-ba 27451 df-sm 27452 df-0v 27453 df-vs 27454 df-nmcv 27455 df-ims 27456 df-lno 27599 df-nmoo 27600 df-blo 27601 df-0o 27602

This theorem is referenced by: lnocni 27661 blocn 27662

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