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Mirrors > Home > HSE Home > Th. List > bdophdi | Structured version Visualization version GIF version |
Description: The difference between two bounded linear operators is bounded. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nmoptri.1 | โข ๐ โ BndLinOp |
nmoptri.2 | โข ๐ โ BndLinOp |
Ref | Expression |
---|---|
bdophdi | โข (๐ โop ๐) โ BndLinOp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nmoptri.1 | . . . 4 โข ๐ โ BndLinOp | |
2 | bdopf 31619 | . . . 4 โข (๐ โ BndLinOp โ ๐: โโถ โ) | |
3 | 1, 2 | ax-mp 5 | . . 3 โข ๐: โโถ โ |
4 | nmoptri.2 | . . . 4 โข ๐ โ BndLinOp | |
5 | bdopf 31619 | . . . 4 โข (๐ โ BndLinOp โ ๐: โโถ โ) | |
6 | 4, 5 | ax-mp 5 | . . 3 โข ๐: โโถ โ |
7 | 3, 6 | honegsubi 31553 | . 2 โข (๐ +op (-1 ยทop ๐)) = (๐ โop ๐) |
8 | neg1cn 12327 | . . . 4 โข -1 โ โ | |
9 | 4 | bdophmi 31789 | . . . 4 โข (-1 โ โ โ (-1 ยทop ๐) โ BndLinOp) |
10 | 8, 9 | ax-mp 5 | . . 3 โข (-1 ยทop ๐) โ BndLinOp |
11 | 1, 10 | bdophsi 31853 | . 2 โข (๐ +op (-1 ยทop ๐)) โ BndLinOp |
12 | 7, 11 | eqeltrri 2824 | 1 โข (๐ โop ๐) โ BndLinOp |
Colors of variables: wff setvar class |
Syntax hints: โ wcel 2098 โถwf 6532 (class class class)co 7404 โcc 11107 1c1 11110 -cneg 11446 โchba 30676 +op chos 30695 ยทop chot 30696 โop chod 30697 BndLinOpcbo 30705 |
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-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-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 |
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-op 4630 df-uni 4903 df-iun 4992 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-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-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-sup 9436 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-n0 12474 df-z 12560 df-uz 12824 df-rp 12978 df-seq 13970 df-exp 14030 df-cj 15049 df-re 15050 df-im 15051 df-sqrt 15185 df-abs 15186 df-grpo 30250 df-gid 30251 df-ablo 30302 df-vc 30316 df-nv 30349 df-va 30352 df-ba 30353 df-sm 30354 df-0v 30355 df-nmcv 30357 df-hnorm 30725 df-hba 30726 df-hvsub 30728 df-hosum 31487 df-homul 31488 df-hodif 31489 df-nmop 31596 df-lnop 31598 df-bdop 31599 |
This theorem is referenced by: unierri 31861 |
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