Nearby theorems
|
|
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
| Mirrors > Home > MPE Home > Th. List > 0nmhm | Structured version Visualization version GIF version | ||
| Description: The zero operator is a bounded linear operator. (Contributed by Mario Carneiro, 20-Oct-2015.) |
| Ref | Expression |
|---|---|
| 0nmhm.1 | β’ π = (Baseβπ) |
| 0nmhm.2 | β’ 0 = (0gβπ) |
| 0nmhm.f | β’ πΉ = (Scalarβπ) |
| 0nmhm.g | β’ πΊ = (Scalarβπ) |
| Ref | Expression |
|---|---|
| 0nmhm | β’ ((π β NrmMod β§ π β NrmMod β§ πΉ = πΊ) β (π Γ { 0 }) β (π NMHom π)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nlmlmod 24417 | . . 3 β’ (π β NrmMod β π β LMod) | |
| 2 | nlmlmod 24417 | . . 3 β’ (π β NrmMod β π β LMod) | |
| 3 | id 22 | . . 3 β’ (πΉ = πΊ β πΉ = πΊ) | |
| 4 | 0nmhm.2 | . . . 4 β’ 0 = (0gβπ) | |
| 5 | 0nmhm.1 | . . . 4 β’ π = (Baseβπ) | |
| 6 | 0nmhm.f | . . . 4 β’ πΉ = (Scalarβπ) | |
| 7 | 0nmhm.g | . . . 4 β’ πΊ = (Scalarβπ) | |
| 8 | 4, 5, 6, 7 | 0lmhm 20797 | . . 3 β’ ((π β LMod β§ π β LMod β§ πΉ = πΊ) β (π Γ { 0 }) β (π LMHom π)) |
| 9 | 1, 2, 3, 8 | syl3an 1158 | . 2 β’ ((π β NrmMod β§ π β NrmMod β§ πΉ = πΊ) β (π Γ { 0 }) β (π LMHom π)) |
| 10 | nlmngp 24416 | . . . 4 β’ (π β NrmMod β π β NrmGrp) | |
| 11 | nlmngp 24416 | . . . 4 β’ (π β NrmMod β π β NrmGrp) | |
| 12 | 5, 4 | 0nghm 24480 | . . . 4 β’ ((π β NrmGrp β§ π β NrmGrp) β (π Γ { 0 }) β (π NGHom π)) |
| 13 | 10, 11, 12 | syl2an 594 | . . 3 β’ ((π β NrmMod β§ π β NrmMod) β (π Γ { 0 }) β (π NGHom π)) |
| 14 | 13 | 3adant3 1130 | . 2 β’ ((π β NrmMod β§ π β NrmMod β§ πΉ = πΊ) β (π Γ { 0 }) β (π NGHom π)) |
| 15 | isnmhm 24485 | . . . 4 β’ ((π Γ { 0 }) β (π NMHom π) β ((π β NrmMod β§ π β NrmMod) β§ ((π Γ { 0 }) β (π LMHom π) β§ (π Γ { 0 }) β (π NGHom π)))) | |
| 16 | 15 | baib 534 | . . 3 β’ ((π β NrmMod β§ π β NrmMod) β ((π Γ { 0 }) β (π NMHom π) β ((π Γ { 0 }) β (π LMHom π) β§ (π Γ { 0 }) β (π NGHom π)))) |
| 17 | 16 | 3adant3 1130 | . 2 β’ ((π β NrmMod β§ π β NrmMod β§ πΉ = πΊ) β ((π Γ { 0 }) β (π NMHom π) β ((π Γ { 0 }) β (π LMHom π) β§ (π Γ { 0 }) β (π NGHom π)))) |
| 18 | 9, 14, 17 | mpbir2and 709 | 1 β’ ((π β NrmMod β§ π β NrmMod β§ πΉ = πΊ) β (π Γ { 0 }) β (π NMHom π)) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β wb 205 β§ wa 394 β§ w3a 1085 = wceq 1539 β wcel 2104 {csn 4629 Γ cxp 5675 βcfv 6544 (class class class)co 7413 Basecbs 17150 Scalarcsca 17206 0gc0g 17391 LModclmod 20616 LMHom clmhm 20776 NrmGrpcngp 24308 NrmModcnlm 24311 NGHom cnghm 24445 NMHom cnmhm 24446 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7729 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 ax-pre-sup 11192 |
| This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-er 8707 df-map 8826 df-en 8944 df-dom 8945 df-sdom 8946 df-sup 9441 df-inf 9442 df-pnf 11256 df-mnf 11257 df-xr 11258 df-ltxr 11259 df-le 11260 df-sub 11452 df-neg 11453 df-div 11878 df-nn 12219 df-2 12281 df-n0 12479 df-z 12565 df-uz 12829 df-q 12939 df-rp 12981 df-xneg 13098 df-xadd 13099 df-xmul 13100 df-ico 13336 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17151 df-plusg 17216 df-0g 17393 df-topgen 17395 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-mhm 18707 df-grp 18860 df-minusg 18861 df-ghm 19130 df-cmn 19693 df-abl 19694 df-mgp 20031 df-rng 20049 df-ur 20078 df-ring 20131 df-lmod 20618 df-lmhm 20779 df-psmet 21138 df-xmet 21139 df-met 21140 df-bl 21141 df-mopn 21142 df-top 22618 df-topon 22635 df-topsp 22657 df-bases 22671 df-xms 24048 df-ms 24049 df-nm 24313 df-ngp 24314 df-nlm 24317 df-nmo 24447 df-nghm 24448 df-nmhm 24449 |
| This theorem is referenced by: (None) |
| Copyright terms: Public domain | W3C validator |