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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 23289 | . . 3 ⊢ (𝑆 ∈ NrmMod → 𝑆 ∈ LMod) | |
2 | nlmlmod 23289 | . . 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 19814 | . . 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 23288 | . . . 4 ⊢ (𝑆 ∈ NrmMod → 𝑆 ∈ NrmGrp) | |
11 | nlmngp 23288 | . . . 4 ⊢ (𝑇 ∈ NrmMod → 𝑇 ∈ NrmGrp) | |
12 | 5, 4 | 0nghm 23352 | . . . 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 23357 | . . . 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 1537 ∈ wcel 2114 {csn 4569 × cxp 5555 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 Scalarcsca 16570 0gc0g 16715 LModclmod 19636 LMHom clmhm 19793 NrmGrpcngp 23189 NrmModcnlm 23192 NGHom cnghm 23317 NMHom cnmhm 23318 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-sup 8908 df-inf 8909 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-n0 11901 df-z 11985 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-ico 12747 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-plusg 16580 df-0g 16717 df-topgen 16719 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-mhm 17958 df-grp 18108 df-ghm 18358 df-mgp 19242 df-ring 19301 df-lmod 19638 df-lmhm 19796 df-psmet 20539 df-xmet 20540 df-met 20541 df-bl 20542 df-mopn 20543 df-top 21504 df-topon 21521 df-topsp 21543 df-bases 21556 df-xms 22932 df-ms 22933 df-nm 23194 df-ngp 23195 df-nlm 23198 df-nmo 23319 df-nghm 23320 df-nmhm 23321 |
This theorem is referenced by: (None) |
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