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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 5851 | . . 3 ⊢ (𝑇 = 0hop → (𝑆 ∘ 𝑇) = (𝑆 ∘ 0hop )) | |
2 | lnopco.1 | . . . . . . . 8 ⊢ 𝑆 ∈ LinOp | |
3 | 0lnop 31741 | . . . . . . . 8 ⊢ 0hop ∈ LinOp | |
4 | 2, 3 | lnopcoi 31760 | . . . . . . 7 ⊢ (𝑆 ∘ 0hop ) ∈ LinOp |
5 | 4 | lnopfi 31726 | . . . . . 6 ⊢ (𝑆 ∘ 0hop ): ℋ⟶ ℋ |
6 | ffn 6710 | . . . . . 6 ⊢ ((𝑆 ∘ 0hop ): ℋ⟶ ℋ → (𝑆 ∘ 0hop ) Fn ℋ) | |
7 | 5, 6 | ax-mp 5 | . . . . 5 ⊢ (𝑆 ∘ 0hop ) Fn ℋ |
8 | ho0f 31508 | . . . . . 6 ⊢ 0hop : ℋ⟶ ℋ | |
9 | ffn 6710 | . . . . . 6 ⊢ ( 0hop : ℋ⟶ ℋ → 0hop Fn ℋ) | |
10 | 8, 9 | ax-mp 5 | . . . . 5 ⊢ 0hop Fn ℋ |
11 | eqfnfv 7025 | . . . . 5 ⊢ (((𝑆 ∘ 0hop ) Fn ℋ ∧ 0hop Fn ℋ) → ((𝑆 ∘ 0hop ) = 0hop ↔ ∀𝑥 ∈ ℋ ((𝑆 ∘ 0hop )‘𝑥) = ( 0hop ‘𝑥))) | |
12 | 7, 10, 11 | mp2an 689 | . . . 4 ⊢ ((𝑆 ∘ 0hop ) = 0hop ↔ ∀𝑥 ∈ ℋ ((𝑆 ∘ 0hop )‘𝑥) = ( 0hop ‘𝑥)) |
13 | ho0val 31507 | . . . . . . 7 ⊢ (𝑥 ∈ ℋ → ( 0hop ‘𝑥) = 0ℎ) | |
14 | 13 | fveq2d 6888 | . . . . . 6 ⊢ (𝑥 ∈ ℋ → (𝑆‘( 0hop ‘𝑥)) = (𝑆‘0ℎ)) |
15 | 2 | lnop0i 31727 | . . . . . 6 ⊢ (𝑆‘0ℎ) = 0ℎ |
16 | 14, 15 | eqtrdi 2782 | . . . . 5 ⊢ (𝑥 ∈ ℋ → (𝑆‘( 0hop ‘𝑥)) = 0ℎ) |
17 | 2 | lnopfi 31726 | . . . . . 6 ⊢ 𝑆: ℋ⟶ ℋ |
18 | 17, 8 | hocoi 31521 | . . . . 5 ⊢ (𝑥 ∈ ℋ → ((𝑆 ∘ 0hop )‘𝑥) = (𝑆‘( 0hop ‘𝑥))) |
19 | 16, 18, 13 | 3eqtr4d 2776 | . . . 4 ⊢ (𝑥 ∈ ℋ → ((𝑆 ∘ 0hop )‘𝑥) = ( 0hop ‘𝑥)) |
20 | 12, 19 | mprgbir 3062 | . . 3 ⊢ (𝑆 ∘ 0hop ) = 0hop |
21 | 1, 20 | eqtrdi 2782 | . 2 ⊢ (𝑇 = 0hop → (𝑆 ∘ 𝑇) = 0hop ) |
22 | lnopco.2 | . . 3 ⊢ 𝑇 ∈ LinOp | |
23 | 22 | nmlnop0iHIL 31753 | . 2 ⊢ ((normop‘𝑇) = 0 ↔ 𝑇 = 0hop ) |
24 | 2, 22 | lnopcoi 31760 | . . 3 ⊢ (𝑆 ∘ 𝑇) ∈ LinOp |
25 | 24 | nmlnop0iHIL 31753 | . 2 ⊢ ((normop‘(𝑆 ∘ 𝑇)) = 0 ↔ (𝑆 ∘ 𝑇) = 0hop ) |
26 | 21, 23, 25 | 3imtr4i 292 | 1 ⊢ ((normop‘𝑇) = 0 → (normop‘(𝑆 ∘ 𝑇)) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1533 ∈ wcel 2098 ∀wral 3055 ∘ ccom 5673 Fn wfn 6531 ⟶wf 6532 ‘cfv 6536 0cc0 11109 ℋchba 30676 0ℎc0v 30681 0hop ch0o 30700 normopcnop 30702 LinOpclo 30704 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-inf2 9635 ax-cc 10429 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 ax-addf 11188 ax-mulf 11189 ax-hilex 30756 ax-hfvadd 30757 ax-hvcom 30758 ax-hvass 30759 ax-hv0cl 30760 ax-hvaddid 30761 ax-hfvmul 30762 ax-hvmulid 30763 ax-hvmulass 30764 ax-hvdistr1 30765 ax-hvdistr2 30766 ax-hvmul0 30767 ax-hfi 30836 ax-his1 30839 ax-his2 30840 ax-his3 30841 ax-his4 30842 ax-hcompl 30959 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8144 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-2o 8465 df-oadd 8468 df-omul 8469 df-er 8702 df-map 8821 df-pm 8822 df-ixp 8891 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fsupp 9361 df-fi 9405 df-sup 9436 df-inf 9437 df-oi 9504 df-card 9933 df-acn 9936 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 df-n0 12474 df-z 12560 df-dec 12679 df-uz 12824 df-q 12934 df-rp 12978 df-xneg 13095 df-xadd 13096 df-xmul 13097 df-ioo 13331 df-ico 13333 df-icc 13334 df-fz 13488 df-fzo 13631 df-fl 13760 df-seq 13970 df-exp 14030 df-hash 14293 df-cj 15049 df-re 15050 df-im 15051 df-sqrt 15185 df-abs 15186 df-clim 15435 df-rlim 15436 df-sum 15636 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17151 df-ress 17180 df-plusg 17216 df-mulr 17217 df-starv 17218 df-sca 17219 df-vsca 17220 df-ip 17221 df-tset 17222 df-ple 17223 df-ds 17225 df-unif 17226 df-hom 17227 df-cco 17228 df-rest 17374 df-topn 17375 df-0g 17393 df-gsum 17394 df-topgen 17395 df-pt 17396 df-prds 17399 df-xrs 17454 df-qtop 17459 df-imas 17460 df-xps 17462 df-mre 17536 df-mrc 17537 df-acs 17539 df-mgm 18570 df-sgrp 18649 df-mnd 18665 df-submnd 18711 df-mulg 18993 df-cntz 19230 df-cmn 19699 df-psmet 21227 df-xmet 21228 df-met 21229 df-bl 21230 df-mopn 21231 df-fbas 21232 df-fg 21233 df-cnfld 21236 df-top 22746 df-topon 22763 df-topsp 22785 df-bases 22799 df-cld 22873 df-ntr 22874 df-cls 22875 df-nei 22952 df-cn 23081 df-cnp 23082 df-lm 23083 df-haus 23169 df-tx 23416 df-hmeo 23609 df-fil 23700 df-fm 23792 df-flim 23793 df-flf 23794 df-xms 24176 df-ms 24177 df-tms 24178 df-cfil 25133 df-cau 25134 df-cmet 25135 df-grpo 30250 df-gid 30251 df-ginv 30252 df-gdiv 30253 df-ablo 30302 df-vc 30316 df-nv 30349 df-va 30352 df-ba 30353 df-sm 30354 df-0v 30355 df-vs 30356 df-nmcv 30357 df-ims 30358 df-dip 30458 df-ssp 30479 df-lno 30501 df-nmoo 30502 df-0o 30504 df-ph 30570 df-cbn 30620 df-hnorm 30725 df-hba 30726 df-hvsub 30728 df-hlim 30729 df-hcau 30730 df-sh 30964 df-ch 30978 df-oc 31009 df-ch0 31010 df-shs 31065 df-pjh 31152 df-h0op 31505 df-nmop 31596 df-lnop 31598 df-hmop 31601 |
This theorem is referenced by: nmopcoi 31852 |
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