dsmmlmod - Metamath Proof Explorer

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Theorem dsmmlmod 21635

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 2726	. . 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 20813	. 2 ⊢ (𝜑 → (𝑆X_s𝑅) ∈ LMod)
7		eqid 2726	. . 3 ⊢ (LSubSp‘(𝑆X_s𝑅)) = (LSubSp‘(𝑆X_s𝑅))
8		eqid 2726	. . 3 ⊢ (Base‘(𝑆 ⊕_m 𝑅)) = (Base‘(𝑆 ⊕_m 𝑅))
9	3, 2, 4, 5, 1, 7, 8	dsmmlss 21634	. 2 ⊢ (𝜑 → (Base‘(𝑆 ⊕_m 𝑅)) ∈ (LSubSp‘(𝑆X_s𝑅)))
10		dsmmlmod.c	. . . 4 ⊢ 𝐶 = (𝑆 ⊕_m 𝑅)
11	8	dsmmval2 21626	. . . 4 ⊢ (𝑆 ⊕_m 𝑅) = ((𝑆X_s𝑅) ↾_s (Base‘(𝑆 ⊕_m 𝑅)))
12	10, 11	eqtri 2754	. . 3 ⊢ 𝐶 = ((𝑆X_s𝑅) ↾_s (Base‘(𝑆 ⊕_m 𝑅)))
13	12, 7	lsslmod 20804	. 2 ⊢ (((𝑆X_s𝑅) ∈ LMod ∧ (Base‘(𝑆 ⊕_m 𝑅)) ∈ (LSubSp‘(𝑆X_s𝑅))) → 𝐶 ∈ LMod)
14	6, 9, 13	syl2anc 583	1 ⊢ (𝜑 → 𝐶 ∈ LMod)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ⟶wf 6532 ‘cfv 6536 (class class class)co 7404 Basecbs 17150 ↾_s cress 17179 Scalarcsca 17206 X_scprds 17397 Ringcrg 20135 LModclmod 20703 LSubSpclss 20775 ⊕_m cdsmm 21621

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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-er 8702 df-map 8821 df-ixp 8891 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-sup 9436 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 df-n0 12474 df-z 12560 df-dec 12679 df-uz 12824 df-fz 13488 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17151 df-ress 17180 df-plusg 17216 df-mulr 17217 df-sca 17219 df-vsca 17220 df-ip 17221 df-tset 17222 df-ple 17223 df-ds 17225 df-hom 17227 df-cco 17228 df-0g 17393 df-prds 17399 df-mgm 18570 df-sgrp 18649 df-mnd 18665 df-grp 18863 df-minusg 18864 df-sbg 18865 df-subg 19047 df-cmn 19699 df-abl 19700 df-mgp 20037 df-rng 20055 df-ur 20084 df-ring 20137 df-lmod 20705 df-lss 20776 df-dsmm 21622

This theorem is referenced by: frlmlmod 21639

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