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Theorem dsmmlmod 21686

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 2728	. . 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 20860	. 2 ⊢ (𝜑 → (𝑆X_s𝑅) ∈ LMod)
7		eqid 2728	. . 3 ⊢ (LSubSp‘(𝑆X_s𝑅)) = (LSubSp‘(𝑆X_s𝑅))
8		eqid 2728	. . 3 ⊢ (Base‘(𝑆 ⊕_m 𝑅)) = (Base‘(𝑆 ⊕_m 𝑅))
9	3, 2, 4, 5, 1, 7, 8	dsmmlss 21685	. 2 ⊢ (𝜑 → (Base‘(𝑆 ⊕_m 𝑅)) ∈ (LSubSp‘(𝑆X_s𝑅)))
10		dsmmlmod.c	. . . 4 ⊢ 𝐶 = (𝑆 ⊕_m 𝑅)
11	8	dsmmval2 21677	. . . 4 ⊢ (𝑆 ⊕_m 𝑅) = ((𝑆X_s𝑅) ↾_s (Base‘(𝑆 ⊕_m 𝑅)))
12	10, 11	eqtri 2756	. . 3 ⊢ 𝐶 = ((𝑆X_s𝑅) ↾_s (Base‘(𝑆 ⊕_m 𝑅)))
13	12, 7	lsslmod 20851	. 2 ⊢ (((𝑆X_s𝑅) ∈ LMod ∧ (Base‘(𝑆 ⊕_m 𝑅)) ∈ (LSubSp‘(𝑆X_s𝑅))) → 𝐶 ∈ LMod)
14	6, 9, 13	syl2anc 582	1 ⊢ (𝜑 → 𝐶 ∈ LMod)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ⟶wf 6549 ‘cfv 6553 (class class class)co 7426 Basecbs 17187 ↾_s cress 17216 Scalarcsca 17243 X_scprds 17434 Ringcrg 20180 LModclmod 20750 LSubSpclss 20822 ⊕_m cdsmm 21672

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-map 8853 df-ixp 8923 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-sup 9473 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-fz 13525 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-mulr 17254 df-sca 17256 df-vsca 17257 df-ip 17258 df-tset 17259 df-ple 17260 df-ds 17262 df-hom 17264 df-cco 17265 df-0g 17430 df-prds 17436 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-grp 18900 df-minusg 18901 df-sbg 18902 df-subg 19085 df-cmn 19744 df-abl 19745 df-mgp 20082 df-rng 20100 df-ur 20129 df-ring 20182 df-lmod 20752 df-lss 20823 df-dsmm 21673

This theorem is referenced by: frlmlmod 21690

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