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Mirrors > Home > MPE Home > Th. List > lss0v | Structured version Visualization version GIF version |
Description: The zero vector in a submodule equals the zero vector in the including module. (Contributed by NM, 15-Mar-2015.) |
Ref | Expression |
---|---|
lss0v.x | ⊢ 𝑋 = (𝑊 ↾s 𝑈) |
lss0v.o | ⊢ 0 = (0g‘𝑊) |
lss0v.z | ⊢ 𝑍 = (0g‘𝑋) |
lss0v.l | ⊢ 𝐿 = (LSubSp‘𝑊) |
Ref | Expression |
---|---|
lss0v | ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿) → 𝑍 = 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ss 4005 | . . . . 5 ⊢ ∅ ⊆ 𝑈 | |
2 | lss0v.x | . . . . . 6 ⊢ 𝑋 = (𝑊 ↾s 𝑈) | |
3 | eqid 2651 | . . . . . 6 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊) | |
4 | eqid 2651 | . . . . . 6 ⊢ (LSpan‘𝑋) = (LSpan‘𝑋) | |
5 | lss0v.l | . . . . . 6 ⊢ 𝐿 = (LSubSp‘𝑊) | |
6 | 2, 3, 4, 5 | lsslsp 19063 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿 ∧ ∅ ⊆ 𝑈) → ((LSpan‘𝑊)‘∅) = ((LSpan‘𝑋)‘∅)) |
7 | 1, 6 | mp3an3 1453 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿) → ((LSpan‘𝑊)‘∅) = ((LSpan‘𝑋)‘∅)) |
8 | lss0v.o | . . . . . 6 ⊢ 0 = (0g‘𝑊) | |
9 | 8, 3 | lsp0 19057 | . . . . 5 ⊢ (𝑊 ∈ LMod → ((LSpan‘𝑊)‘∅) = { 0 }) |
10 | 9 | adantr 480 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿) → ((LSpan‘𝑊)‘∅) = { 0 }) |
11 | 2, 5 | lsslmod 19008 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿) → 𝑋 ∈ LMod) |
12 | lss0v.z | . . . . . 6 ⊢ 𝑍 = (0g‘𝑋) | |
13 | 12, 4 | lsp0 19057 | . . . . 5 ⊢ (𝑋 ∈ LMod → ((LSpan‘𝑋)‘∅) = {𝑍}) |
14 | 11, 13 | syl 17 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿) → ((LSpan‘𝑋)‘∅) = {𝑍}) |
15 | 7, 10, 14 | 3eqtr3rd 2694 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿) → {𝑍} = { 0 }) |
16 | 15 | unieqd 4478 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿) → ∪ {𝑍} = ∪ { 0 }) |
17 | fvex 6239 | . . . 4 ⊢ (0g‘𝑋) ∈ V | |
18 | 12, 17 | eqeltri 2726 | . . 3 ⊢ 𝑍 ∈ V |
19 | 18 | unisn 4483 | . 2 ⊢ ∪ {𝑍} = 𝑍 |
20 | fvex 6239 | . . . 4 ⊢ (0g‘𝑊) ∈ V | |
21 | 8, 20 | eqeltri 2726 | . . 3 ⊢ 0 ∈ V |
22 | 21 | unisn 4483 | . 2 ⊢ ∪ { 0 } = 0 |
23 | 16, 19, 22 | 3eqtr3g 2708 | 1 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿) → 𝑍 = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1523 ∈ wcel 2030 Vcvv 3231 ⊆ wss 3607 ∅c0 3948 {csn 4210 ∪ cuni 4468 ‘cfv 5926 (class class class)co 6690 ↾s cress 15905 0gc0g 16147 LModclmod 18911 LSubSpclss 18980 LSpanclspn 19019 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-5 11120 df-6 11121 df-ndx 15907 df-slot 15908 df-base 15910 df-sets 15911 df-ress 15912 df-plusg 16001 df-sca 16004 df-vsca 16005 df-0g 16149 df-mgm 17289 df-sgrp 17331 df-mnd 17342 df-grp 17472 df-minusg 17473 df-sbg 17474 df-subg 17638 df-mgp 18536 df-ur 18548 df-ring 18595 df-lmod 18913 df-lss 18981 df-lsp 19020 |
This theorem is referenced by: lcd0v 37217 |
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