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Mirrors > Home > MPE Home > Th. List > lsmpr | Structured version Visualization version GIF version |
Description: The span of a pair of vectors equals the sum of the spans of their singletons. (Contributed by NM, 13-Jan-2015.) |
Ref | Expression |
---|---|
lsmpr.v | ⊢ 𝑉 = (Base‘𝑊) |
lsmpr.n | ⊢ 𝑁 = (LSpan‘𝑊) |
lsmpr.p | ⊢ ⊕ = (LSSum‘𝑊) |
lsmpr.w | ⊢ (𝜑 → 𝑊 ∈ LMod) |
lsmpr.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
lsmpr.y | ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
Ref | Expression |
---|---|
lsmpr | ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) = ((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lsmpr.w | . . 3 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
2 | lsmpr.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
3 | 2 | snssd 4814 | . . 3 ⊢ (𝜑 → {𝑋} ⊆ 𝑉) |
4 | lsmpr.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
5 | 4 | snssd 4814 | . . 3 ⊢ (𝜑 → {𝑌} ⊆ 𝑉) |
6 | lsmpr.v | . . . 4 ⊢ 𝑉 = (Base‘𝑊) | |
7 | lsmpr.n | . . . 4 ⊢ 𝑁 = (LSpan‘𝑊) | |
8 | 6, 7 | lspun 20883 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ {𝑋} ⊆ 𝑉 ∧ {𝑌} ⊆ 𝑉) → (𝑁‘({𝑋} ∪ {𝑌})) = (𝑁‘((𝑁‘{𝑋}) ∪ (𝑁‘{𝑌})))) |
9 | 1, 3, 5, 8 | syl3anc 1368 | . 2 ⊢ (𝜑 → (𝑁‘({𝑋} ∪ {𝑌})) = (𝑁‘((𝑁‘{𝑋}) ∪ (𝑁‘{𝑌})))) |
10 | df-pr 4633 | . . . 4 ⊢ {𝑋, 𝑌} = ({𝑋} ∪ {𝑌}) | |
11 | 10 | fveq2i 6899 | . . 3 ⊢ (𝑁‘{𝑋, 𝑌}) = (𝑁‘({𝑋} ∪ {𝑌})) |
12 | 11 | a1i 11 | . 2 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) = (𝑁‘({𝑋} ∪ {𝑌}))) |
13 | eqid 2725 | . . . . 5 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
14 | 6, 13, 7 | lspsncl 20873 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑊)) |
15 | 1, 2, 14 | syl2anc 582 | . . 3 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑊)) |
16 | 6, 13, 7 | lspsncl 20873 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊)) |
17 | 1, 4, 16 | syl2anc 582 | . . 3 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊)) |
18 | lsmpr.p | . . . 4 ⊢ ⊕ = (LSSum‘𝑊) | |
19 | 13, 7, 18 | lsmsp 20983 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ (𝑁‘{𝑋}) ∈ (LSubSp‘𝑊) ∧ (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊)) → ((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) = (𝑁‘((𝑁‘{𝑋}) ∪ (𝑁‘{𝑌})))) |
20 | 1, 15, 17, 19 | syl3anc 1368 | . 2 ⊢ (𝜑 → ((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) = (𝑁‘((𝑁‘{𝑋}) ∪ (𝑁‘{𝑌})))) |
21 | 9, 12, 20 | 3eqtr4d 2775 | 1 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) = ((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ∪ cun 3942 ⊆ wss 3944 {csn 4630 {cpr 4632 ‘cfv 6549 (class class class)co 7419 Basecbs 17183 LSSumclsm 19601 LModclmod 20755 LSubSpclss 20827 LSpanclspn 20867 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-nn 12246 df-2 12308 df-sets 17136 df-slot 17154 df-ndx 17166 df-base 17184 df-ress 17213 df-plusg 17249 df-0g 17426 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-submnd 18744 df-grp 18901 df-minusg 18902 df-sbg 18903 df-subg 19086 df-cntz 19280 df-lsm 19603 df-cmn 19749 df-abl 19750 df-mgp 20087 df-ur 20134 df-ring 20187 df-lmod 20757 df-lss 20828 df-lsp 20868 |
This theorem is referenced by: lsppreli 20987 lsmelpr 20988 lsppr0 20989 lspprabs 20992 lspabs2 21020 lspindpi 21032 lsmsat 38610 dvh4dimlem 41046 dvh3dim3N 41052 lclkrlem2c 41112 lcfrlem20 41165 lcfrlem23 41168 mapdindp 41274 mapdindp2 41324 mapdindp4 41326 mapdh6dN 41342 lspindp5 41373 hdmap1l6d 41416 hdmaprnlem3eN 41461 |
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