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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 2610 | . 2 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀) | |
3 | eqid 2610 | . 2 ⊢ ( ·𝑠 ‘𝑁) = ( ·𝑠 ‘𝑁) | |
4 | 0lmhm.s | . 2 ⊢ 𝑆 = (Scalar‘𝑀) | |
5 | 0lmhm.t | . 2 ⊢ 𝑇 = (Scalar‘𝑁) | |
6 | eqid 2610 | . 2 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
7 | simp1 1054 | . 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑁 ∈ LMod ∧ 𝑆 = 𝑇) → 𝑀 ∈ LMod) | |
8 | simp2 1055 | . 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑁 ∈ LMod ∧ 𝑆 = 𝑇) → 𝑁 ∈ LMod) | |
9 | simp3 1056 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑁 ∈ LMod ∧ 𝑆 = 𝑇) → 𝑆 = 𝑇) | |
10 | 9 | eqcomd 2616 | . 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑁 ∈ LMod ∧ 𝑆 = 𝑇) → 𝑇 = 𝑆) |
11 | lmodgrp 18693 | . . . 4 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ Grp) | |
12 | lmodgrp 18693 | . . . 4 ⊢ (𝑁 ∈ LMod → 𝑁 ∈ Grp) | |
13 | 0lmhm.z | . . . . 5 ⊢ 0 = (0g‘𝑁) | |
14 | 13, 1 | 0ghm 17497 | . . . 4 ⊢ ((𝑀 ∈ Grp ∧ 𝑁 ∈ Grp) → (𝐵 × { 0 }) ∈ (𝑀 GrpHom 𝑁)) |
15 | 11, 12, 14 | syl2an 493 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑁 ∈ LMod) → (𝐵 × { 0 }) ∈ (𝑀 GrpHom 𝑁)) |
16 | 15 | 3adant3 1074 | . 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑁 ∈ LMod ∧ 𝑆 = 𝑇) → (𝐵 × { 0 }) ∈ (𝑀 GrpHom 𝑁)) |
17 | simpl2 1058 | . . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑁 ∈ LMod ∧ 𝑆 = 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ 𝐵)) → 𝑁 ∈ LMod) | |
18 | simprl 790 | . . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑁 ∈ LMod ∧ 𝑆 = 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ (Base‘𝑆)) | |
19 | simpl3 1059 | . . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑁 ∈ LMod ∧ 𝑆 = 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ 𝐵)) → 𝑆 = 𝑇) | |
20 | 19 | fveq2d 6107 | . . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑁 ∈ LMod ∧ 𝑆 = 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ 𝐵)) → (Base‘𝑆) = (Base‘𝑇)) |
21 | 18, 20 | eleqtrd 2690 | . . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑁 ∈ LMod ∧ 𝑆 = 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ (Base‘𝑇)) |
22 | eqid 2610 | . . . . 5 ⊢ (Base‘𝑇) = (Base‘𝑇) | |
23 | 5, 3, 22, 13 | lmodvs0 18720 | . . . 4 ⊢ ((𝑁 ∈ LMod ∧ 𝑥 ∈ (Base‘𝑇)) → (𝑥( ·𝑠 ‘𝑁) 0 ) = 0 ) |
24 | 17, 21, 23 | syl2anc 691 | . . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑁 ∈ LMod ∧ 𝑆 = 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝑁) 0 ) = 0 ) |
25 | fvex 6113 | . . . . . . 7 ⊢ (0g‘𝑁) ∈ V | |
26 | 13, 25 | eqeltri 2684 | . . . . . 6 ⊢ 0 ∈ V |
27 | 26 | fvconst2 6374 | . . . . 5 ⊢ (𝑦 ∈ 𝐵 → ((𝐵 × { 0 })‘𝑦) = 0 ) |
28 | 27 | oveq2d 6565 | . . . 4 ⊢ (𝑦 ∈ 𝐵 → (𝑥( ·𝑠 ‘𝑁)((𝐵 × { 0 })‘𝑦)) = (𝑥( ·𝑠 ‘𝑁) 0 )) |
29 | 28 | ad2antll 761 | . . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑁 ∈ LMod ∧ 𝑆 = 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝑁)((𝐵 × { 0 })‘𝑦)) = (𝑥( ·𝑠 ‘𝑁) 0 )) |
30 | simpl1 1057 | . . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑁 ∈ LMod ∧ 𝑆 = 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ 𝐵)) → 𝑀 ∈ LMod) | |
31 | simprr 792 | . . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑁 ∈ LMod ∧ 𝑆 = 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ 𝐵) | |
32 | 1, 4, 2, 6 | lmodvscl 18703 | . . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ 𝐵) → (𝑥( ·𝑠 ‘𝑀)𝑦) ∈ 𝐵) |
33 | 30, 18, 31, 32 | syl3anc 1318 | . . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑁 ∈ LMod ∧ 𝑆 = 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝑀)𝑦) ∈ 𝐵) |
34 | 26 | fvconst2 6374 | . . . 4 ⊢ ((𝑥( ·𝑠 ‘𝑀)𝑦) ∈ 𝐵 → ((𝐵 × { 0 })‘(𝑥( ·𝑠 ‘𝑀)𝑦)) = 0 ) |
35 | 33, 34 | syl 17 | . . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑁 ∈ LMod ∧ 𝑆 = 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ 𝐵)) → ((𝐵 × { 0 })‘(𝑥( ·𝑠 ‘𝑀)𝑦)) = 0 ) |
36 | 24, 29, 35 | 3eqtr4rd 2655 | . 2 ⊢ (((𝑀 ∈ LMod ∧ 𝑁 ∈ LMod ∧ 𝑆 = 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ 𝐵)) → ((𝐵 × { 0 })‘(𝑥( ·𝑠 ‘𝑀)𝑦)) = (𝑥( ·𝑠 ‘𝑁)((𝐵 × { 0 })‘𝑦))) |
37 | 1, 2, 3, 4, 5, 6, 7, 8, 10, 16, 36 | islmhmd 18860 | 1 ⊢ ((𝑀 ∈ LMod ∧ 𝑁 ∈ LMod ∧ 𝑆 = 𝑇) → (𝐵 × { 0 }) ∈ (𝑀 LMHom 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 Vcvv 3173 {csn 4125 × cxp 5036 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 Scalarcsca 15771 ·𝑠 cvsca 15772 0gc0g 15923 Grpcgrp 17245 GrpHom cghm 17480 LModclmod 18686 LMHom clmhm 18840 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-plusg 15781 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-mhm 17158 df-grp 17248 df-ghm 17481 df-mgp 18313 df-ring 18372 df-lmod 18688 df-lmhm 18843 |
This theorem is referenced by: 0nmhm 22369 mendring 36781 |
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