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Mirrors > Home > MPE Home > Th. List > dsmmlmod | Structured version Visualization version GIF version |
Description: The direct sum of a family of modules is a module. See also the remark in [Lang] p. 128. (Contributed by Stefan O'Rear, 11-Jan-2015.) |
Ref | Expression |
---|---|
dsmmlss.i | ⊢ (𝜑 → 𝐼 ∈ 𝑊) |
dsmmlss.s | ⊢ (𝜑 → 𝑆 ∈ Ring) |
dsmmlss.r | ⊢ (𝜑 → 𝑅:𝐼⟶LMod) |
dsmmlss.k | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (Scalar‘(𝑅‘𝑥)) = 𝑆) |
dsmmlmod.c | ⊢ 𝐶 = (𝑆 ⊕m 𝑅) |
Ref | Expression |
---|---|
dsmmlmod | ⊢ (𝜑 → 𝐶 ∈ LMod) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2622 | . . 3 ⊢ (𝑆Xs𝑅) = (𝑆Xs𝑅) | |
2 | dsmmlss.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ Ring) | |
3 | dsmmlss.i | . . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑊) | |
4 | dsmmlss.r | . . 3 ⊢ (𝜑 → 𝑅:𝐼⟶LMod) | |
5 | dsmmlss.k | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (Scalar‘(𝑅‘𝑥)) = 𝑆) | |
6 | 1, 2, 3, 4, 5 | prdslmodd 18969 | . 2 ⊢ (𝜑 → (𝑆Xs𝑅) ∈ LMod) |
7 | eqid 2622 | . . 3 ⊢ (LSubSp‘(𝑆Xs𝑅)) = (LSubSp‘(𝑆Xs𝑅)) | |
8 | eqid 2622 | . . 3 ⊢ (Base‘(𝑆 ⊕m 𝑅)) = (Base‘(𝑆 ⊕m 𝑅)) | |
9 | 3, 2, 4, 5, 1, 7, 8 | dsmmlss 20088 | . 2 ⊢ (𝜑 → (Base‘(𝑆 ⊕m 𝑅)) ∈ (LSubSp‘(𝑆Xs𝑅))) |
10 | dsmmlmod.c | . . . 4 ⊢ 𝐶 = (𝑆 ⊕m 𝑅) | |
11 | 8 | dsmmval2 20080 | . . . 4 ⊢ (𝑆 ⊕m 𝑅) = ((𝑆Xs𝑅) ↾s (Base‘(𝑆 ⊕m 𝑅))) |
12 | 10, 11 | eqtri 2644 | . . 3 ⊢ 𝐶 = ((𝑆Xs𝑅) ↾s (Base‘(𝑆 ⊕m 𝑅))) |
13 | 12, 7 | lsslmod 18960 | . 2 ⊢ (((𝑆Xs𝑅) ∈ LMod ∧ (Base‘(𝑆 ⊕m 𝑅)) ∈ (LSubSp‘(𝑆Xs𝑅))) → 𝐶 ∈ LMod) |
14 | 6, 9, 13 | syl2anc 693 | 1 ⊢ (𝜑 → 𝐶 ∈ LMod) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 ↾s cress 15858 Scalarcsca 15944 Xscprds 16106 Ringcrg 18547 LModclmod 18863 LSubSpclss 18932 ⊕m cdsmm 20075 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-fz 12327 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-hom 15966 df-cco 15967 df-0g 16102 df-prds 16108 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 df-sbg 17427 df-subg 17591 df-mgp 18490 df-ur 18502 df-ring 18549 df-lmod 18865 df-lss 18933 df-dsmm 20076 |
This theorem is referenced by: frlmlmod 20093 |
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