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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 | ⊢ 𝑉 = (Base‘𝑆) |
0nmhm.2 | ⊢ 0 = (0g‘𝑇) |
0nmhm.f | ⊢ 𝐹 = (Scalar‘𝑆) |
0nmhm.g | ⊢ 𝐺 = (Scalar‘𝑇) |
Ref | Expression |
---|---|
0nmhm | ⊢ ((𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod ∧ 𝐹 = 𝐺) → (𝑉 × { 0 }) ∈ (𝑆 NMHom 𝑇)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nlmlmod 23284 | . . 3 ⊢ (𝑆 ∈ NrmMod → 𝑆 ∈ LMod) | |
2 | nlmlmod 23284 | . . 3 ⊢ (𝑇 ∈ NrmMod → 𝑇 ∈ LMod) | |
3 | id 22 | . . 3 ⊢ (𝐹 = 𝐺 → 𝐹 = 𝐺) | |
4 | 0nmhm.2 | . . . 4 ⊢ 0 = (0g‘𝑇) | |
5 | 0nmhm.1 | . . . 4 ⊢ 𝑉 = (Base‘𝑆) | |
6 | 0nmhm.f | . . . 4 ⊢ 𝐹 = (Scalar‘𝑆) | |
7 | 0nmhm.g | . . . 4 ⊢ 𝐺 = (Scalar‘𝑇) | |
8 | 4, 5, 6, 7 | 0lmhm 19805 | . . 3 ⊢ ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod ∧ 𝐹 = 𝐺) → (𝑉 × { 0 }) ∈ (𝑆 LMHom 𝑇)) |
9 | 1, 2, 3, 8 | syl3an 1157 | . 2 ⊢ ((𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod ∧ 𝐹 = 𝐺) → (𝑉 × { 0 }) ∈ (𝑆 LMHom 𝑇)) |
10 | nlmngp 23283 | . . . 4 ⊢ (𝑆 ∈ NrmMod → 𝑆 ∈ NrmGrp) | |
11 | nlmngp 23283 | . . . 4 ⊢ (𝑇 ∈ NrmMod → 𝑇 ∈ NrmGrp) | |
12 | 5, 4 | 0nghm 23347 | . . . 4 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (𝑉 × { 0 }) ∈ (𝑆 NGHom 𝑇)) |
13 | 10, 11, 12 | syl2an 598 | . . 3 ⊢ ((𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod) → (𝑉 × { 0 }) ∈ (𝑆 NGHom 𝑇)) |
14 | 13 | 3adant3 1129 | . 2 ⊢ ((𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod ∧ 𝐹 = 𝐺) → (𝑉 × { 0 }) ∈ (𝑆 NGHom 𝑇)) |
15 | isnmhm 23352 | . . . 4 ⊢ ((𝑉 × { 0 }) ∈ (𝑆 NMHom 𝑇) ↔ ((𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod) ∧ ((𝑉 × { 0 }) ∈ (𝑆 LMHom 𝑇) ∧ (𝑉 × { 0 }) ∈ (𝑆 NGHom 𝑇)))) | |
16 | 15 | baib 539 | . . 3 ⊢ ((𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod) → ((𝑉 × { 0 }) ∈ (𝑆 NMHom 𝑇) ↔ ((𝑉 × { 0 }) ∈ (𝑆 LMHom 𝑇) ∧ (𝑉 × { 0 }) ∈ (𝑆 NGHom 𝑇)))) |
17 | 16 | 3adant3 1129 | . 2 ⊢ ((𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod ∧ 𝐹 = 𝐺) → ((𝑉 × { 0 }) ∈ (𝑆 NMHom 𝑇) ↔ ((𝑉 × { 0 }) ∈ (𝑆 LMHom 𝑇) ∧ (𝑉 × { 0 }) ∈ (𝑆 NGHom 𝑇)))) |
18 | 9, 14, 17 | mpbir2and 712 | 1 ⊢ ((𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod ∧ 𝐹 = 𝐺) → (𝑉 × { 0 }) ∈ (𝑆 NMHom 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 {csn 4525 × cxp 5517 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 Scalarcsca 16560 0gc0g 16705 LModclmod 19627 LMHom clmhm 19784 NrmGrpcngp 23184 NrmModcnlm 23187 NGHom cnghm 23312 NMHom cnmhm 23313 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-sup 8890 df-inf 8891 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-n0 11886 df-z 11970 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ico 12732 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-plusg 16570 df-0g 16707 df-topgen 16709 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-mhm 17948 df-grp 18098 df-ghm 18348 df-mgp 19233 df-ring 19292 df-lmod 19629 df-lmhm 19787 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-mopn 20087 df-top 21499 df-topon 21516 df-topsp 21538 df-bases 21551 df-xms 22927 df-ms 22928 df-nm 23189 df-ngp 23190 df-nlm 23193 df-nmo 23314 df-nghm 23315 df-nmhm 23316 |
This theorem is referenced by: (None) |
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