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| Description: The constant zero linear function between two modules. (Contributed by Stefan O'Rear, 5-Sep-2015.) | 
| Ref | Expression | 
|---|---|
| 0lmhm.z | ⊢ 0 = (0g‘𝑁) | 
| 0lmhm.b | ⊢ 𝐵 = (Base‘𝑀) | 
| 0lmhm.s | ⊢ 𝑆 = (Scalar‘𝑀) | 
| 0lmhm.t | ⊢ 𝑇 = (Scalar‘𝑁) | 
| Ref | Expression | 
|---|---|
| 0lmhm | ⊢ ((𝑀 ∈ LMod ∧ 𝑁 ∈ LMod ∧ 𝑆 = 𝑇) → (𝐵 × { 0 }) ∈ (𝑀 LMHom 𝑁)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | 0lmhm.b | . 2 ⊢ 𝐵 = (Base‘𝑀) | |
| 2 | eqid 2736 | . 2 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀) | |
| 3 | eqid 2736 | . 2 ⊢ ( ·𝑠 ‘𝑁) = ( ·𝑠 ‘𝑁) | |
| 4 | 0lmhm.s | . 2 ⊢ 𝑆 = (Scalar‘𝑀) | |
| 5 | 0lmhm.t | . 2 ⊢ 𝑇 = (Scalar‘𝑁) | |
| 6 | eqid 2736 | . 2 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
| 7 | simp1 1136 | . 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑁 ∈ LMod ∧ 𝑆 = 𝑇) → 𝑀 ∈ LMod) | |
| 8 | simp2 1137 | . 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑁 ∈ LMod ∧ 𝑆 = 𝑇) → 𝑁 ∈ LMod) | |
| 9 | simp3 1138 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑁 ∈ LMod ∧ 𝑆 = 𝑇) → 𝑆 = 𝑇) | |
| 10 | 9 | eqcomd 2742 | . 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑁 ∈ LMod ∧ 𝑆 = 𝑇) → 𝑇 = 𝑆) | 
| 11 | lmodgrp 20866 | . . . 4 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ Grp) | |
| 12 | lmodgrp 20866 | . . . 4 ⊢ (𝑁 ∈ LMod → 𝑁 ∈ Grp) | |
| 13 | 0lmhm.z | . . . . 5 ⊢ 0 = (0g‘𝑁) | |
| 14 | 13, 1 | 0ghm 19249 | . . . 4 ⊢ ((𝑀 ∈ Grp ∧ 𝑁 ∈ Grp) → (𝐵 × { 0 }) ∈ (𝑀 GrpHom 𝑁)) | 
| 15 | 11, 12, 14 | syl2an 596 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑁 ∈ LMod) → (𝐵 × { 0 }) ∈ (𝑀 GrpHom 𝑁)) | 
| 16 | 15 | 3adant3 1132 | . 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑁 ∈ LMod ∧ 𝑆 = 𝑇) → (𝐵 × { 0 }) ∈ (𝑀 GrpHom 𝑁)) | 
| 17 | simpl2 1192 | . . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑁 ∈ LMod ∧ 𝑆 = 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ 𝐵)) → 𝑁 ∈ LMod) | |
| 18 | simprl 770 | . . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑁 ∈ LMod ∧ 𝑆 = 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ (Base‘𝑆)) | |
| 19 | simpl3 1193 | . . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑁 ∈ LMod ∧ 𝑆 = 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ 𝐵)) → 𝑆 = 𝑇) | |
| 20 | 19 | fveq2d 6909 | . . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑁 ∈ LMod ∧ 𝑆 = 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ 𝐵)) → (Base‘𝑆) = (Base‘𝑇)) | 
| 21 | 18, 20 | eleqtrd 2842 | . . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑁 ∈ LMod ∧ 𝑆 = 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ (Base‘𝑇)) | 
| 22 | eqid 2736 | . . . . 5 ⊢ (Base‘𝑇) = (Base‘𝑇) | |
| 23 | 5, 3, 22, 13 | lmodvs0 20895 | . . . 4 ⊢ ((𝑁 ∈ LMod ∧ 𝑥 ∈ (Base‘𝑇)) → (𝑥( ·𝑠 ‘𝑁) 0 ) = 0 ) | 
| 24 | 17, 21, 23 | syl2anc 584 | . . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑁 ∈ LMod ∧ 𝑆 = 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝑁) 0 ) = 0 ) | 
| 25 | 13 | fvexi 6919 | . . . . . 6 ⊢ 0 ∈ V | 
| 26 | 25 | fvconst2 7225 | . . . . 5 ⊢ (𝑦 ∈ 𝐵 → ((𝐵 × { 0 })‘𝑦) = 0 ) | 
| 27 | 26 | oveq2d 7448 | . . . 4 ⊢ (𝑦 ∈ 𝐵 → (𝑥( ·𝑠 ‘𝑁)((𝐵 × { 0 })‘𝑦)) = (𝑥( ·𝑠 ‘𝑁) 0 )) | 
| 28 | 27 | ad2antll 729 | . . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑁 ∈ LMod ∧ 𝑆 = 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝑁)((𝐵 × { 0 })‘𝑦)) = (𝑥( ·𝑠 ‘𝑁) 0 )) | 
| 29 | simpl1 1191 | . . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑁 ∈ LMod ∧ 𝑆 = 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ 𝐵)) → 𝑀 ∈ LMod) | |
| 30 | simprr 772 | . . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑁 ∈ LMod ∧ 𝑆 = 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ 𝐵) | |
| 31 | 1, 4, 2, 6 | lmodvscl 20877 | . . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ 𝐵) → (𝑥( ·𝑠 ‘𝑀)𝑦) ∈ 𝐵) | 
| 32 | 29, 18, 30, 31 | syl3anc 1372 | . . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑁 ∈ LMod ∧ 𝑆 = 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝑀)𝑦) ∈ 𝐵) | 
| 33 | 25 | fvconst2 7225 | . . . 4 ⊢ ((𝑥( ·𝑠 ‘𝑀)𝑦) ∈ 𝐵 → ((𝐵 × { 0 })‘(𝑥( ·𝑠 ‘𝑀)𝑦)) = 0 ) | 
| 34 | 32, 33 | syl 17 | . . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑁 ∈ LMod ∧ 𝑆 = 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ 𝐵)) → ((𝐵 × { 0 })‘(𝑥( ·𝑠 ‘𝑀)𝑦)) = 0 ) | 
| 35 | 24, 28, 34 | 3eqtr4rd 2787 | . 2 ⊢ (((𝑀 ∈ LMod ∧ 𝑁 ∈ LMod ∧ 𝑆 = 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ 𝐵)) → ((𝐵 × { 0 })‘(𝑥( ·𝑠 ‘𝑀)𝑦)) = (𝑥( ·𝑠 ‘𝑁)((𝐵 × { 0 })‘𝑦))) | 
| 36 | 1, 2, 3, 4, 5, 6, 7, 8, 10, 16, 35 | islmhmd 21039 | 1 ⊢ ((𝑀 ∈ LMod ∧ 𝑁 ∈ LMod ∧ 𝑆 = 𝑇) → (𝐵 × { 0 }) ∈ (𝑀 LMHom 𝑁)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 {csn 4625 × cxp 5682 ‘cfv 6560 (class class class)co 7432 Basecbs 17248 Scalarcsca 17301 ·𝑠 cvsca 17302 0gc0g 17485 Grpcgrp 18952 GrpHom cghm 19231 LModclmod 20859 LMHom clmhm 21019 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-er 8746 df-map 8869 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-2 12330 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-plusg 17311 df-0g 17487 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-mhm 18797 df-grp 18955 df-minusg 18956 df-ghm 19232 df-cmn 19801 df-abl 19802 df-mgp 20139 df-rng 20151 df-ur 20180 df-ring 20233 df-lmod 20861 df-lmhm 21022 | 
| This theorem is referenced by: 0nmhm 24777 mendring 43205 | 
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