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Mirrors > Home > MPE Home > Th. List > 0lmhm | Structured version Visualization version GIF version |
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 2798 | . 2 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀) | |
3 | eqid 2798 | . 2 ⊢ ( ·𝑠 ‘𝑁) = ( ·𝑠 ‘𝑁) | |
4 | 0lmhm.s | . 2 ⊢ 𝑆 = (Scalar‘𝑀) | |
5 | 0lmhm.t | . 2 ⊢ 𝑇 = (Scalar‘𝑁) | |
6 | eqid 2798 | . 2 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
7 | simp1 1133 | . 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑁 ∈ LMod ∧ 𝑆 = 𝑇) → 𝑀 ∈ LMod) | |
8 | simp2 1134 | . 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑁 ∈ LMod ∧ 𝑆 = 𝑇) → 𝑁 ∈ LMod) | |
9 | simp3 1135 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑁 ∈ LMod ∧ 𝑆 = 𝑇) → 𝑆 = 𝑇) | |
10 | 9 | eqcomd 2804 | . 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑁 ∈ LMod ∧ 𝑆 = 𝑇) → 𝑇 = 𝑆) |
11 | lmodgrp 19634 | . . . 4 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ Grp) | |
12 | lmodgrp 19634 | . . . 4 ⊢ (𝑁 ∈ LMod → 𝑁 ∈ Grp) | |
13 | 0lmhm.z | . . . . 5 ⊢ 0 = (0g‘𝑁) | |
14 | 13, 1 | 0ghm 18364 | . . . 4 ⊢ ((𝑀 ∈ Grp ∧ 𝑁 ∈ Grp) → (𝐵 × { 0 }) ∈ (𝑀 GrpHom 𝑁)) |
15 | 11, 12, 14 | syl2an 598 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑁 ∈ LMod) → (𝐵 × { 0 }) ∈ (𝑀 GrpHom 𝑁)) |
16 | 15 | 3adant3 1129 | . 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑁 ∈ LMod ∧ 𝑆 = 𝑇) → (𝐵 × { 0 }) ∈ (𝑀 GrpHom 𝑁)) |
17 | simpl2 1189 | . . . 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 1190 | . . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑁 ∈ LMod ∧ 𝑆 = 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ 𝐵)) → 𝑆 = 𝑇) | |
20 | 19 | fveq2d 6649 | . . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑁 ∈ LMod ∧ 𝑆 = 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ 𝐵)) → (Base‘𝑆) = (Base‘𝑇)) |
21 | 18, 20 | eleqtrd 2892 | . . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑁 ∈ LMod ∧ 𝑆 = 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ (Base‘𝑇)) |
22 | eqid 2798 | . . . . 5 ⊢ (Base‘𝑇) = (Base‘𝑇) | |
23 | 5, 3, 22, 13 | lmodvs0 19661 | . . . 4 ⊢ ((𝑁 ∈ LMod ∧ 𝑥 ∈ (Base‘𝑇)) → (𝑥( ·𝑠 ‘𝑁) 0 ) = 0 ) |
24 | 17, 21, 23 | syl2anc 587 | . . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑁 ∈ LMod ∧ 𝑆 = 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝑁) 0 ) = 0 ) |
25 | 13 | fvexi 6659 | . . . . . 6 ⊢ 0 ∈ V |
26 | 25 | fvconst2 6943 | . . . . 5 ⊢ (𝑦 ∈ 𝐵 → ((𝐵 × { 0 })‘𝑦) = 0 ) |
27 | 26 | oveq2d 7151 | . . . 4 ⊢ (𝑦 ∈ 𝐵 → (𝑥( ·𝑠 ‘𝑁)((𝐵 × { 0 })‘𝑦)) = (𝑥( ·𝑠 ‘𝑁) 0 )) |
28 | 27 | ad2antll 728 | . . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑁 ∈ LMod ∧ 𝑆 = 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝑁)((𝐵 × { 0 })‘𝑦)) = (𝑥( ·𝑠 ‘𝑁) 0 )) |
29 | simpl1 1188 | . . . . 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 19644 | . . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ 𝐵) → (𝑥( ·𝑠 ‘𝑀)𝑦) ∈ 𝐵) |
32 | 29, 18, 30, 31 | syl3anc 1368 | . . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑁 ∈ LMod ∧ 𝑆 = 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝑀)𝑦) ∈ 𝐵) |
33 | 25 | fvconst2 6943 | . . . 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 2844 | . 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 19804 | 1 ⊢ ((𝑀 ∈ LMod ∧ 𝑁 ∈ LMod ∧ 𝑆 = 𝑇) → (𝐵 × { 0 }) ∈ (𝑀 LMHom 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 {csn 4525 × cxp 5517 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 Scalarcsca 16560 ·𝑠 cvsca 16561 0gc0g 16705 Grpcgrp 18095 GrpHom cghm 18347 LModclmod 19627 LMHom clmhm 19784 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-plusg 16570 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-mhm 17948 df-grp 18098 df-ghm 18348 df-mgp 19233 df-ring 19292 df-lmod 19629 df-lmhm 19787 |
This theorem is referenced by: 0nmhm 23361 mendring 40136 |
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