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Mirrors > Home > MPE Home > Th. List > lssincl | Structured version Visualization version GIF version |
Description: The intersection of two subspaces is a subspace. (Contributed by NM, 7-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) |
Ref | Expression |
---|---|
lssintcl.s | ⊢ 𝑆 = (LSubSp‘𝑊) |
Ref | Expression |
---|---|
lssincl | ⊢ ((𝑊 ∈ LMod ∧ 𝑇 ∈ 𝑆 ∧ 𝑈 ∈ 𝑆) → (𝑇 ∩ 𝑈) ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | intprg 4983 | . . 3 ⊢ ((𝑇 ∈ 𝑆 ∧ 𝑈 ∈ 𝑆) → ∩ {𝑇, 𝑈} = (𝑇 ∩ 𝑈)) | |
2 | 1 | 3adant1 1131 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑇 ∈ 𝑆 ∧ 𝑈 ∈ 𝑆) → ∩ {𝑇, 𝑈} = (𝑇 ∩ 𝑈)) |
3 | simp1 1137 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑇 ∈ 𝑆 ∧ 𝑈 ∈ 𝑆) → 𝑊 ∈ LMod) | |
4 | prssi 4822 | . . . 4 ⊢ ((𝑇 ∈ 𝑆 ∧ 𝑈 ∈ 𝑆) → {𝑇, 𝑈} ⊆ 𝑆) | |
5 | 4 | 3adant1 1131 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑇 ∈ 𝑆 ∧ 𝑈 ∈ 𝑆) → {𝑇, 𝑈} ⊆ 𝑆) |
6 | prnzg 4780 | . . . 4 ⊢ (𝑇 ∈ 𝑆 → {𝑇, 𝑈} ≠ ∅) | |
7 | 6 | 3ad2ant2 1135 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑇 ∈ 𝑆 ∧ 𝑈 ∈ 𝑆) → {𝑇, 𝑈} ≠ ∅) |
8 | lssintcl.s | . . . 4 ⊢ 𝑆 = (LSubSp‘𝑊) | |
9 | 8 | lssintcl 20562 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ {𝑇, 𝑈} ⊆ 𝑆 ∧ {𝑇, 𝑈} ≠ ∅) → ∩ {𝑇, 𝑈} ∈ 𝑆) |
10 | 3, 5, 7, 9 | syl3anc 1372 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑇 ∈ 𝑆 ∧ 𝑈 ∈ 𝑆) → ∩ {𝑇, 𝑈} ∈ 𝑆) |
11 | 2, 10 | eqeltrrd 2835 | 1 ⊢ ((𝑊 ∈ LMod ∧ 𝑇 ∈ 𝑆 ∧ 𝑈 ∈ 𝑆) → (𝑇 ∩ 𝑈) ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∩ cin 3945 ⊆ wss 3946 ∅c0 4320 {cpr 4628 ∩ cint 4948 ‘cfv 6539 LModclmod 20458 LSubSpclss 20529 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5297 ax-nul 5304 ax-pow 5361 ax-pr 5425 ax-un 7719 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4527 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4907 df-int 4949 df-iun 4997 df-br 5147 df-opab 5209 df-mpt 5230 df-tr 5264 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6296 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6491 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-riota 7359 df-ov 7406 df-oprab 7407 df-mpo 7408 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8260 df-wrecs 8291 df-recs 8365 df-rdg 8404 df-er 8698 df-en 8935 df-dom 8936 df-sdom 8937 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11441 df-neg 11442 df-nn 12208 df-2 12270 df-sets 17092 df-slot 17110 df-ndx 17122 df-base 17140 df-plusg 17205 df-0g 17382 df-mgm 18556 df-sgrp 18605 df-mnd 18621 df-grp 18817 df-minusg 18818 df-sbg 18819 df-mgp 19979 df-ur 19996 df-ring 20048 df-lmod 20460 df-lss 20530 |
This theorem is referenced by: ocvin 21210 lshpdisj 37794 lcvexchlem2 37842 lcvexchlem4 37844 lcvexchlem5 37845 lcvp 37847 lsatcvat3 37859 dihmeetlem13N 40127 dochnoncon 40199 dochexmidlem5 40272 lclkrlem2f 40320 lcfrlem25 40375 mapdincl 40469 mapdin 40470 lmhmlnmsplit 41761 |
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