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Theorem lsp0 20944

Description: Span of the empty set. (Contributed by Mario Carneiro, 5-Sep-2014.)

**Hypotheses**
Ref	Expression
lspsn0.z	⊢ 0 = (0_g‘𝑊)
lspsn0.n	⊢ 𝑁 = (LSpan‘𝑊)

**Assertion**
Ref	Expression
lsp0	⊢ (𝑊 ∈ LMod → (𝑁‘∅) = { 0 })

**Proof of Theorem lsp0**
Step	Hyp	Ref	Expression
1		lspsn0.z	. . . 4 ⊢ 0 = (0_g‘𝑊)
2		eqid 2733	. . . 4 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊)
3	1, 2	lsssn0 20883	. . 3 ⊢ (𝑊 ∈ LMod → { 0 } ∈ (LSubSp‘𝑊))
4		0ss 4349	. . . 4 ⊢ ∅ ⊆ { 0 }
5		lspsn0.n	. . . . 5 ⊢ 𝑁 = (LSpan‘𝑊)
6	2, 5	lspssp 20923	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ { 0 } ∈ (LSubSp‘𝑊) ∧ ∅ ⊆ { 0 }) → (𝑁‘∅) ⊆ { 0 })
7	4, 6	mp3an3 1452	. . 3 ⊢ ((𝑊 ∈ LMod ∧ { 0 } ∈ (LSubSp‘𝑊)) → (𝑁‘∅) ⊆ { 0 })
8	3, 7	mpdan 687	. 2 ⊢ (𝑊 ∈ LMod → (𝑁‘∅) ⊆ { 0 })
9		0ss 4349	. . . 4 ⊢ ∅ ⊆ (Base‘𝑊)
10		eqid 2733	. . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊)
11	10, 2, 5	lspcl 20911	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ ∅ ⊆ (Base‘𝑊)) → (𝑁‘∅) ∈ (LSubSp‘𝑊))
12	9, 11	mpan2 691	. . 3 ⊢ (𝑊 ∈ LMod → (𝑁‘∅) ∈ (LSubSp‘𝑊))
13	1, 2	lss0ss 20884	. . 3 ⊢ ((𝑊 ∈ LMod ∧ (𝑁‘∅) ∈ (LSubSp‘𝑊)) → { 0 } ⊆ (𝑁‘∅))
14	12, 13	mpdan 687	. 2 ⊢ (𝑊 ∈ LMod → { 0 } ⊆ (𝑁‘∅))
15	8, 14	eqssd 3948	1 ⊢ (𝑊 ∈ LMod → (𝑁‘∅) = { 0 })

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2113 ⊆ wss 3898 ∅c0 4282 {csn 4575 ‘cfv 6486 Basecbs 17122 0_gc0g 17345 LModclmod 20795 LSubSpclss 20866 LSpanclspn 20906

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5219 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674 ax-cnex 11069 ax-resscn 11070 ax-1cn 11071 ax-icn 11072 ax-addcl 11073 ax-addrcl 11074 ax-mulcl 11075 ax-mulrcl 11076 ax-mulcom 11077 ax-addass 11078 ax-mulass 11079 ax-distr 11080 ax-i2m1 11081 ax-1ne0 11082 ax-1rid 11083 ax-rnegex 11084 ax-rrecex 11085 ax-cnre 11086 ax-pre-lttri 11087 ax-pre-lttrn 11088 ax-pre-ltadd 11089 ax-pre-mulgt0 11090

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-op 4582 df-uni 4859 df-int 4898 df-iun 4943 df-br 5094 df-opab 5156 df-mpt 5175 df-tr 5201 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7309 df-ov 7355 df-oprab 7356 df-mpo 7357 df-om 7803 df-1st 7927 df-2nd 7928 df-frecs 8217 df-wrecs 8248 df-recs 8297 df-rdg 8335 df-er 8628 df-en 8876 df-dom 8877 df-sdom 8878 df-pnf 11155 df-mnf 11156 df-xr 11157 df-ltxr 11158 df-le 11159 df-sub 11353 df-neg 11354 df-nn 12133 df-2 12195 df-sets 17077 df-slot 17095 df-ndx 17107 df-base 17123 df-plusg 17176 df-0g 17347 df-mgm 18550 df-sgrp 18629 df-mnd 18645 df-grp 18851 df-minusg 18852 df-sbg 18853 df-cmn 19696 df-abl 19697 df-mgp 20061 df-rng 20073 df-ur 20102 df-ring 20155 df-lmod 20797 df-lss 20867 df-lsp 20907

This theorem is referenced by: lspuni0 20945 lss0v 20952 lspsnat 21084 lsppratlem3 21088 ocvz 21617 lindssn 33350 lvecdim0i 33639 lbslsat 33650

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