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Mirrors > Home > HSE Home > Th. List > lnopco0i | Structured version Visualization version GIF version |
Description: The composition of a linear operator with one whose norm is zero. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
lnopco.1 | ⊢ 𝑆 ∈ LinOp |
lnopco.2 | ⊢ 𝑇 ∈ LinOp |
Ref | Expression |
---|---|
lnopco0i | ⊢ ((normop‘𝑇) = 0 → (normop‘(𝑆 ∘ 𝑇)) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | coeq2 5856 | . . 3 ⊢ (𝑇 = 0hop → (𝑆 ∘ 𝑇) = (𝑆 ∘ 0hop )) | |
2 | lnopco.1 | . . . . . . . 8 ⊢ 𝑆 ∈ LinOp | |
3 | 0lnop 31224 | . . . . . . . 8 ⊢ 0hop ∈ LinOp | |
4 | 2, 3 | lnopcoi 31243 | . . . . . . 7 ⊢ (𝑆 ∘ 0hop ) ∈ LinOp |
5 | 4 | lnopfi 31209 | . . . . . 6 ⊢ (𝑆 ∘ 0hop ): ℋ⟶ ℋ |
6 | ffn 6714 | . . . . . 6 ⊢ ((𝑆 ∘ 0hop ): ℋ⟶ ℋ → (𝑆 ∘ 0hop ) Fn ℋ) | |
7 | 5, 6 | ax-mp 5 | . . . . 5 ⊢ (𝑆 ∘ 0hop ) Fn ℋ |
8 | ho0f 30991 | . . . . . 6 ⊢ 0hop : ℋ⟶ ℋ | |
9 | ffn 6714 | . . . . . 6 ⊢ ( 0hop : ℋ⟶ ℋ → 0hop Fn ℋ) | |
10 | 8, 9 | ax-mp 5 | . . . . 5 ⊢ 0hop Fn ℋ |
11 | eqfnfv 7029 | . . . . 5 ⊢ (((𝑆 ∘ 0hop ) Fn ℋ ∧ 0hop Fn ℋ) → ((𝑆 ∘ 0hop ) = 0hop ↔ ∀𝑥 ∈ ℋ ((𝑆 ∘ 0hop )‘𝑥) = ( 0hop ‘𝑥))) | |
12 | 7, 10, 11 | mp2an 690 | . . . 4 ⊢ ((𝑆 ∘ 0hop ) = 0hop ↔ ∀𝑥 ∈ ℋ ((𝑆 ∘ 0hop )‘𝑥) = ( 0hop ‘𝑥)) |
13 | ho0val 30990 | . . . . . . 7 ⊢ (𝑥 ∈ ℋ → ( 0hop ‘𝑥) = 0ℎ) | |
14 | 13 | fveq2d 6892 | . . . . . 6 ⊢ (𝑥 ∈ ℋ → (𝑆‘( 0hop ‘𝑥)) = (𝑆‘0ℎ)) |
15 | 2 | lnop0i 31210 | . . . . . 6 ⊢ (𝑆‘0ℎ) = 0ℎ |
16 | 14, 15 | eqtrdi 2788 | . . . . 5 ⊢ (𝑥 ∈ ℋ → (𝑆‘( 0hop ‘𝑥)) = 0ℎ) |
17 | 2 | lnopfi 31209 | . . . . . 6 ⊢ 𝑆: ℋ⟶ ℋ |
18 | 17, 8 | hocoi 31004 | . . . . 5 ⊢ (𝑥 ∈ ℋ → ((𝑆 ∘ 0hop )‘𝑥) = (𝑆‘( 0hop ‘𝑥))) |
19 | 16, 18, 13 | 3eqtr4d 2782 | . . . 4 ⊢ (𝑥 ∈ ℋ → ((𝑆 ∘ 0hop )‘𝑥) = ( 0hop ‘𝑥)) |
20 | 12, 19 | mprgbir 3068 | . . 3 ⊢ (𝑆 ∘ 0hop ) = 0hop |
21 | 1, 20 | eqtrdi 2788 | . 2 ⊢ (𝑇 = 0hop → (𝑆 ∘ 𝑇) = 0hop ) |
22 | lnopco.2 | . . 3 ⊢ 𝑇 ∈ LinOp | |
23 | 22 | nmlnop0iHIL 31236 | . 2 ⊢ ((normop‘𝑇) = 0 ↔ 𝑇 = 0hop ) |
24 | 2, 22 | lnopcoi 31243 | . . 3 ⊢ (𝑆 ∘ 𝑇) ∈ LinOp |
25 | 24 | nmlnop0iHIL 31236 | . 2 ⊢ ((normop‘(𝑆 ∘ 𝑇)) = 0 ↔ (𝑆 ∘ 𝑇) = 0hop ) |
26 | 21, 23, 25 | 3imtr4i 291 | 1 ⊢ ((normop‘𝑇) = 0 → (normop‘(𝑆 ∘ 𝑇)) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1541 ∈ wcel 2106 ∀wral 3061 ∘ ccom 5679 Fn wfn 6535 ⟶wf 6536 ‘cfv 6540 0cc0 11106 ℋchba 30159 0ℎc0v 30164 0hop ch0o 30183 normopcnop 30185 LinOpclo 30187 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632 ax-cc 10426 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185 ax-mulf 11186 ax-hilex 30239 ax-hfvadd 30240 ax-hvcom 30241 ax-hvass 30242 ax-hv0cl 30243 ax-hvaddid 30244 ax-hfvmul 30245 ax-hvmulid 30246 ax-hvmulass 30247 ax-hvdistr1 30248 ax-hvdistr2 30249 ax-hvmul0 30250 ax-hfi 30319 ax-his1 30322 ax-his2 30323 ax-his3 30324 ax-his4 30325 ax-hcompl 30442 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-oadd 8466 df-omul 8467 df-er 8699 df-map 8818 df-pm 8819 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-fi 9402 df-sup 9433 df-inf 9434 df-oi 9501 df-card 9930 df-acn 9933 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-ioo 13324 df-ico 13326 df-icc 13327 df-fz 13481 df-fzo 13624 df-fl 13753 df-seq 13963 df-exp 14024 df-hash 14287 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-clim 15428 df-rlim 15429 df-sum 15629 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-starv 17208 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-hom 17217 df-cco 17218 df-rest 17364 df-topn 17365 df-0g 17383 df-gsum 17384 df-topgen 17385 df-pt 17386 df-prds 17389 df-xrs 17444 df-qtop 17449 df-imas 17450 df-xps 17452 df-mre 17526 df-mrc 17527 df-acs 17529 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-submnd 18668 df-mulg 18945 df-cntz 19175 df-cmn 19644 df-psmet 20928 df-xmet 20929 df-met 20930 df-bl 20931 df-mopn 20932 df-fbas 20933 df-fg 20934 df-cnfld 20937 df-top 22387 df-topon 22404 df-topsp 22426 df-bases 22440 df-cld 22514 df-ntr 22515 df-cls 22516 df-nei 22593 df-cn 22722 df-cnp 22723 df-lm 22724 df-haus 22810 df-tx 23057 df-hmeo 23250 df-fil 23341 df-fm 23433 df-flim 23434 df-flf 23435 df-xms 23817 df-ms 23818 df-tms 23819 df-cfil 24763 df-cau 24764 df-cmet 24765 df-grpo 29733 df-gid 29734 df-ginv 29735 df-gdiv 29736 df-ablo 29785 df-vc 29799 df-nv 29832 df-va 29835 df-ba 29836 df-sm 29837 df-0v 29838 df-vs 29839 df-nmcv 29840 df-ims 29841 df-dip 29941 df-ssp 29962 df-lno 29984 df-nmoo 29985 df-0o 29987 df-ph 30053 df-cbn 30103 df-hnorm 30208 df-hba 30209 df-hvsub 30211 df-hlim 30212 df-hcau 30213 df-sh 30447 df-ch 30461 df-oc 30492 df-ch0 30493 df-shs 30548 df-pjh 30635 df-h0op 30988 df-nmop 31079 df-lnop 31081 df-hmop 31084 |
This theorem is referenced by: nmopcoi 31335 |
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