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Mirrors > Home > MPE Home > Th. List > lsppr0 | Structured version Visualization version GIF version |
Description: The span of a vector paired with zero equals the span of the singleton of the vector. (Contributed by NM, 29-Aug-2014.) |
Ref | Expression |
---|---|
lsppr0.v | ⊢ 𝑉 = (Base‘𝑊) |
lsppr0.z | ⊢ 0 = (0g‘𝑊) |
lsppr0.n | ⊢ 𝑁 = (LSpan‘𝑊) |
lsppr0.w | ⊢ (𝜑 → 𝑊 ∈ LMod) |
lsppr0.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
Ref | Expression |
---|---|
lsppr0 | ⊢ (𝜑 → (𝑁‘{𝑋, 0 }) = (𝑁‘{𝑋})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lsppr0.v | . . 3 ⊢ 𝑉 = (Base‘𝑊) | |
2 | lsppr0.n | . . 3 ⊢ 𝑁 = (LSpan‘𝑊) | |
3 | eqid 2825 | . . 3 ⊢ (LSSum‘𝑊) = (LSSum‘𝑊) | |
4 | lsppr0.w | . . 3 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
5 | lsppr0.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
6 | lsppr0.z | . . . . 5 ⊢ 0 = (0g‘𝑊) | |
7 | 1, 6 | lmod0vcl 19248 | . . . 4 ⊢ (𝑊 ∈ LMod → 0 ∈ 𝑉) |
8 | 4, 7 | syl 17 | . . 3 ⊢ (𝜑 → 0 ∈ 𝑉) |
9 | 1, 2, 3, 4, 5, 8 | lsmpr 19448 | . 2 ⊢ (𝜑 → (𝑁‘{𝑋, 0 }) = ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{ 0 }))) |
10 | 6, 2 | lspsn0 19367 | . . . 4 ⊢ (𝑊 ∈ LMod → (𝑁‘{ 0 }) = { 0 }) |
11 | 4, 10 | syl 17 | . . 3 ⊢ (𝜑 → (𝑁‘{ 0 }) = { 0 }) |
12 | 11 | oveq2d 6921 | . 2 ⊢ (𝜑 → ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{ 0 })) = ((𝑁‘{𝑋})(LSSum‘𝑊){ 0 })) |
13 | 1, 2 | lspsnsubg 19339 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ (SubGrp‘𝑊)) |
14 | 4, 5, 13 | syl2anc 581 | . . 3 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ (SubGrp‘𝑊)) |
15 | 6, 3 | lsm01 18435 | . . 3 ⊢ ((𝑁‘{𝑋}) ∈ (SubGrp‘𝑊) → ((𝑁‘{𝑋})(LSSum‘𝑊){ 0 }) = (𝑁‘{𝑋})) |
16 | 14, 15 | syl 17 | . 2 ⊢ (𝜑 → ((𝑁‘{𝑋})(LSSum‘𝑊){ 0 }) = (𝑁‘{𝑋})) |
17 | 9, 12, 16 | 3eqtrd 2865 | 1 ⊢ (𝜑 → (𝑁‘{𝑋, 0 }) = (𝑁‘{𝑋})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1658 ∈ wcel 2166 {csn 4397 {cpr 4399 ‘cfv 6123 (class class class)co 6905 Basecbs 16222 0gc0g 16453 SubGrpcsubg 17939 LSSumclsm 18400 LModclmod 19219 LSpanclspn 19330 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-1st 7428 df-2nd 7429 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-er 8009 df-en 8223 df-dom 8224 df-sdom 8225 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-nn 11351 df-2 11414 df-ndx 16225 df-slot 16226 df-base 16228 df-sets 16229 df-ress 16230 df-plusg 16318 df-0g 16455 df-mgm 17595 df-sgrp 17637 df-mnd 17648 df-submnd 17689 df-grp 17779 df-minusg 17780 df-sbg 17781 df-subg 17942 df-cntz 18100 df-lsm 18402 df-cmn 18548 df-abl 18549 df-mgp 18844 df-ur 18856 df-ring 18903 df-lmod 19221 df-lss 19289 df-lsp 19331 |
This theorem is referenced by: lspfixed 19487 lspfixedOLD 19488 dihprrn 37501 dvh3dim 37521 mapdindp2 37796 hdmap11lem2 37917 |
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