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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 24550 | . . 3 ⊢ (𝑆 ∈ NrmMod → 𝑆 ∈ LMod) | |
2 | nlmlmod 24550 | . . 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 20888 | . . 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 24549 | . . . 4 ⊢ (𝑆 ∈ NrmMod → 𝑆 ∈ NrmGrp) | |
11 | nlmngp 24549 | . . . 4 ⊢ (𝑇 ∈ NrmMod → 𝑇 ∈ NrmGrp) | |
12 | 5, 4 | 0nghm 24613 | . . . 4 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (𝑉 × { 0 }) ∈ (𝑆 NGHom 𝑇)) |
13 | 10, 11, 12 | syl2an 595 | . . 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 24618 | . . . 4 ⊢ ((𝑉 × { 0 }) ∈ (𝑆 NMHom 𝑇) ↔ ((𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod) ∧ ((𝑉 × { 0 }) ∈ (𝑆 LMHom 𝑇) ∧ (𝑉 × { 0 }) ∈ (𝑆 NGHom 𝑇)))) | |
16 | 15 | baib 535 | . . 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 710 | 1 ⊢ ((𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod ∧ 𝐹 = 𝐺) → (𝑉 × { 0 }) ∈ (𝑆 NMHom 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 {csn 4623 × cxp 5667 ‘cfv 6537 (class class class)co 7405 Basecbs 17153 Scalarcsca 17209 0gc0g 17394 LModclmod 20706 LMHom clmhm 20867 NrmGrpcngp 24441 NrmModcnlm 24444 NGHom cnghm 24578 NMHom cnmhm 24579 |
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 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 |
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 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-sup 9439 df-inf 9440 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-n0 12477 df-z 12563 df-uz 12827 df-q 12937 df-rp 12981 df-xneg 13098 df-xadd 13099 df-xmul 13100 df-ico 13336 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-plusg 17219 df-0g 17396 df-topgen 17398 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-mhm 18713 df-grp 18866 df-minusg 18867 df-ghm 19139 df-cmn 19702 df-abl 19703 df-mgp 20040 df-rng 20058 df-ur 20087 df-ring 20140 df-lmod 20708 df-lmhm 20870 df-psmet 21232 df-xmet 21233 df-met 21234 df-bl 21235 df-mopn 21236 df-top 22751 df-topon 22768 df-topsp 22790 df-bases 22804 df-xms 24181 df-ms 24182 df-nm 24446 df-ngp 24447 df-nlm 24450 df-nmo 24580 df-nghm 24581 df-nmhm 24582 |
This theorem is referenced by: (None) |
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