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Mirrors > Home > MPE Home > Th. List > frlmsslss2 | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
frlmsslss.y | ⊢ 𝑌 = (𝑅 freeLMod 𝐼) |
frlmsslss.u | ⊢ 𝑈 = (LSubSp‘𝑌) |
frlmsslss.b | ⊢ 𝐵 = (Base‘𝑌) |
frlmsslss.z | ⊢ 0 = (0g‘𝑅) |
frlmsslss2.c | ⊢ 𝐶 = {𝑥 ∈ 𝐵 ∣ (𝑥 supp 0 ) ⊆ 𝐽} |
Ref | Expression |
---|---|
frlmsslss2 | ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → 𝐶 ∈ 𝑈) |
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 2738 | . . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
4 | frlmsslss.b | . . . . . . . . 9 ⊢ 𝐵 = (Base‘𝑌) | |
5 | 2, 3, 4 | frlmbasf 20576 | . . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → 𝑥:𝐼⟶(Base‘𝑅)) |
6 | 5 | 3ad2antl2 1187 | . . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑥 ∈ 𝐵) → 𝑥:𝐼⟶(Base‘𝑅)) |
7 | 6 | ffnd 6505 | . . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑥 ∈ 𝐵) → 𝑥 Fn 𝐼) |
8 | simpl3 1194 | . . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑥 ∈ 𝐵) → 𝐽 ⊆ 𝐼) | |
9 | undif 4371 | . . . . . . . 8 ⊢ (𝐽 ⊆ 𝐼 ↔ (𝐽 ∪ (𝐼 ∖ 𝐽)) = 𝐼) | |
10 | 8, 9 | sylib 221 | . . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑥 ∈ 𝐵) → (𝐽 ∪ (𝐼 ∖ 𝐽)) = 𝐼) |
11 | 10 | fneq2d 6432 | . . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑥 ∈ 𝐵) → (𝑥 Fn (𝐽 ∪ (𝐼 ∖ 𝐽)) ↔ 𝑥 Fn 𝐼)) |
12 | 7, 11 | mpbird 260 | . . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑥 ∈ 𝐵) → 𝑥 Fn (𝐽 ∪ (𝐼 ∖ 𝐽))) |
13 | simpr 488 | . . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵) | |
14 | frlmsslss.z | . . . . . . 7 ⊢ 0 = (0g‘𝑅) | |
15 | 14 | fvexi 6688 | . . . . . 6 ⊢ 0 ∈ V |
16 | 15 | a1i 11 | . . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑥 ∈ 𝐵) → 0 ∈ V) |
17 | disjdif 4361 | . . . . . 6 ⊢ (𝐽 ∩ (𝐼 ∖ 𝐽)) = ∅ | |
18 | 17 | a1i 11 | . . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑥 ∈ 𝐵) → (𝐽 ∩ (𝐼 ∖ 𝐽)) = ∅) |
19 | fnsuppres 7886 | . . . . 5 ⊢ ((𝑥 Fn (𝐽 ∪ (𝐼 ∖ 𝐽)) ∧ (𝑥 ∈ 𝐵 ∧ 0 ∈ V) ∧ (𝐽 ∩ (𝐼 ∖ 𝐽)) = ∅) → ((𝑥 supp 0 ) ⊆ 𝐽 ↔ (𝑥 ↾ (𝐼 ∖ 𝐽)) = ((𝐼 ∖ 𝐽) × { 0 }))) | |
20 | 12, 13, 16, 18, 19 | syl121anc 1376 | . . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑥 ∈ 𝐵) → ((𝑥 supp 0 ) ⊆ 𝐽 ↔ (𝑥 ↾ (𝐼 ∖ 𝐽)) = ((𝐼 ∖ 𝐽) × { 0 }))) |
21 | 20 | rabbidva 3379 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → {𝑥 ∈ 𝐵 ∣ (𝑥 supp 0 ) ⊆ 𝐽} = {𝑥 ∈ 𝐵 ∣ (𝑥 ↾ (𝐼 ∖ 𝐽)) = ((𝐼 ∖ 𝐽) × { 0 })}) |
22 | 1, 21 | syl5eq 2785 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → 𝐶 = {𝑥 ∈ 𝐵 ∣ (𝑥 ↾ (𝐼 ∖ 𝐽)) = ((𝐼 ∖ 𝐽) × { 0 })}) |
23 | difssd 4023 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → (𝐼 ∖ 𝐽) ⊆ 𝐼) | |
24 | frlmsslss.u | . . . 4 ⊢ 𝑈 = (LSubSp‘𝑌) | |
25 | eqid 2738 | . . . 4 ⊢ {𝑥 ∈ 𝐵 ∣ (𝑥 ↾ (𝐼 ∖ 𝐽)) = ((𝐼 ∖ 𝐽) × { 0 })} = {𝑥 ∈ 𝐵 ∣ (𝑥 ↾ (𝐼 ∖ 𝐽)) = ((𝐼 ∖ 𝐽) × { 0 })} | |
26 | 2, 24, 4, 14, 25 | frlmsslss 20590 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ (𝐼 ∖ 𝐽) ⊆ 𝐼) → {𝑥 ∈ 𝐵 ∣ (𝑥 ↾ (𝐼 ∖ 𝐽)) = ((𝐼 ∖ 𝐽) × { 0 })} ∈ 𝑈) |
27 | 23, 26 | syld3an3 1410 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → {𝑥 ∈ 𝐵 ∣ (𝑥 ↾ (𝐼 ∖ 𝐽)) = ((𝐼 ∖ 𝐽) × { 0 })} ∈ 𝑈) |
28 | 22, 27 | eqeltrd 2833 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → 𝐶 ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1088 = wceq 1542 ∈ wcel 2114 {crab 3057 Vcvv 3398 ∖ cdif 3840 ∪ cun 3841 ∩ cin 3842 ⊆ wss 3843 ∅c0 4211 {csn 4516 × cxp 5523 ↾ cres 5527 Fn wfn 6334 ⟶wf 6335 ‘cfv 6339 (class class class)co 7170 supp csupp 7856 Basecbs 16586 0gc0g 16816 Ringcrg 19416 LSubSpclss 19822 freeLMod cfrlm 20562 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-of 7425 df-om 7600 df-1st 7714 df-2nd 7715 df-supp 7857 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-1o 8131 df-er 8320 df-map 8439 df-ixp 8508 df-en 8556 df-dom 8557 df-sdom 8558 df-fin 8559 df-fsupp 8907 df-sup 8979 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-nn 11717 df-2 11779 df-3 11780 df-4 11781 df-5 11782 df-6 11783 df-7 11784 df-8 11785 df-9 11786 df-n0 11977 df-z 12063 df-dec 12180 df-uz 12325 df-fz 12982 df-struct 16588 df-ndx 16589 df-slot 16590 df-base 16592 df-sets 16593 df-ress 16594 df-plusg 16681 df-mulr 16682 df-sca 16684 df-vsca 16685 df-ip 16686 df-tset 16687 df-ple 16688 df-ds 16690 df-hom 16692 df-cco 16693 df-0g 16818 df-prds 16824 df-pws 16826 df-mgm 17968 df-sgrp 18017 df-mnd 18028 df-mhm 18072 df-submnd 18073 df-grp 18222 df-minusg 18223 df-sbg 18224 df-subg 18394 df-ghm 18474 df-mgp 19359 df-ur 19371 df-ring 19418 df-subrg 19652 df-lmod 19755 df-lss 19823 df-lmhm 19913 df-sra 20063 df-rgmod 20064 df-dsmm 20548 df-frlm 20563 |
This theorem is referenced by: frlmssuvc1 20610 frlmsslsp 20612 |
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