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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 19663 | . . . . 5 ⊢ (𝑊 ∈ LMod → 0 ∈ 𝑉) |
6 | 1, 5 | syl 17 | . . . 4 ⊢ (𝜑 → 0 ∈ 𝑉) |
7 | 6 | snssd 4742 | . . 3 ⊢ (𝜑 → { 0 } ⊆ 𝑉) |
8 | lspun0.n | . . . 4 ⊢ 𝑁 = (LSpan‘𝑊) | |
9 | 3, 8 | lspun 19759 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ⊆ 𝑉 ∧ { 0 } ⊆ 𝑉) → (𝑁‘(𝑋 ∪ { 0 })) = (𝑁‘((𝑁‘𝑋) ∪ (𝑁‘{ 0 })))) |
10 | 1, 2, 7, 9 | syl3anc 1367 | . 2 ⊢ (𝜑 → (𝑁‘(𝑋 ∪ { 0 })) = (𝑁‘((𝑁‘𝑋) ∪ (𝑁‘{ 0 })))) |
11 | 4, 8 | lspsn0 19780 | . . . . . . 7 ⊢ (𝑊 ∈ LMod → (𝑁‘{ 0 }) = { 0 }) |
12 | 1, 11 | syl 17 | . . . . . 6 ⊢ (𝜑 → (𝑁‘{ 0 }) = { 0 }) |
13 | 12 | uneq2d 4139 | . . . . 5 ⊢ (𝜑 → ((𝑁‘𝑋) ∪ (𝑁‘{ 0 })) = ((𝑁‘𝑋) ∪ { 0 })) |
14 | eqid 2821 | . . . . . . . . 9 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
15 | 3, 14, 8 | lspcl 19748 | . . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ⊆ 𝑉) → (𝑁‘𝑋) ∈ (LSubSp‘𝑊)) |
16 | 1, 2, 15 | syl2anc 586 | . . . . . . 7 ⊢ (𝜑 → (𝑁‘𝑋) ∈ (LSubSp‘𝑊)) |
17 | 4, 14 | lss0ss 19720 | . . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ (𝑁‘𝑋) ∈ (LSubSp‘𝑊)) → { 0 } ⊆ (𝑁‘𝑋)) |
18 | 1, 16, 17 | syl2anc 586 | . . . . . 6 ⊢ (𝜑 → { 0 } ⊆ (𝑁‘𝑋)) |
19 | ssequn2 4159 | . . . . . 6 ⊢ ({ 0 } ⊆ (𝑁‘𝑋) ↔ ((𝑁‘𝑋) ∪ { 0 }) = (𝑁‘𝑋)) | |
20 | 18, 19 | sylib 220 | . . . . 5 ⊢ (𝜑 → ((𝑁‘𝑋) ∪ { 0 }) = (𝑁‘𝑋)) |
21 | 13, 20 | eqtrd 2856 | . . . 4 ⊢ (𝜑 → ((𝑁‘𝑋) ∪ (𝑁‘{ 0 })) = (𝑁‘𝑋)) |
22 | 21 | fveq2d 6674 | . . 3 ⊢ (𝜑 → (𝑁‘((𝑁‘𝑋) ∪ (𝑁‘{ 0 }))) = (𝑁‘(𝑁‘𝑋))) |
23 | 3, 8 | lspidm 19758 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ⊆ 𝑉) → (𝑁‘(𝑁‘𝑋)) = (𝑁‘𝑋)) |
24 | 1, 2, 23 | syl2anc 586 | . . 3 ⊢ (𝜑 → (𝑁‘(𝑁‘𝑋)) = (𝑁‘𝑋)) |
25 | 22, 24 | eqtrd 2856 | . 2 ⊢ (𝜑 → (𝑁‘((𝑁‘𝑋) ∪ (𝑁‘{ 0 }))) = (𝑁‘𝑋)) |
26 | 10, 25 | eqtrd 2856 | 1 ⊢ (𝜑 → (𝑁‘(𝑋 ∪ { 0 })) = (𝑁‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ∪ cun 3934 ⊆ wss 3936 {csn 4567 ‘cfv 6355 Basecbs 16483 0gc0g 16713 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-plusg 16578 df-0g 16715 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-grp 18106 df-minusg 18107 df-sbg 18108 df-mgp 19240 df-ur 19252 df-ring 19299 df-lmod 19636 df-lss 19704 df-lsp 19744 |
This theorem is referenced by: dvh4dimN 38598 |
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