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Theorem cnlnadjlem8 31956

Description: Lemma for cnlnadji 31958. 𝐹 is continuous. (Contributed by NM, 17-Feb-2006.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
cnlnadjlem.1	⊢ 𝑇 ∈ LinOp
cnlnadjlem.2	⊢ 𝑇 ∈ ContOp
cnlnadjlem.3	⊢ 𝐺 = (𝑔 ∈ ℋ ↦ ((𝑇‘𝑔) ·_ih 𝑦))
cnlnadjlem.4	⊢ 𝐵 = (℩𝑤 ∈ ℋ ∀𝑣 ∈ ℋ ((𝑇‘𝑣) ·_ih 𝑦) = (𝑣 ·_ih 𝑤))
cnlnadjlem.5	⊢ 𝐹 = (𝑦 ∈ ℋ ↦ 𝐵)

**Assertion**
Ref	Expression
cnlnadjlem8	⊢ 𝐹 ∈ ContOp

Distinct variable groups: 𝑣,𝑔,𝑤,𝑦 𝑤,𝐹 𝑇,𝑔,𝑣,𝑤,𝑦 𝑣,𝐺,𝑤 Allowed substitution hints: 𝐵(𝑦,𝑤,𝑣,𝑔) 𝐹(𝑦,𝑣,𝑔) 𝐺(𝑦,𝑔)

Proof of Theorem cnlnadjlem8 Dummy variables 𝑧 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnlnadjlem.1	. . . 4 ⊢ 𝑇 ∈ LinOp
2		cnlnadjlem.2	. . . 4 ⊢ 𝑇 ∈ ContOp
3	1, 2	nmcopexi 31909	. . 3 ⊢ (norm_op‘𝑇) ∈ ℝ
4		cnlnadjlem.3	. . . . 5 ⊢ 𝐺 = (𝑔 ∈ ℋ ↦ ((𝑇‘𝑔) ·_ih 𝑦))
5		cnlnadjlem.4	. . . . 5 ⊢ 𝐵 = (℩𝑤 ∈ ℋ ∀𝑣 ∈ ℋ ((𝑇‘𝑣) ·_ih 𝑦) = (𝑣 ·_ih 𝑤))
6		cnlnadjlem.5	. . . . 5 ⊢ 𝐹 = (𝑦 ∈ ℋ ↦ 𝐵)
7	1, 2, 4, 5, 6	cnlnadjlem7 31955	. . . 4 ⊢ (𝑧 ∈ ℋ → (norm_ℎ‘(𝐹‘𝑧)) ≤ ((norm_op‘𝑇) · (norm_ℎ‘𝑧)))
8	7	rgen 3052	. . 3 ⊢ ∀𝑧 ∈ ℋ (norm_ℎ‘(𝐹‘𝑧)) ≤ ((norm_op‘𝑇) · (norm_ℎ‘𝑧))
9		oveq1 7426	. . . . . 6 ⊢ (𝑥 = (norm_op‘𝑇) → (𝑥 · (norm_ℎ‘𝑧)) = ((norm_op‘𝑇) · (norm_ℎ‘𝑧)))
10	9	breq2d 5161	. . . . 5 ⊢ (𝑥 = (norm_op‘𝑇) → ((norm_ℎ‘(𝐹‘𝑧)) ≤ (𝑥 · (norm_ℎ‘𝑧)) ↔ (norm_ℎ‘(𝐹‘𝑧)) ≤ ((norm_op‘𝑇) · (norm_ℎ‘𝑧))))
11	10	ralbidv 3167	. . . 4 ⊢ (𝑥 = (norm_op‘𝑇) → (∀𝑧 ∈ ℋ (norm_ℎ‘(𝐹‘𝑧)) ≤ (𝑥 · (norm_ℎ‘𝑧)) ↔ ∀𝑧 ∈ ℋ (norm_ℎ‘(𝐹‘𝑧)) ≤ ((norm_op‘𝑇) · (norm_ℎ‘𝑧))))
12	11	rspcev 3606	. . 3 ⊢ (((norm_op‘𝑇) ∈ ℝ ∧ ∀𝑧 ∈ ℋ (norm_ℎ‘(𝐹‘𝑧)) ≤ ((norm_op‘𝑇) · (norm_ℎ‘𝑧))) → ∃𝑥 ∈ ℝ ∀𝑧 ∈ ℋ (norm_ℎ‘(𝐹‘𝑧)) ≤ (𝑥 · (norm_ℎ‘𝑧)))
13	3, 8, 12	mp2an 690	. 2 ⊢ ∃𝑥 ∈ ℝ ∀𝑧 ∈ ℋ (norm_ℎ‘(𝐹‘𝑧)) ≤ (𝑥 · (norm_ℎ‘𝑧))
14	1, 2, 4, 5, 6	cnlnadjlem6 31954	. . 3 ⊢ 𝐹 ∈ LinOp
15	14	lnopconi 31916	. 2 ⊢ (𝐹 ∈ ContOp ↔ ∃𝑥 ∈ ℝ ∀𝑧 ∈ ℋ (norm_ℎ‘(𝐹‘𝑧)) ≤ (𝑥 · (norm_ℎ‘𝑧)))
16	13, 15	mpbir 230	1 ⊢ 𝐹 ∈ ContOp

Colors of variables: wff setvar class

Syntax hints: = wceq 1533 ∈ wcel 2098 ∀wral 3050 ∃wrex 3059 class class class wbr 5149 ↦ cmpt 5232 ‘cfv 6549 ℩crio 7374 (class class class)co 7419 ℝcr 11139 · cmul 11145 ≤ cle 11281 ℋchba 30801 ·_ih csp 30804 norm_ℎcno 30805 norm_opcnop 30827 ContOpccop 30828 LinOpclo 30829

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-inf2 9666 ax-cc 10460 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 ax-pre-sup 11218 ax-addf 11219 ax-mulf 11220 ax-hilex 30881 ax-hfvadd 30882 ax-hvcom 30883 ax-hvass 30884 ax-hv0cl 30885 ax-hvaddid 30886 ax-hfvmul 30887 ax-hvmulid 30888 ax-hvmulass 30889 ax-hvdistr1 30890 ax-hvdistr2 30891 ax-hvmul0 30892 ax-hfi 30961 ax-his1 30964 ax-his2 30965 ax-his3 30966 ax-his4 30967 ax-hcompl 31084

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-iin 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-isom 6558 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7685 df-om 7872 df-1st 7994 df-2nd 7995 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-oadd 8491 df-omul 8492 df-er 8725 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9388 df-fi 9436 df-sup 9467 df-inf 9468 df-oi 9535 df-card 9964 df-acn 9967 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-div 11904 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12506 df-z 12592 df-dec 12711 df-uz 12856 df-q 12966 df-rp 13010 df-xneg 13127 df-xadd 13128 df-xmul 13129 df-ioo 13363 df-ico 13365 df-icc 13366 df-fz 13520 df-fzo 13663 df-fl 13793 df-seq 14003 df-exp 14063 df-hash 14326 df-cj 15082 df-re 15083 df-im 15084 df-sqrt 15218 df-abs 15219 df-clim 15468 df-rlim 15469 df-sum 15669 df-struct 17119 df-sets 17136 df-slot 17154 df-ndx 17166 df-base 17184 df-ress 17213 df-plusg 17249 df-mulr 17250 df-starv 17251 df-sca 17252 df-vsca 17253 df-ip 17254 df-tset 17255 df-ple 17256 df-ds 17258 df-unif 17259 df-hom 17260 df-cco 17261 df-rest 17407 df-topn 17408 df-0g 17426 df-gsum 17427 df-topgen 17428 df-pt 17429 df-prds 17432 df-xrs 17487 df-qtop 17492 df-imas 17493 df-xps 17495 df-mre 17569 df-mrc 17570 df-acs 17572 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-submnd 18744 df-mulg 19032 df-cntz 19280 df-cmn 19749 df-psmet 21288 df-xmet 21289 df-met 21290 df-bl 21291 df-mopn 21292 df-fbas 21293 df-fg 21294 df-cnfld 21297 df-top 22840 df-topon 22857 df-topsp 22879 df-bases 22893 df-cld 22967 df-ntr 22968 df-cls 22969 df-nei 23046 df-cn 23175 df-cnp 23176 df-lm 23177 df-t1 23262 df-haus 23263 df-tx 23510 df-hmeo 23703 df-fil 23794 df-fm 23886 df-flim 23887 df-flf 23888 df-xms 24270 df-ms 24271 df-tms 24272 df-cfil 25227 df-cau 25228 df-cmet 25229 df-grpo 30375 df-gid 30376 df-ginv 30377 df-gdiv 30378 df-ablo 30427 df-vc 30441 df-nv 30474 df-va 30477 df-ba 30478 df-sm 30479 df-0v 30480 df-vs 30481 df-nmcv 30482 df-ims 30483 df-dip 30583 df-ssp 30604 df-ph 30695 df-cbn 30745 df-hnorm 30850 df-hba 30851 df-hvsub 30853 df-hlim 30854 df-hcau 30855 df-sh 31089 df-ch 31103 df-oc 31134 df-ch0 31135 df-nmop 31721 df-cnop 31722 df-lnop 31723 df-unop 31725 df-nmfn 31727 df-nlfn 31728 df-cnfn 31729 df-lnfn 31730

This theorem is referenced by: cnlnadjlem9 31957

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