frlmlss - Metamath Proof Explorer

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Theorem frlmlss 21702

Description: The base set of the free module is a subspace of the power module. (Contributed by Stefan O'Rear, 1-Feb-2015.)

**Hypotheses**
Ref	Expression
frlmval.f	⊢ 𝐹 = (𝑅 freeLMod 𝐼)
frlmpws.b	⊢ 𝐵 = (Base‘𝐹)
frlmlss.u	⊢ 𝑈 = (LSubSp‘((ringLMod‘𝑅) ↑_s 𝐼))

**Assertion**
Ref	Expression
frlmlss	⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → 𝐵 ∈ 𝑈)

Proof of Theorem frlmlss Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		frlmpws.b	. . 3 ⊢ 𝐵 = (Base‘𝐹)
2		frlmval.f	. . . . 5 ⊢ 𝐹 = (𝑅 freeLMod 𝐼)
3	2	frlmval 21699	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → 𝐹 = (𝑅 ⊕_m (𝐼 × {(ringLMod‘𝑅)})))
4	3	fveq2d 6900	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → (Base‘𝐹) = (Base‘(𝑅 ⊕_m (𝐼 × {(ringLMod‘𝑅)}))))
5	1, 4	eqtrid 2777	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → 𝐵 = (Base‘(𝑅 ⊕_m (𝐼 × {(ringLMod‘𝑅)}))))
6		simpr 483	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → 𝐼 ∈ 𝑊)
7		simpl 481	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → 𝑅 ∈ Ring)
8		rlmlmod 21108	. . . . . 6 ⊢ (𝑅 ∈ Ring → (ringLMod‘𝑅) ∈ LMod)
9	8	adantr 479	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → (ringLMod‘𝑅) ∈ LMod)
10		fconst6g 6786	. . . . 5 ⊢ ((ringLMod‘𝑅) ∈ LMod → (𝐼 × {(ringLMod‘𝑅)}):𝐼⟶LMod)
11	9, 10	syl 17	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → (𝐼 × {(ringLMod‘𝑅)}):𝐼⟶LMod)
12		fvex 6909	. . . . . . . 8 ⊢ (ringLMod‘𝑅) ∈ V
13	12	fvconst2 7216	. . . . . . 7 ⊢ (𝑖 ∈ 𝐼 → ((𝐼 × {(ringLMod‘𝑅)})‘𝑖) = (ringLMod‘𝑅))
14	13	adantl 480	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) ∧ 𝑖 ∈ 𝐼) → ((𝐼 × {(ringLMod‘𝑅)})‘𝑖) = (ringLMod‘𝑅))
15	14	fveq2d 6900	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) ∧ 𝑖 ∈ 𝐼) → (Scalar‘((𝐼 × {(ringLMod‘𝑅)})‘𝑖)) = (Scalar‘(ringLMod‘𝑅)))
16		rlmsca 21103	. . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘(ringLMod‘𝑅)))
17	16	ad2antrr 724	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) ∧ 𝑖 ∈ 𝐼) → 𝑅 = (Scalar‘(ringLMod‘𝑅)))
18	15, 17	eqtr4d 2768	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) ∧ 𝑖 ∈ 𝐼) → (Scalar‘((𝐼 × {(ringLMod‘𝑅)})‘𝑖)) = 𝑅)
19		eqid 2725	. . . 4 ⊢ (𝑅X_s(𝐼 × {(ringLMod‘𝑅)})) = (𝑅X_s(𝐼 × {(ringLMod‘𝑅)}))
20		eqid 2725	. . . 4 ⊢ (LSubSp‘(𝑅X_s(𝐼 × {(ringLMod‘𝑅)}))) = (LSubSp‘(𝑅X_s(𝐼 × {(ringLMod‘𝑅)})))
21		eqid 2725	. . . 4 ⊢ (Base‘(𝑅 ⊕_m (𝐼 × {(ringLMod‘𝑅)}))) = (Base‘(𝑅 ⊕_m (𝐼 × {(ringLMod‘𝑅)})))
22	6, 7, 11, 18, 19, 20, 21	dsmmlss 21695	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → (Base‘(𝑅 ⊕_m (𝐼 × {(ringLMod‘𝑅)}))) ∈ (LSubSp‘(𝑅X_s(𝐼 × {(ringLMod‘𝑅)}))))
23		eqid 2725	. . . . . . . . 9 ⊢ ((ringLMod‘𝑅) ↑_s 𝐼) = ((ringLMod‘𝑅) ↑_s 𝐼)
24		eqid 2725	. . . . . . . . 9 ⊢ (Scalar‘(ringLMod‘𝑅)) = (Scalar‘(ringLMod‘𝑅))
25	23, 24	pwsval 17471	. . . . . . . 8 ⊢ (((ringLMod‘𝑅) ∈ V ∧ 𝐼 ∈ 𝑊) → ((ringLMod‘𝑅) ↑_s 𝐼) = ((Scalar‘(ringLMod‘𝑅))X_s(𝐼 × {(ringLMod‘𝑅)})))
26	12, 25	mpan 688	. . . . . . 7 ⊢ (𝐼 ∈ 𝑊 → ((ringLMod‘𝑅) ↑_s 𝐼) = ((Scalar‘(ringLMod‘𝑅))X_s(𝐼 × {(ringLMod‘𝑅)})))
27	26	adantl 480	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → ((ringLMod‘𝑅) ↑_s 𝐼) = ((Scalar‘(ringLMod‘𝑅))X_s(𝐼 × {(ringLMod‘𝑅)})))
28	16	eqcomd 2731	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → (Scalar‘(ringLMod‘𝑅)) = 𝑅)
29	28	adantr 479	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → (Scalar‘(ringLMod‘𝑅)) = 𝑅)
30	29	oveq1d 7434	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → ((Scalar‘(ringLMod‘𝑅))X_s(𝐼 × {(ringLMod‘𝑅)})) = (𝑅X_s(𝐼 × {(ringLMod‘𝑅)})))
31	27, 30	eqtr2d 2766	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → (𝑅X_s(𝐼 × {(ringLMod‘𝑅)})) = ((ringLMod‘𝑅) ↑_s 𝐼))
32	31	fveq2d 6900	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → (LSubSp‘(𝑅X_s(𝐼 × {(ringLMod‘𝑅)}))) = (LSubSp‘((ringLMod‘𝑅) ↑_s 𝐼)))
33		frlmlss.u	. . . 4 ⊢ 𝑈 = (LSubSp‘((ringLMod‘𝑅) ↑_s 𝐼))
34	32, 33	eqtr4di 2783	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → (LSubSp‘(𝑅X_s(𝐼 × {(ringLMod‘𝑅)}))) = 𝑈)
35	22, 34	eleqtrd 2827	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → (Base‘(𝑅 ⊕_m (𝐼 × {(ringLMod‘𝑅)}))) ∈ 𝑈)
36	5, 35	eqeltrd 2825	1 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → 𝐵 ∈ 𝑈)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 Vcvv 3461 {csn 4630 × cxp 5676 ⟶wf 6545 ‘cfv 6549 (class class class)co 7419 Basecbs 17183 Scalarcsca 17239 X_scprds 17430 ↑_s cpws 17431 Ringcrg 20185 LModclmod 20755 LSubSpclss 20827 ringLModcrglmod 21069 ⊕_m cdsmm 21682 freeLMod cfrlm 21697

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 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217

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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-sup 9467 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12506 df-z 12592 df-dec 12711 df-uz 12856 df-fz 13520 df-struct 17119 df-sets 17136 df-slot 17154 df-ndx 17166 df-base 17184 df-ress 17213 df-plusg 17249 df-mulr 17250 df-sca 17252 df-vsca 17253 df-ip 17254 df-tset 17255 df-ple 17256 df-ds 17258 df-hom 17260 df-cco 17261 df-0g 17426 df-prds 17432 df-pws 17434 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-grp 18901 df-minusg 18902 df-sbg 18903 df-subg 19086 df-cmn 19749 df-abl 19750 df-mgp 20087 df-rng 20105 df-ur 20134 df-ring 20187 df-subrg 20520 df-lmod 20757 df-lss 20828 df-sra 21070 df-rgmod 21071 df-dsmm 21683 df-frlm 21698

This theorem is referenced by: frlm0 21705 frlmsubgval 21716 frlmgsum 21723 frlmsplit2 21724

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