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Mirrors > Home > ILE Home > Th. List > lspid | GIF version |
Description: The span of a subspace is itself. (Contributed by NM, 15-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.) |
Ref | Expression |
---|---|
lspid.s | ⊢ 𝑆 = (LSubSp‘𝑊) |
lspid.n | ⊢ 𝑁 = (LSpan‘𝑊) |
Ref | Expression |
---|---|
lspid | ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → (𝑁‘𝑈) = 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2189 | . . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
2 | lspid.s | . . . 4 ⊢ 𝑆 = (LSubSp‘𝑊) | |
3 | 1, 2 | lssssg 13693 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → 𝑈 ⊆ (Base‘𝑊)) |
4 | lspid.n | . . . 4 ⊢ 𝑁 = (LSpan‘𝑊) | |
5 | 1, 2, 4 | lspval 13723 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ⊆ (Base‘𝑊)) → (𝑁‘𝑈) = ∩ {𝑡 ∈ 𝑆 ∣ 𝑈 ⊆ 𝑡}) |
6 | 3, 5 | syldan 282 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → (𝑁‘𝑈) = ∩ {𝑡 ∈ 𝑆 ∣ 𝑈 ⊆ 𝑡}) |
7 | intmin 3879 | . . 3 ⊢ (𝑈 ∈ 𝑆 → ∩ {𝑡 ∈ 𝑆 ∣ 𝑈 ⊆ 𝑡} = 𝑈) | |
8 | 7 | adantl 277 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → ∩ {𝑡 ∈ 𝑆 ∣ 𝑈 ⊆ 𝑡} = 𝑈) |
9 | 6, 8 | eqtrd 2222 | 1 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → (𝑁‘𝑈) = 𝑈) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1364 ∈ wcel 2160 {crab 2472 ⊆ wss 3144 ∩ cint 3859 ‘cfv 5235 Basecbs 12515 LModclmod 13620 LSubSpclss 13685 LSpanclspn 13719 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-cnex 7933 ax-resscn 7934 ax-1re 7936 ax-addrcl 7939 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-fv 5243 df-riota 5852 df-ov 5900 df-inn 8951 df-2 9009 df-3 9010 df-4 9011 df-5 9012 df-6 9013 df-ndx 12518 df-slot 12519 df-base 12521 df-plusg 12605 df-mulr 12606 df-sca 12608 df-vsca 12609 df-0g 12766 df-mgm 12835 df-sgrp 12880 df-mnd 12893 df-grp 12963 df-lmod 13622 df-lssm 13686 df-lsp 13720 |
This theorem is referenced by: lspidm 13734 lspssp 13736 lspsn0 13755 |
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