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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 23286 | . . 3 ⊢ (𝑆 ∈ NrmMod → 𝑆 ∈ LMod) | |
2 | nlmlmod 23286 | . . 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 19811 | . . 3 ⊢ ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod ∧ 𝐹 = 𝐺) → (𝑉 × { 0 }) ∈ (𝑆 LMHom 𝑇)) |
9 | 1, 2, 3, 8 | syl3an 1156 | . 2 ⊢ ((𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod ∧ 𝐹 = 𝐺) → (𝑉 × { 0 }) ∈ (𝑆 LMHom 𝑇)) |
10 | nlmngp 23285 | . . . 4 ⊢ (𝑆 ∈ NrmMod → 𝑆 ∈ NrmGrp) | |
11 | nlmngp 23285 | . . . 4 ⊢ (𝑇 ∈ NrmMod → 𝑇 ∈ NrmGrp) | |
12 | 5, 4 | 0nghm 23349 | . . . 4 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (𝑉 × { 0 }) ∈ (𝑆 NGHom 𝑇)) |
13 | 10, 11, 12 | syl2an 597 | . . 3 ⊢ ((𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod) → (𝑉 × { 0 }) ∈ (𝑆 NGHom 𝑇)) |
14 | 13 | 3adant3 1128 | . 2 ⊢ ((𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod ∧ 𝐹 = 𝐺) → (𝑉 × { 0 }) ∈ (𝑆 NGHom 𝑇)) |
15 | isnmhm 23354 | . . . 4 ⊢ ((𝑉 × { 0 }) ∈ (𝑆 NMHom 𝑇) ↔ ((𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod) ∧ ((𝑉 × { 0 }) ∈ (𝑆 LMHom 𝑇) ∧ (𝑉 × { 0 }) ∈ (𝑆 NGHom 𝑇)))) | |
16 | 15 | baib 538 | . . 3 ⊢ ((𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod) → ((𝑉 × { 0 }) ∈ (𝑆 NMHom 𝑇) ↔ ((𝑉 × { 0 }) ∈ (𝑆 LMHom 𝑇) ∧ (𝑉 × { 0 }) ∈ (𝑆 NGHom 𝑇)))) |
17 | 16 | 3adant3 1128 | . 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 711 | 1 ⊢ ((𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod ∧ 𝐹 = 𝐺) → (𝑉 × { 0 }) ∈ (𝑆 NMHom 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 {csn 4566 × cxp 5552 ‘cfv 6354 (class class class)co 7155 Basecbs 16482 Scalarcsca 16567 0gc0g 16712 LModclmod 19633 LMHom clmhm 19790 NrmGrpcngp 23186 NrmModcnlm 23189 NGHom cnghm 23314 NMHom cnmhm 23315 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-pre-sup 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-er 8288 df-map 8407 df-en 8509 df-dom 8510 df-sdom 8511 df-sup 8905 df-inf 8906 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-nn 11638 df-2 11699 df-n0 11897 df-z 11981 df-uz 12243 df-q 12348 df-rp 12389 df-xneg 12506 df-xadd 12507 df-xmul 12508 df-ico 12743 df-ndx 16485 df-slot 16486 df-base 16488 df-sets 16489 df-plusg 16577 df-0g 16714 df-topgen 16716 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-mhm 17955 df-grp 18105 df-ghm 18355 df-mgp 19239 df-ring 19298 df-lmod 19635 df-lmhm 19793 df-psmet 20536 df-xmet 20537 df-met 20538 df-bl 20539 df-mopn 20540 df-top 21501 df-topon 21518 df-topsp 21540 df-bases 21553 df-xms 22929 df-ms 22930 df-nm 23191 df-ngp 23192 df-nlm 23195 df-nmo 23316 df-nghm 23317 df-nmhm 23318 |
This theorem is referenced by: (None) |
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