frlmsslss2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > frlmsslss2		Structured version Visualization version GIF version

Theorem frlmsslss2 20028

Description: A subset of a free module obtained by restricting the support set is a submodule. 𝐽 is the set of permitted unit vectors. (Contributed by Stefan O'Rear, 5-Feb-2015.) (Revised by AV, 23-Jun-2019.)

**Hypotheses**
Ref	Expression
frlmsslss.y	⊢ 𝑌 = (𝑅 freeLMod 𝐼)
frlmsslss.u	⊢ 𝑈 = (LSubSp‘𝑌)
frlmsslss.b	⊢ 𝐵 = (Base‘𝑌)
frlmsslss.z	⊢ 0 = (0_g‘𝑅)
frlmsslss2.c	⊢ 𝐶 = {𝑥 ∈ 𝐵 ∣ (𝑥 supp 0 ) ⊆ 𝐽}

**Assertion**
Ref	Expression
frlmsslss2	⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → 𝐶 ∈ 𝑈)

Distinct variable groups: 𝑥,𝐵 𝑥,𝐼 𝑥,𝐽 𝑥,𝑅 𝑥,𝑈 𝑥, 0 𝑥,𝑉 𝑥,𝑌 Allowed substitution hint: 𝐶(𝑥)

**Proof of Theorem frlmsslss2**
Step	Hyp	Ref	Expression
1		frlmsslss2.c	. . 3 ⊢ 𝐶 = {𝑥 ∈ 𝐵 ∣ (𝑥 supp 0 ) ⊆ 𝐽}
2		frlmsslss.y	. . . . . . . . 9 ⊢ 𝑌 = (𝑅 freeLMod 𝐼)
3		eqid 2626	. . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅)
4		frlmsslss.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝑌)
5	2, 3, 4	frlmbasf 20018	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → 𝑥:𝐼⟶(Base‘𝑅))
6	5	3ad2antl2 1222	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑥 ∈ 𝐵) → 𝑥:𝐼⟶(Base‘𝑅))
7		ffn 6004	. . . . . . 7 ⊢ (𝑥:𝐼⟶(Base‘𝑅) → 𝑥 Fn 𝐼)
8	6, 7	syl 17	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑥 ∈ 𝐵) → 𝑥 Fn 𝐼)
9		simpl3 1064	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑥 ∈ 𝐵) → 𝐽 ⊆ 𝐼)
10		undif 4026	. . . . . . . 8 ⊢ (𝐽 ⊆ 𝐼 ↔ (𝐽 ∪ (𝐼 ∖ 𝐽)) = 𝐼)
11	9, 10	sylib 208	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑥 ∈ 𝐵) → (𝐽 ∪ (𝐼 ∖ 𝐽)) = 𝐼)
12	11	fneq2d 5942	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑥 ∈ 𝐵) → (𝑥 Fn (𝐽 ∪ (𝐼 ∖ 𝐽)) ↔ 𝑥 Fn 𝐼))
13	8, 12	mpbird 247	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑥 ∈ 𝐵) → 𝑥 Fn (𝐽 ∪ (𝐼 ∖ 𝐽)))
14		simpr 477	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
15		frlmsslss.z	. . . . . . 7 ⊢ 0 = (0_g‘𝑅)
16		fvex 6160	. . . . . . 7 ⊢ (0_g‘𝑅) ∈ V
17	15, 16	eqeltri 2700	. . . . . 6 ⊢ 0 ∈ V
18	17	a1i 11	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑥 ∈ 𝐵) → 0 ∈ V)
19		disjdif 4017	. . . . . 6 ⊢ (𝐽 ∩ (𝐼 ∖ 𝐽)) = ∅
20	19	a1i 11	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑥 ∈ 𝐵) → (𝐽 ∩ (𝐼 ∖ 𝐽)) = ∅)
21		fnsuppres 7268	. . . . 5 ⊢ ((𝑥 Fn (𝐽 ∪ (𝐼 ∖ 𝐽)) ∧ (𝑥 ∈ 𝐵 ∧ 0 ∈ V) ∧ (𝐽 ∩ (𝐼 ∖ 𝐽)) = ∅) → ((𝑥 supp 0 ) ⊆ 𝐽 ↔ (𝑥 ↾ (𝐼 ∖ 𝐽)) = ((𝐼 ∖ 𝐽) × { 0 })))
22	13, 14, 18, 20, 21	syl121anc 1328	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑥 ∈ 𝐵) → ((𝑥 supp 0 ) ⊆ 𝐽 ↔ (𝑥 ↾ (𝐼 ∖ 𝐽)) = ((𝐼 ∖ 𝐽) × { 0 })))
23	22	rabbidva 3181	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → {𝑥 ∈ 𝐵 ∣ (𝑥 supp 0 ) ⊆ 𝐽} = {𝑥 ∈ 𝐵 ∣ (𝑥 ↾ (𝐼 ∖ 𝐽)) = ((𝐼 ∖ 𝐽) × { 0 })})
24	1, 23	syl5eq 2672	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → 𝐶 = {𝑥 ∈ 𝐵 ∣ (𝑥 ↾ (𝐼 ∖ 𝐽)) = ((𝐼 ∖ 𝐽) × { 0 })})
25		difssd 3721	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → (𝐼 ∖ 𝐽) ⊆ 𝐼)
26		frlmsslss.u	. . . 4 ⊢ 𝑈 = (LSubSp‘𝑌)
27		eqid 2626	. . . 4 ⊢ {𝑥 ∈ 𝐵 ∣ (𝑥 ↾ (𝐼 ∖ 𝐽)) = ((𝐼 ∖ 𝐽) × { 0 })} = {𝑥 ∈ 𝐵 ∣ (𝑥 ↾ (𝐼 ∖ 𝐽)) = ((𝐼 ∖ 𝐽) × { 0 })}
28	2, 26, 4, 15, 27	frlmsslss 20027	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ (𝐼 ∖ 𝐽) ⊆ 𝐼) → {𝑥 ∈ 𝐵 ∣ (𝑥 ↾ (𝐼 ∖ 𝐽)) = ((𝐼 ∖ 𝐽) × { 0 })} ∈ 𝑈)
29	25, 28	syld3an3 1368	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → {𝑥 ∈ 𝐵 ∣ (𝑥 ↾ (𝐼 ∖ 𝐽)) = ((𝐼 ∖ 𝐽) × { 0 })} ∈ 𝑈)
30	24, 29	eqeltrd 2704	1 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → 𝐶 ∈ 𝑈)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1036 = wceq 1480 ∈ wcel 1992 {crab 2916 Vcvv 3191 ∖ cdif 3557 ∪ cun 3558 ∩ cin 3559 ⊆ wss 3560 ∅c0 3896 {csn 4153 × cxp 5077 ↾ cres 5081 Fn wfn 5845 ⟶wf 5846 ‘cfv 5850 (class class class)co 6605 supp csupp 7241 Basecbs 15776 0_gc0g 16016 Ringcrg 18463 LSubSpclss 18846 freeLMod cfrlm 20004

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1719 ax-4 1734 ax-5 1841 ax-6 1890 ax-7 1937 ax-8 1994 ax-9 2001 ax-10 2021 ax-11 2036 ax-12 2049 ax-13 2250 ax-ext 2606 ax-rep 4736 ax-sep 4746 ax-nul 4754 ax-pow 4808 ax-pr 4872 ax-un 6903 ax-cnex 9937 ax-resscn 9938 ax-1cn 9939 ax-icn 9940 ax-addcl 9941 ax-addrcl 9942 ax-mulcl 9943 ax-mulrcl 9944 ax-mulcom 9945 ax-addass 9946 ax-mulass 9947 ax-distr 9948 ax-i2m1 9949 ax-1ne0 9950 ax-1rid 9951 ax-rnegex 9952 ax-rrecex 9953 ax-cnre 9954 ax-pre-lttri 9955 ax-pre-lttrn 9956 ax-pre-ltadd 9957 ax-pre-mulgt0 9958

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1037 df-3an 1038 df-tru 1483 df-ex 1702 df-nf 1707 df-sb 1883 df-eu 2478 df-mo 2479 df-clab 2613 df-cleq 2619 df-clel 2622 df-nfc 2756 df-ne 2797 df-nel 2900 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3193 df-sbc 3423 df-csb 3520 df-dif 3563 df-un 3565 df-in 3567 df-ss 3574 df-pss 3576 df-nul 3897 df-if 4064 df-pw 4137 df-sn 4154 df-pr 4156 df-tp 4158 df-op 4160 df-uni 4408 df-int 4446 df-iun 4492 df-br 4619 df-opab 4679 df-mpt 4680 df-tr 4718 df-eprel 4990 df-id 4994 df-po 5000 df-so 5001 df-fr 5038 df-we 5040 df-xp 5085 df-rel 5086 df-cnv 5087 df-co 5088 df-dm 5089 df-rn 5090 df-res 5091 df-ima 5092 df-pred 5642 df-ord 5688 df-on 5689 df-lim 5690 df-suc 5691 df-iota 5813 df-fun 5852 df-fn 5853 df-f 5854 df-f1 5855 df-fo 5856 df-f1o 5857 df-fv 5858 df-riota 6566 df-ov 6608 df-oprab 6609 df-mpt2 6610 df-of 6851 df-om 7014 df-1st 7116 df-2nd 7117 df-supp 7242 df-wrecs 7353 df-recs 7414 df-rdg 7452 df-1o 7506 df-oadd 7510 df-er 7688 df-map 7805 df-ixp 7854 df-en 7901 df-dom 7902 df-sdom 7903 df-fin 7904 df-fsupp 8221 df-sup 8293 df-pnf 10021 df-mnf 10022 df-xr 10023 df-ltxr 10024 df-le 10025 df-sub 10213 df-neg 10214 df-nn 10966 df-2 11024 df-3 11025 df-4 11026 df-5 11027 df-6 11028 df-7 11029 df-8 11030 df-9 11031 df-n0 11238 df-z 11323 df-dec 11438 df-uz 11632 df-fz 12266 df-struct 15778 df-ndx 15779 df-slot 15780 df-base 15781 df-sets 15782 df-ress 15783 df-plusg 15870 df-mulr 15871 df-sca 15873 df-vsca 15874 df-ip 15875 df-tset 15876 df-ple 15877 df-ds 15880 df-hom 15882 df-cco 15883 df-0g 16018 df-prds 16024 df-pws 16026 df-mgm 17158 df-sgrp 17200 df-mnd 17211 df-mhm 17251 df-submnd 17252 df-grp 17341 df-minusg 17342 df-sbg 17343 df-subg 17507 df-ghm 17574 df-mgp 18406 df-ur 18418 df-ring 18465 df-subrg 18694 df-lmod 18781 df-lss 18847 df-lmhm 18936 df-sra 19086 df-rgmod 19087 df-dsmm 19990 df-frlm 20005

This theorem is referenced by: frlmssuvc1 20047 frlmsslsp 20049

W3C validator