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Theorem dsmmlmod 21727

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.)

**Hypotheses**
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)

**Assertion**
Ref	Expression
dsmmlmod	⊢ (𝜑 → 𝐶 ∈ LMod)

Distinct variable groups: 𝜑,𝑥 𝑥,𝑆 𝑥,𝑅 𝑥,𝐼 Allowed substitution hints: 𝐶(𝑥) 𝑊(𝑥)

**Proof of Theorem dsmmlmod**
Step	Hyp	Ref	Expression
1		eqid 2740	. . 3 ⊢ (𝑆X_s𝑅) = (𝑆X_s𝑅)
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 20966	. 2 ⊢ (𝜑 → (𝑆X_s𝑅) ∈ LMod)
7		eqid 2740	. . 3 ⊢ (LSubSp‘(𝑆X_s𝑅)) = (LSubSp‘(𝑆X_s𝑅))
8		eqid 2740	. . 3 ⊢ (Base‘(𝑆 ⊕_m 𝑅)) = (Base‘(𝑆 ⊕_m 𝑅))
9	3, 2, 4, 5, 1, 7, 8	dsmmlss 21726	. 2 ⊢ (𝜑 → (Base‘(𝑆 ⊕_m 𝑅)) ∈ (LSubSp‘(𝑆X_s𝑅)))
10		dsmmlmod.c	. . . 4 ⊢ 𝐶 = (𝑆 ⊕_m 𝑅)
11	8	dsmmval2 21718	. . . 4 ⊢ (𝑆 ⊕_m 𝑅) = ((𝑆X_s𝑅) ↾_s (Base‘(𝑆 ⊕_m 𝑅)))
12	10, 11	eqtri 2763	. . 3 ⊢ 𝐶 = ((𝑆X_s𝑅) ↾_s (Base‘(𝑆 ⊕_m 𝑅)))
13	12, 7	lsslmod 20957	. 2 ⊢ (((𝑆X_s𝑅) ∈ LMod ∧ (Base‘(𝑆 ⊕_m 𝑅)) ∈ (LSubSp‘(𝑆X_s𝑅))) → 𝐶 ∈ LMod)
14	6, 9, 13	syl2anc 590	1 ⊢ (𝜑 → 𝐶 ∈ LMod)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1547 ∈ wcel 2119 ⟶wf 6488 ‘cfv 6492 (class class class)co 7363 Basecbs 17177 ↾_s cress 17198 Scalarcsca 17221 X_scprds 17406 Ringcrg 20212 LModclmod 20857 LSubSpclss 20928 ⊕_m cdsmm 21713

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-rep 5206 ax-sep 5225 ax-nul 5235 ax-pow 5301 ax-pr 5369 ax-un 7685 ax-cnex 11092 ax-resscn 11093 ax-1cn 11094 ax-icn 11095 ax-addcl 11096 ax-addrcl 11097 ax-mulcl 11098 ax-mulrcl 11099 ax-mulcom 11100 ax-addass 11101 ax-mulass 11102 ax-distr 11103 ax-i2m1 11104 ax-1ne0 11105 ax-1rid 11106 ax-rnegex 11107 ax-rrecex 11108 ax-cnre 11109 ax-pre-lttri 11110 ax-pre-lttrn 11111 ax-pre-ltadd 11112 ax-pre-mulgt0 11113

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-nel 3040 df-ral 3055 df-rex 3065 df-rmo 3345 df-reu 3346 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-tp 4567 df-op 4569 df-uni 4846 df-iun 4930 df-br 5080 df-opab 5142 df-mpt 5161 df-tr 5187 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7320 df-ov 7366 df-oprab 7367 df-mpo 7368 df-om 7814 df-1st 7938 df-2nd 7939 df-frecs 8228 df-wrecs 8259 df-recs 8308 df-rdg 8346 df-1o 8402 df-er 8640 df-map 8772 df-ixp 8843 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-sup 9352 df-pnf 11179 df-mnf 11180 df-xr 11181 df-ltxr 11182 df-le 11183 df-sub 11377 df-neg 11378 df-nn 12173 df-2 12242 df-3 12243 df-4 12244 df-5 12245 df-6 12246 df-7 12247 df-8 12248 df-9 12249 df-n0 12436 df-z 12523 df-dec 12643 df-uz 12787 df-fz 13460 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17178 df-ress 17199 df-plusg 17231 df-mulr 17232 df-sca 17234 df-vsca 17235 df-ip 17236 df-tset 17237 df-ple 17238 df-ds 17240 df-hom 17242 df-cco 17243 df-0g 17402 df-prds 17408 df-mgm 18606 df-sgrp 18685 df-mnd 18701 df-grp 18910 df-minusg 18911 df-sbg 18912 df-subg 19097 df-cmn 19755 df-abl 19756 df-mgp 20120 df-rng 20132 df-ur 20161 df-ring 20214 df-lmod 20859 df-lss 20929 df-dsmm 21714

This theorem is referenced by: frlmlmod 21731

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