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Mirrors > Home > MPE Home > Th. List > lspun0 | Structured version Visualization version GIF version |
Description: The span of a union with the zero subspace. (Contributed by NM, 22-May-2015.) |
Ref | Expression |
---|---|
lspun0.v | ⊢ 𝑉 = (Base‘𝑊) |
lspun0.o | ⊢ 0 = (0g‘𝑊) |
lspun0.n | ⊢ 𝑁 = (LSpan‘𝑊) |
lspun0.w | ⊢ (𝜑 → 𝑊 ∈ LMod) |
lspun0.x | ⊢ (𝜑 → 𝑋 ⊆ 𝑉) |
Ref | Expression |
---|---|
lspun0 | ⊢ (𝜑 → (𝑁‘(𝑋 ∪ { 0 })) = (𝑁‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lspun0.w | . . 3 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
2 | lspun0.x | . . 3 ⊢ (𝜑 → 𝑋 ⊆ 𝑉) | |
3 | lspun0.v | . . . . . 6 ⊢ 𝑉 = (Base‘𝑊) | |
4 | lspun0.o | . . . . . 6 ⊢ 0 = (0g‘𝑊) | |
5 | 3, 4 | lmod0vcl 19656 | . . . . 5 ⊢ (𝑊 ∈ LMod → 0 ∈ 𝑉) |
6 | 1, 5 | syl 17 | . . . 4 ⊢ (𝜑 → 0 ∈ 𝑉) |
7 | 6 | snssd 4702 | . . 3 ⊢ (𝜑 → { 0 } ⊆ 𝑉) |
8 | lspun0.n | . . . 4 ⊢ 𝑁 = (LSpan‘𝑊) | |
9 | 3, 8 | lspun 19752 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ⊆ 𝑉 ∧ { 0 } ⊆ 𝑉) → (𝑁‘(𝑋 ∪ { 0 })) = (𝑁‘((𝑁‘𝑋) ∪ (𝑁‘{ 0 })))) |
10 | 1, 2, 7, 9 | syl3anc 1368 | . 2 ⊢ (𝜑 → (𝑁‘(𝑋 ∪ { 0 })) = (𝑁‘((𝑁‘𝑋) ∪ (𝑁‘{ 0 })))) |
11 | 4, 8 | lspsn0 19773 | . . . . . . 7 ⊢ (𝑊 ∈ LMod → (𝑁‘{ 0 }) = { 0 }) |
12 | 1, 11 | syl 17 | . . . . . 6 ⊢ (𝜑 → (𝑁‘{ 0 }) = { 0 }) |
13 | 12 | uneq2d 4090 | . . . . 5 ⊢ (𝜑 → ((𝑁‘𝑋) ∪ (𝑁‘{ 0 })) = ((𝑁‘𝑋) ∪ { 0 })) |
14 | eqid 2798 | . . . . . . . . 9 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
15 | 3, 14, 8 | lspcl 19741 | . . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ⊆ 𝑉) → (𝑁‘𝑋) ∈ (LSubSp‘𝑊)) |
16 | 1, 2, 15 | syl2anc 587 | . . . . . . 7 ⊢ (𝜑 → (𝑁‘𝑋) ∈ (LSubSp‘𝑊)) |
17 | 4, 14 | lss0ss 19713 | . . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ (𝑁‘𝑋) ∈ (LSubSp‘𝑊)) → { 0 } ⊆ (𝑁‘𝑋)) |
18 | 1, 16, 17 | syl2anc 587 | . . . . . 6 ⊢ (𝜑 → { 0 } ⊆ (𝑁‘𝑋)) |
19 | ssequn2 4110 | . . . . . 6 ⊢ ({ 0 } ⊆ (𝑁‘𝑋) ↔ ((𝑁‘𝑋) ∪ { 0 }) = (𝑁‘𝑋)) | |
20 | 18, 19 | sylib 221 | . . . . 5 ⊢ (𝜑 → ((𝑁‘𝑋) ∪ { 0 }) = (𝑁‘𝑋)) |
21 | 13, 20 | eqtrd 2833 | . . . 4 ⊢ (𝜑 → ((𝑁‘𝑋) ∪ (𝑁‘{ 0 })) = (𝑁‘𝑋)) |
22 | 21 | fveq2d 6649 | . . 3 ⊢ (𝜑 → (𝑁‘((𝑁‘𝑋) ∪ (𝑁‘{ 0 }))) = (𝑁‘(𝑁‘𝑋))) |
23 | 3, 8 | lspidm 19751 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ⊆ 𝑉) → (𝑁‘(𝑁‘𝑋)) = (𝑁‘𝑋)) |
24 | 1, 2, 23 | syl2anc 587 | . . 3 ⊢ (𝜑 → (𝑁‘(𝑁‘𝑋)) = (𝑁‘𝑋)) |
25 | 22, 24 | eqtrd 2833 | . 2 ⊢ (𝜑 → (𝑁‘((𝑁‘𝑋) ∪ (𝑁‘{ 0 }))) = (𝑁‘𝑋)) |
26 | 10, 25 | eqtrd 2833 | 1 ⊢ (𝜑 → (𝑁‘(𝑋 ∪ { 0 })) = (𝑁‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ∪ cun 3879 ⊆ wss 3881 {csn 4525 ‘cfv 6324 Basecbs 16475 0gc0g 16705 LModclmod 19627 LSubSpclss 19696 LSpanclspn 19736 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-plusg 16570 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-minusg 18099 df-sbg 18100 df-mgp 19233 df-ur 19245 df-ring 19292 df-lmod 19629 df-lss 19697 df-lsp 19737 |
This theorem is referenced by: dvh4dimN 38743 |
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