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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 23823 | . . 3 ⊢ (𝑆 ∈ NrmMod → 𝑆 ∈ LMod) | |
2 | nlmlmod 23823 | . . 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 20283 | . . 3 ⊢ ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod ∧ 𝐹 = 𝐺) → (𝑉 × { 0 }) ∈ (𝑆 LMHom 𝑇)) |
9 | 1, 2, 3, 8 | syl3an 1158 | . 2 ⊢ ((𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod ∧ 𝐹 = 𝐺) → (𝑉 × { 0 }) ∈ (𝑆 LMHom 𝑇)) |
10 | nlmngp 23822 | . . . 4 ⊢ (𝑆 ∈ NrmMod → 𝑆 ∈ NrmGrp) | |
11 | nlmngp 23822 | . . . 4 ⊢ (𝑇 ∈ NrmMod → 𝑇 ∈ NrmGrp) | |
12 | 5, 4 | 0nghm 23886 | . . . 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 1130 | . 2 ⊢ ((𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod ∧ 𝐹 = 𝐺) → (𝑉 × { 0 }) ∈ (𝑆 NGHom 𝑇)) |
15 | isnmhm 23891 | . . . 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 1130 | . 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 709 | 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 1085 = wceq 1541 ∈ wcel 2109 {csn 4566 × cxp 5586 ‘cfv 6430 (class class class)co 7268 Basecbs 16893 Scalarcsca 16946 0gc0g 17131 LModclmod 20104 LMHom clmhm 20262 NrmGrpcngp 23714 NrmModcnlm 23717 NGHom cnghm 23851 NMHom cnmhm 23852 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-rep 5213 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-cnex 10911 ax-resscn 10912 ax-1cn 10913 ax-icn 10914 ax-addcl 10915 ax-addrcl 10916 ax-mulcl 10917 ax-mulrcl 10918 ax-mulcom 10919 ax-addass 10920 ax-mulass 10921 ax-distr 10922 ax-i2m1 10923 ax-1ne0 10924 ax-1rid 10925 ax-rnegex 10926 ax-rrecex 10927 ax-cnre 10928 ax-pre-lttri 10929 ax-pre-lttrn 10930 ax-pre-ltadd 10931 ax-pre-mulgt0 10932 ax-pre-sup 10933 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3072 df-rmo 3073 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-pss 3910 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4845 df-iun 4931 df-br 5079 df-opab 5141 df-mpt 5162 df-tr 5196 df-id 5488 df-eprel 5494 df-po 5502 df-so 5503 df-fr 5543 df-we 5545 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-pred 6199 df-ord 6266 df-on 6267 df-lim 6268 df-suc 6269 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-om 7701 df-1st 7817 df-2nd 7818 df-frecs 8081 df-wrecs 8112 df-recs 8186 df-rdg 8225 df-er 8472 df-map 8591 df-en 8708 df-dom 8709 df-sdom 8710 df-sup 9162 df-inf 9163 df-pnf 10995 df-mnf 10996 df-xr 10997 df-ltxr 10998 df-le 10999 df-sub 11190 df-neg 11191 df-div 11616 df-nn 11957 df-2 12019 df-n0 12217 df-z 12303 df-uz 12565 df-q 12671 df-rp 12713 df-xneg 12830 df-xadd 12831 df-xmul 12832 df-ico 13067 df-sets 16846 df-slot 16864 df-ndx 16876 df-base 16894 df-plusg 16956 df-0g 17133 df-topgen 17135 df-mgm 18307 df-sgrp 18356 df-mnd 18367 df-mhm 18411 df-grp 18561 df-ghm 18813 df-mgp 19702 df-ring 19766 df-lmod 20106 df-lmhm 20265 df-psmet 20570 df-xmet 20571 df-met 20572 df-bl 20573 df-mopn 20574 df-top 22024 df-topon 22041 df-topsp 22063 df-bases 22077 df-xms 23454 df-ms 23455 df-nm 23719 df-ngp 23720 df-nlm 23723 df-nmo 23853 df-nghm 23854 df-nmhm 23855 |
This theorem is referenced by: (None) |
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