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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 4742 | . . 3 ⊢ (𝜑 → {𝑋} ⊆ 𝑉) |
4 | lsmpr.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
5 | 4 | snssd 4742 | . . 3 ⊢ (𝜑 → {𝑌} ⊆ 𝑉) |
6 | lsmpr.v | . . . 4 ⊢ 𝑉 = (Base‘𝑊) | |
7 | lsmpr.n | . . . 4 ⊢ 𝑁 = (LSpan‘𝑊) | |
8 | 6, 7 | lspun 19759 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ {𝑋} ⊆ 𝑉 ∧ {𝑌} ⊆ 𝑉) → (𝑁‘({𝑋} ∪ {𝑌})) = (𝑁‘((𝑁‘{𝑋}) ∪ (𝑁‘{𝑌})))) |
9 | 1, 3, 5, 8 | syl3anc 1367 | . 2 ⊢ (𝜑 → (𝑁‘({𝑋} ∪ {𝑌})) = (𝑁‘((𝑁‘{𝑋}) ∪ (𝑁‘{𝑌})))) |
10 | df-pr 4570 | . . . 4 ⊢ {𝑋, 𝑌} = ({𝑋} ∪ {𝑌}) | |
11 | 10 | fveq2i 6673 | . . 3 ⊢ (𝑁‘{𝑋, 𝑌}) = (𝑁‘({𝑋} ∪ {𝑌})) |
12 | 11 | a1i 11 | . 2 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) = (𝑁‘({𝑋} ∪ {𝑌}))) |
13 | eqid 2821 | . . . . 5 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
14 | 6, 13, 7 | lspsncl 19749 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑊)) |
15 | 1, 2, 14 | syl2anc 586 | . . 3 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑊)) |
16 | 6, 13, 7 | lspsncl 19749 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊)) |
17 | 1, 4, 16 | syl2anc 586 | . . 3 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊)) |
18 | lsmpr.p | . . . 4 ⊢ ⊕ = (LSSum‘𝑊) | |
19 | 13, 7, 18 | lsmsp 19858 | . . 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 1367 | . 2 ⊢ (𝜑 → ((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) = (𝑁‘((𝑁‘{𝑋}) ∪ (𝑁‘{𝑌})))) |
21 | 9, 12, 20 | 3eqtr4d 2866 | 1 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) = ((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ∪ cun 3934 ⊆ wss 3936 {csn 4567 {cpr 4569 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 LSSumclsm 18759 LModclmod 19634 LSubSpclss 19703 LSpanclspn 19743 |
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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-0g 16715 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-grp 18106 df-minusg 18107 df-sbg 18108 df-subg 18276 df-cntz 18447 df-lsm 18761 df-cmn 18908 df-abl 18909 df-mgp 19240 df-ur 19252 df-ring 19299 df-lmod 19636 df-lss 19704 df-lsp 19744 |
This theorem is referenced by: lsppreli 19862 lsmelpr 19863 lsppr0 19864 lspprabs 19867 lspabs2 19892 lspindpi 19904 lsmsat 36159 dvh4dimlem 38594 dvh3dim3N 38600 lclkrlem2c 38660 lcfrlem20 38713 lcfrlem23 38716 mapdindp 38822 mapdindp2 38872 mapdindp4 38874 mapdh6dN 38890 lspindp5 38921 hdmap1l6d 38964 hdmaprnlem3eN 39009 |
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