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Mirrors > Home > MPE Home > Th. List > 0nghm | Structured version Visualization version GIF version |
Description: The zero operator is a normed group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.) |
Ref | Expression |
---|---|
0nghm.2 | ⊢ 𝑉 = (Base‘𝑆) |
0nghm.3 | ⊢ 0 = (0g‘𝑇) |
Ref | Expression |
---|---|
0nghm | ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (𝑉 × { 0 }) ∈ (𝑆 NGHom 𝑇)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2738 | . . . 4 ⊢ (𝑆 normOp 𝑇) = (𝑆 normOp 𝑇) | |
2 | 0nghm.2 | . . . 4 ⊢ 𝑉 = (Base‘𝑆) | |
3 | 0nghm.3 | . . . 4 ⊢ 0 = (0g‘𝑇) | |
4 | 1, 2, 3 | nmo0 23887 | . . 3 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → ((𝑆 normOp 𝑇)‘(𝑉 × { 0 })) = 0) |
5 | 0re 10965 | . . 3 ⊢ 0 ∈ ℝ | |
6 | 4, 5 | eqeltrdi 2847 | . 2 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → ((𝑆 normOp 𝑇)‘(𝑉 × { 0 })) ∈ ℝ) |
7 | ngpgrp 23743 | . . . 4 ⊢ (𝑆 ∈ NrmGrp → 𝑆 ∈ Grp) | |
8 | ngpgrp 23743 | . . . 4 ⊢ (𝑇 ∈ NrmGrp → 𝑇 ∈ Grp) | |
9 | 3, 2 | 0ghm 18836 | . . . 4 ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → (𝑉 × { 0 }) ∈ (𝑆 GrpHom 𝑇)) |
10 | 7, 8, 9 | syl2an 596 | . . 3 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (𝑉 × { 0 }) ∈ (𝑆 GrpHom 𝑇)) |
11 | 1 | isnghm2 23876 | . . 3 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ (𝑉 × { 0 }) ∈ (𝑆 GrpHom 𝑇)) → ((𝑉 × { 0 }) ∈ (𝑆 NGHom 𝑇) ↔ ((𝑆 normOp 𝑇)‘(𝑉 × { 0 })) ∈ ℝ)) |
12 | 10, 11 | mpd3an3 1461 | . 2 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → ((𝑉 × { 0 }) ∈ (𝑆 NGHom 𝑇) ↔ ((𝑆 normOp 𝑇)‘(𝑉 × { 0 })) ∈ ℝ)) |
13 | 6, 12 | mpbird 256 | 1 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (𝑉 × { 0 }) ∈ (𝑆 NGHom 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 {csn 4562 × cxp 5583 ‘cfv 6427 (class class class)co 7268 ℝcr 10858 0cc0 10859 Basecbs 16900 0gc0g 17138 Grpcgrp 18565 GrpHom cghm 18819 NrmGrpcngp 23721 normOp cnmo 23857 NGHom cnghm 23858 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5222 ax-nul 5229 ax-pow 5287 ax-pr 5351 ax-un 7579 ax-cnex 10915 ax-resscn 10916 ax-1cn 10917 ax-icn 10918 ax-addcl 10919 ax-addrcl 10920 ax-mulcl 10921 ax-mulrcl 10922 ax-mulcom 10923 ax-addass 10924 ax-mulass 10925 ax-distr 10926 ax-i2m1 10927 ax-1ne0 10928 ax-1rid 10929 ax-rnegex 10930 ax-rrecex 10931 ax-cnre 10932 ax-pre-lttri 10933 ax-pre-lttrn 10934 ax-pre-ltadd 10935 ax-pre-mulgt0 10936 ax-pre-sup 10937 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rmo 3072 df-rab 3073 df-v 3432 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4258 df-if 4461 df-pw 4536 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4841 df-iun 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5485 df-eprel 5491 df-po 5499 df-so 5500 df-fr 5540 df-we 5542 df-xp 5591 df-rel 5592 df-cnv 5593 df-co 5594 df-dm 5595 df-rn 5596 df-res 5597 df-ima 5598 df-pred 6196 df-ord 6263 df-on 6264 df-lim 6265 df-suc 6266 df-iota 6385 df-fun 6429 df-fn 6430 df-f 6431 df-f1 6432 df-fo 6433 df-f1o 6434 df-fv 6435 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-om 7704 df-1st 7821 df-2nd 7822 df-frecs 8085 df-wrecs 8116 df-recs 8190 df-rdg 8229 df-er 8486 df-map 8605 df-en 8722 df-dom 8723 df-sdom 8724 df-sup 9189 df-inf 9190 df-pnf 10999 df-mnf 11000 df-xr 11001 df-ltxr 11002 df-le 11003 df-sub 11195 df-neg 11196 df-div 11621 df-nn 11962 df-2 12024 df-n0 12222 df-z 12308 df-uz 12571 df-q 12677 df-rp 12719 df-xneg 12836 df-xadd 12837 df-xmul 12838 df-ico 13073 df-0g 17140 df-topgen 17142 df-mgm 18314 df-sgrp 18363 df-mnd 18374 df-mhm 18418 df-grp 18568 df-ghm 18820 df-psmet 20577 df-xmet 20578 df-met 20579 df-bl 20580 df-mopn 20581 df-top 22031 df-topon 22048 df-topsp 22070 df-bases 22084 df-xms 23461 df-ms 23462 df-nm 23726 df-ngp 23727 df-nmo 23860 df-nghm 23861 |
This theorem is referenced by: 0nmhm 23907 |
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