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Theorem lsp0 21004

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 2736	. . . 4 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊)
3	1, 2	lsssn0 20943	. . 3 ⊢ (𝑊 ∈ LMod → { 0 } ∈ (LSubSp‘𝑊))
4		0ss 4340	. . . 4 ⊢ ∅ ⊆ { 0 }
5		lspsn0.n	. . . . 5 ⊢ 𝑁 = (LSpan‘𝑊)
6	2, 5	lspssp 20983	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ { 0 } ∈ (LSubSp‘𝑊) ∧ ∅ ⊆ { 0 }) → (𝑁‘∅) ⊆ { 0 })
7	4, 6	mp3an3 1453	. . 3 ⊢ ((𝑊 ∈ LMod ∧ { 0 } ∈ (LSubSp‘𝑊)) → (𝑁‘∅) ⊆ { 0 })
8	3, 7	mpdan 688	. 2 ⊢ (𝑊 ∈ LMod → (𝑁‘∅) ⊆ { 0 })
9		0ss 4340	. . . 4 ⊢ ∅ ⊆ (Base‘𝑊)
10		eqid 2736	. . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊)
11	10, 2, 5	lspcl 20971	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ ∅ ⊆ (Base‘𝑊)) → (𝑁‘∅) ∈ (LSubSp‘𝑊))
12	9, 11	mpan2 692	. . 3 ⊢ (𝑊 ∈ LMod → (𝑁‘∅) ∈ (LSubSp‘𝑊))
13	1, 2	lss0ss 20944	. . 3 ⊢ ((𝑊 ∈ LMod ∧ (𝑁‘∅) ∈ (LSubSp‘𝑊)) → { 0 } ⊆ (𝑁‘∅))
14	12, 13	mpdan 688	. 2 ⊢ (𝑊 ∈ LMod → { 0 } ⊆ (𝑁‘∅))
15	8, 14	eqssd 3939	1 ⊢ (𝑊 ∈ LMod → (𝑁‘∅) = { 0 })

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 ⊆ wss 3889 ∅c0 4273 {csn 4567 ‘cfv 6498 Basecbs 17179 0_gc0g 17402 LModclmod 20855 LSubSpclss 20926 LSpanclspn 20966

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-om 7818 df-1st 7942 df-2nd 7943 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-er 8643 df-en 8894 df-dom 8895 df-sdom 8896 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-nn 12175 df-2 12244 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-plusg 17233 df-0g 17404 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-grp 18912 df-minusg 18913 df-sbg 18914 df-cmn 19757 df-abl 19758 df-mgp 20122 df-rng 20134 df-ur 20163 df-ring 20216 df-lmod 20857 df-lss 20927 df-lsp 20967

This theorem is referenced by: lspuni0 21005 lss0v 21011 lspsnat 21143 lsppratlem3 21147 ocvz 21658 lindssn 33438 lvecdim0i 33750 lbslsat 33760

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