frlmlss - Metamath Proof Explorer

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Theorem frlmlss 20939

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 20936	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → 𝐹 = (𝑅 ⊕_m (𝐼 × {(ringLMod‘𝑅)})))
4	3	fveq2d 6772	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → (Base‘𝐹) = (Base‘(𝑅 ⊕_m (𝐼 × {(ringLMod‘𝑅)}))))
5	1, 4	eqtrid 2791	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → 𝐵 = (Base‘(𝑅 ⊕_m (𝐼 × {(ringLMod‘𝑅)}))))
6		simpr 484	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → 𝐼 ∈ 𝑊)
7		simpl 482	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → 𝑅 ∈ Ring)
8		rlmlmod 20456	. . . . . 6 ⊢ (𝑅 ∈ Ring → (ringLMod‘𝑅) ∈ LMod)
9	8	adantr 480	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → (ringLMod‘𝑅) ∈ LMod)
10		fconst6g 6659	. . . . 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 6781	. . . . . . . 8 ⊢ (ringLMod‘𝑅) ∈ V
13	12	fvconst2 7073	. . . . . . 7 ⊢ (𝑖 ∈ 𝐼 → ((𝐼 × {(ringLMod‘𝑅)})‘𝑖) = (ringLMod‘𝑅))
14	13	adantl 481	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) ∧ 𝑖 ∈ 𝐼) → ((𝐼 × {(ringLMod‘𝑅)})‘𝑖) = (ringLMod‘𝑅))
15	14	fveq2d 6772	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) ∧ 𝑖 ∈ 𝐼) → (Scalar‘((𝐼 × {(ringLMod‘𝑅)})‘𝑖)) = (Scalar‘(ringLMod‘𝑅)))
16		rlmsca 20451	. . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘(ringLMod‘𝑅)))
17	16	ad2antrr 722	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) ∧ 𝑖 ∈ 𝐼) → 𝑅 = (Scalar‘(ringLMod‘𝑅)))
18	15, 17	eqtr4d 2782	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) ∧ 𝑖 ∈ 𝐼) → (Scalar‘((𝐼 × {(ringLMod‘𝑅)})‘𝑖)) = 𝑅)
19		eqid 2739	. . . 4 ⊢ (𝑅X_s(𝐼 × {(ringLMod‘𝑅)})) = (𝑅X_s(𝐼 × {(ringLMod‘𝑅)}))
20		eqid 2739	. . . 4 ⊢ (LSubSp‘(𝑅X_s(𝐼 × {(ringLMod‘𝑅)}))) = (LSubSp‘(𝑅X_s(𝐼 × {(ringLMod‘𝑅)})))
21		eqid 2739	. . . 4 ⊢ (Base‘(𝑅 ⊕_m (𝐼 × {(ringLMod‘𝑅)}))) = (Base‘(𝑅 ⊕_m (𝐼 × {(ringLMod‘𝑅)})))
22	6, 7, 11, 18, 19, 20, 21	dsmmlss 20932	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → (Base‘(𝑅 ⊕_m (𝐼 × {(ringLMod‘𝑅)}))) ∈ (LSubSp‘(𝑅X_s(𝐼 × {(ringLMod‘𝑅)}))))
23		eqid 2739	. . . . . . . . 9 ⊢ ((ringLMod‘𝑅) ↑_s 𝐼) = ((ringLMod‘𝑅) ↑_s 𝐼)
24		eqid 2739	. . . . . . . . 9 ⊢ (Scalar‘(ringLMod‘𝑅)) = (Scalar‘(ringLMod‘𝑅))
25	23, 24	pwsval 17178	. . . . . . . 8 ⊢ (((ringLMod‘𝑅) ∈ V ∧ 𝐼 ∈ 𝑊) → ((ringLMod‘𝑅) ↑_s 𝐼) = ((Scalar‘(ringLMod‘𝑅))X_s(𝐼 × {(ringLMod‘𝑅)})))
26	12, 25	mpan 686	. . . . . . 7 ⊢ (𝐼 ∈ 𝑊 → ((ringLMod‘𝑅) ↑_s 𝐼) = ((Scalar‘(ringLMod‘𝑅))X_s(𝐼 × {(ringLMod‘𝑅)})))
27	26	adantl 481	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → ((ringLMod‘𝑅) ↑_s 𝐼) = ((Scalar‘(ringLMod‘𝑅))X_s(𝐼 × {(ringLMod‘𝑅)})))
28	16	eqcomd 2745	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → (Scalar‘(ringLMod‘𝑅)) = 𝑅)
29	28	adantr 480	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → (Scalar‘(ringLMod‘𝑅)) = 𝑅)
30	29	oveq1d 7283	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → ((Scalar‘(ringLMod‘𝑅))X_s(𝐼 × {(ringLMod‘𝑅)})) = (𝑅X_s(𝐼 × {(ringLMod‘𝑅)})))
31	27, 30	eqtr2d 2780	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → (𝑅X_s(𝐼 × {(ringLMod‘𝑅)})) = ((ringLMod‘𝑅) ↑_s 𝐼))
32	31	fveq2d 6772	. . . 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 2797	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → (LSubSp‘(𝑅X_s(𝐼 × {(ringLMod‘𝑅)}))) = 𝑈)
35	22, 34	eleqtrd 2842	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → (Base‘(𝑅 ⊕_m (𝐼 × {(ringLMod‘𝑅)}))) ∈ 𝑈)
36	5, 35	eqeltrd 2840	1 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → 𝐵 ∈ 𝑈)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2109 Vcvv 3430 {csn 4566 × cxp 5586 ⟶wf 6426 ‘cfv 6430 (class class class)co 7268 Basecbs 16893 Scalarcsca 16946 X_scprds 17137 ↑_s cpws 17138 Ringcrg 19764 LModclmod 20104 LSubSpclss 20174 ringLModcrglmod 20412 ⊕_m cdsmm 20919 freeLMod cfrlm 20934

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-rep 5213 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-cnex 10911 ax-resscn 10912 ax-1cn 10913 ax-icn 10914 ax-addcl 10915 ax-addrcl 10916 ax-mulcl 10917 ax-mulrcl 10918 ax-mulcom 10919 ax-addass 10920 ax-mulass 10921 ax-distr 10922 ax-i2m1 10923 ax-1ne0 10924 ax-1rid 10925 ax-rnegex 10926 ax-rrecex 10927 ax-cnre 10928 ax-pre-lttri 10929 ax-pre-lttrn 10930 ax-pre-ltadd 10931 ax-pre-mulgt0 10932

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3072 df-rmo 3073 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-pss 3910 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4845 df-iun 4931 df-br 5079 df-opab 5141 df-mpt 5162 df-tr 5196 df-id 5488 df-eprel 5494 df-po 5502 df-so 5503 df-fr 5543 df-we 5545 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-pred 6199 df-ord 6266 df-on 6267 df-lim 6268 df-suc 6269 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-om 7701 df-1st 7817 df-2nd 7818 df-frecs 8081 df-wrecs 8112 df-recs 8186 df-rdg 8225 df-1o 8281 df-er 8472 df-map 8591 df-ixp 8660 df-en 8708 df-dom 8709 df-sdom 8710 df-fin 8711 df-sup 9162 df-pnf 10995 df-mnf 10996 df-xr 10997 df-ltxr 10998 df-le 10999 df-sub 11190 df-neg 11191 df-nn 11957 df-2 12019 df-3 12020 df-4 12021 df-5 12022 df-6 12023 df-7 12024 df-8 12025 df-9 12026 df-n0 12217 df-z 12303 df-dec 12420 df-uz 12565 df-fz 13222 df-struct 16829 df-sets 16846 df-slot 16864 df-ndx 16876 df-base 16894 df-ress 16923 df-plusg 16956 df-mulr 16957 df-sca 16959 df-vsca 16960 df-ip 16961 df-tset 16962 df-ple 16963 df-ds 16965 df-hom 16967 df-cco 16968 df-0g 17133 df-prds 17139 df-pws 17141 df-mgm 18307 df-sgrp 18356 df-mnd 18367 df-grp 18561 df-minusg 18562 df-sbg 18563 df-subg 18733 df-mgp 19702 df-ur 19719 df-ring 19766 df-subrg 20003 df-lmod 20106 df-lss 20175 df-sra 20415 df-rgmod 20416 df-dsmm 20920 df-frlm 20935

This theorem is referenced by: frlm0 20942 frlmsubgval 20953 frlmgsum 20960 frlmsplit2 20961

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