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Theorem dsmmlmod 21735

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 2737	. . 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 20955	. 2 ⊢ (𝜑 → (𝑆X_s𝑅) ∈ LMod)
7		eqid 2737	. . 3 ⊢ (LSubSp‘(𝑆X_s𝑅)) = (LSubSp‘(𝑆X_s𝑅))
8		eqid 2737	. . 3 ⊢ (Base‘(𝑆 ⊕_m 𝑅)) = (Base‘(𝑆 ⊕_m 𝑅))
9	3, 2, 4, 5, 1, 7, 8	dsmmlss 21734	. 2 ⊢ (𝜑 → (Base‘(𝑆 ⊕_m 𝑅)) ∈ (LSubSp‘(𝑆X_s𝑅)))
10		dsmmlmod.c	. . . 4 ⊢ 𝐶 = (𝑆 ⊕_m 𝑅)
11	8	dsmmval2 21726	. . . 4 ⊢ (𝑆 ⊕_m 𝑅) = ((𝑆X_s𝑅) ↾_s (Base‘(𝑆 ⊕_m 𝑅)))
12	10, 11	eqtri 2760	. . 3 ⊢ 𝐶 = ((𝑆X_s𝑅) ↾_s (Base‘(𝑆 ⊕_m 𝑅)))
13	12, 7	lsslmod 20946	. 2 ⊢ (((𝑆X_s𝑅) ∈ LMod ∧ (Base‘(𝑆 ⊕_m 𝑅)) ∈ (LSubSp‘(𝑆X_s𝑅))) → 𝐶 ∈ LMod)
14	6, 9, 13	syl2anc 585	1 ⊢ (𝜑 → 𝐶 ∈ LMod)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ⟶wf 6488 ‘cfv 6492 (class class class)co 7360 Basecbs 17170 ↾_s cress 17191 Scalarcsca 17214 X_scprds 17399 Ringcrg 20205 LModclmod 20846 LSubSpclss 20917 ⊕_m cdsmm 21721

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 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 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-1o 8398 df-er 8636 df-map 8768 df-ixp 8839 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-sup 9348 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-z 12516 df-dec 12636 df-uz 12780 df-fz 13453 df-struct 17108 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-mulr 17225 df-sca 17227 df-vsca 17228 df-ip 17229 df-tset 17230 df-ple 17231 df-ds 17233 df-hom 17235 df-cco 17236 df-0g 17395 df-prds 17401 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-grp 18903 df-minusg 18904 df-sbg 18905 df-subg 19090 df-cmn 19748 df-abl 19749 df-mgp 20113 df-rng 20125 df-ur 20154 df-ring 20207 df-lmod 20848 df-lss 20918 df-dsmm 21722

This theorem is referenced by: frlmlmod 21739

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