![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > lsp0 | Structured version Visualization version GIF version |
Description: Span of the empty set. (Contributed by Mario Carneiro, 5-Sep-2014.) |
Ref | Expression |
---|---|
lspsn0.z | ⊢ 0 = (0g‘𝑊) |
lspsn0.n | ⊢ 𝑁 = (LSpan‘𝑊) |
Ref | Expression |
---|---|
lsp0 | ⊢ (𝑊 ∈ LMod → (𝑁‘∅) = { 0 }) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lspsn0.z | . . . 4 ⊢ 0 = (0g‘𝑊) | |
2 | eqid 2726 | . . . 4 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
3 | 1, 2 | lsssn0 20918 | . . 3 ⊢ (𝑊 ∈ LMod → { 0 } ∈ (LSubSp‘𝑊)) |
4 | 0ss 4394 | . . . 4 ⊢ ∅ ⊆ { 0 } | |
5 | lspsn0.n | . . . . 5 ⊢ 𝑁 = (LSpan‘𝑊) | |
6 | 2, 5 | lspssp 20958 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ { 0 } ∈ (LSubSp‘𝑊) ∧ ∅ ⊆ { 0 }) → (𝑁‘∅) ⊆ { 0 }) |
7 | 4, 6 | mp3an3 1447 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ { 0 } ∈ (LSubSp‘𝑊)) → (𝑁‘∅) ⊆ { 0 }) |
8 | 3, 7 | mpdan 685 | . 2 ⊢ (𝑊 ∈ LMod → (𝑁‘∅) ⊆ { 0 }) |
9 | 0ss 4394 | . . . 4 ⊢ ∅ ⊆ (Base‘𝑊) | |
10 | eqid 2726 | . . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
11 | 10, 2, 5 | lspcl 20946 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ ∅ ⊆ (Base‘𝑊)) → (𝑁‘∅) ∈ (LSubSp‘𝑊)) |
12 | 9, 11 | mpan2 689 | . . 3 ⊢ (𝑊 ∈ LMod → (𝑁‘∅) ∈ (LSubSp‘𝑊)) |
13 | 1, 2 | lss0ss 20919 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ (𝑁‘∅) ∈ (LSubSp‘𝑊)) → { 0 } ⊆ (𝑁‘∅)) |
14 | 12, 13 | mpdan 685 | . 2 ⊢ (𝑊 ∈ LMod → { 0 } ⊆ (𝑁‘∅)) |
15 | 8, 14 | eqssd 3996 | 1 ⊢ (𝑊 ∈ LMod → (𝑁‘∅) = { 0 }) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ⊆ wss 3946 ∅c0 4322 {csn 4623 ‘cfv 6543 Basecbs 17205 0gc0g 17446 LModclmod 20829 LSubSpclss 20901 LSpanclspn 20941 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7735 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-int 4947 df-iun 4995 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6302 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-1st 7992 df-2nd 7993 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-er 8723 df-en 8964 df-dom 8965 df-sdom 8966 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12256 df-2 12318 df-sets 17158 df-slot 17176 df-ndx 17188 df-base 17206 df-plusg 17271 df-0g 17448 df-mgm 18625 df-sgrp 18704 df-mnd 18720 df-grp 18923 df-minusg 18924 df-sbg 18925 df-cmn 19773 df-abl 19774 df-mgp 20111 df-rng 20129 df-ur 20158 df-ring 20211 df-lmod 20831 df-lss 20902 df-lsp 20942 |
This theorem is referenced by: lspuni0 20980 lss0v 20987 lspsnat 21119 lsppratlem3 21123 ocvz 21667 lindssn 33256 lvecdim0i 33503 lbslsat 33514 |
Copyright terms: Public domain | W3C validator |