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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 2610 | . . 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 18790 | . 2 ⊢ (𝜑 → (𝑆Xs𝑅) ∈ LMod) |
7 | eqid 2610 | . . 3 ⊢ (LSubSp‘(𝑆Xs𝑅)) = (LSubSp‘(𝑆Xs𝑅)) | |
8 | eqid 2610 | . . 3 ⊢ (Base‘(𝑆 ⊕m 𝑅)) = (Base‘(𝑆 ⊕m 𝑅)) | |
9 | 3, 2, 4, 5, 1, 7, 8 | dsmmlss 19907 | . 2 ⊢ (𝜑 → (Base‘(𝑆 ⊕m 𝑅)) ∈ (LSubSp‘(𝑆Xs𝑅))) |
10 | dsmmlmod.c | . . . 4 ⊢ 𝐶 = (𝑆 ⊕m 𝑅) | |
11 | 8 | dsmmval2 19899 | . . . 4 ⊢ (𝑆 ⊕m 𝑅) = ((𝑆Xs𝑅) ↾s (Base‘(𝑆 ⊕m 𝑅))) |
12 | 10, 11 | eqtri 2632 | . . 3 ⊢ 𝐶 = ((𝑆Xs𝑅) ↾s (Base‘(𝑆 ⊕m 𝑅))) |
13 | 12, 7 | lsslmod 18781 | . 2 ⊢ (((𝑆Xs𝑅) ∈ LMod ∧ (Base‘(𝑆 ⊕m 𝑅)) ∈ (LSubSp‘(𝑆Xs𝑅))) → 𝐶 ∈ LMod) |
14 | 6, 9, 13 | syl2anc 691 | 1 ⊢ (𝜑 → 𝐶 ∈ LMod) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 ↾s cress 15696 Scalarcsca 15771 Xscprds 15929 Ringcrg 18370 LModclmod 18686 LSubSpclss 18753 ⊕m cdsmm 19894 |
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-int 4411 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-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 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-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-fz 12198 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-hom 15793 df-cco 15794 df-0g 15925 df-prds 15931 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-minusg 17249 df-sbg 17250 df-subg 17414 df-mgp 18313 df-ur 18325 df-ring 18372 df-lmod 18688 df-lss 18754 df-dsmm 19895 |
This theorem is referenced by: frlmlmod 19912 |
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