dsmmlmod - Metamath Proof Explorer

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Theorem dsmmlmod 21523

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 2731	. . 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 20728	. 2 ⊢ (𝜑 → (𝑆X_s𝑅) ∈ LMod)
7		eqid 2731	. . 3 ⊢ (LSubSp‘(𝑆X_s𝑅)) = (LSubSp‘(𝑆X_s𝑅))
8		eqid 2731	. . 3 ⊢ (Base‘(𝑆 ⊕_m 𝑅)) = (Base‘(𝑆 ⊕_m 𝑅))
9	3, 2, 4, 5, 1, 7, 8	dsmmlss 21522	. 2 ⊢ (𝜑 → (Base‘(𝑆 ⊕_m 𝑅)) ∈ (LSubSp‘(𝑆X_s𝑅)))
10		dsmmlmod.c	. . . 4 ⊢ 𝐶 = (𝑆 ⊕_m 𝑅)
11	8	dsmmval2 21514	. . . 4 ⊢ (𝑆 ⊕_m 𝑅) = ((𝑆X_s𝑅) ↾_s (Base‘(𝑆 ⊕_m 𝑅)))
12	10, 11	eqtri 2759	. . 3 ⊢ 𝐶 = ((𝑆X_s𝑅) ↾_s (Base‘(𝑆 ⊕_m 𝑅)))
13	12, 7	lsslmod 20719	. 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 1540 ∈ wcel 2105 ⟶wf 6539 ‘cfv 6543 (class class class)co 7412 Basecbs 17151 ↾_s cress 17180 Scalarcsca 17207 X_scprds 17398 Ringcrg 20131 LModclmod 20618 LSubSpclss 20690 ⊕_m cdsmm 21509

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-map 8828 df-ixp 8898 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-sup 9443 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12480 df-z 12566 df-dec 12685 df-uz 12830 df-fz 13492 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-sca 17220 df-vsca 17221 df-ip 17222 df-tset 17223 df-ple 17224 df-ds 17226 df-hom 17228 df-cco 17229 df-0g 17394 df-prds 17400 df-mgm 18568 df-sgrp 18647 df-mnd 18663 df-grp 18861 df-minusg 18862 df-sbg 18863 df-subg 19043 df-cmn 19695 df-abl 19696 df-mgp 20033 df-rng 20051 df-ur 20080 df-ring 20133 df-lmod 20620 df-lss 20691 df-dsmm 21510

This theorem is referenced by: frlmlmod 21527

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