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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 3972 | . . . . 5 ⊢ ∅ ⊆ 𝑈 | |
2 | lss0v.x | . . . . . 6 ⊢ 𝑋 = (𝑊 ↾s 𝑈) | |
3 | eqid 2622 | . . . . . 6 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊) | |
4 | eqid 2622 | . . . . . 6 ⊢ (LSpan‘𝑋) = (LSpan‘𝑋) | |
5 | lss0v.l | . . . . . 6 ⊢ 𝐿 = (LSubSp‘𝑊) | |
6 | 2, 3, 4, 5 | lsslsp 19015 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿 ∧ ∅ ⊆ 𝑈) → ((LSpan‘𝑊)‘∅) = ((LSpan‘𝑋)‘∅)) |
7 | 1, 6 | mp3an3 1413 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿) → ((LSpan‘𝑊)‘∅) = ((LSpan‘𝑋)‘∅)) |
8 | lss0v.o | . . . . . 6 ⊢ 0 = (0g‘𝑊) | |
9 | 8, 3 | lsp0 19009 | . . . . 5 ⊢ (𝑊 ∈ LMod → ((LSpan‘𝑊)‘∅) = { 0 }) |
10 | 9 | adantr 481 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿) → ((LSpan‘𝑊)‘∅) = { 0 }) |
11 | 2, 5 | lsslmod 18960 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿) → 𝑋 ∈ LMod) |
12 | lss0v.z | . . . . . 6 ⊢ 𝑍 = (0g‘𝑋) | |
13 | 12, 4 | lsp0 19009 | . . . . 5 ⊢ (𝑋 ∈ LMod → ((LSpan‘𝑋)‘∅) = {𝑍}) |
14 | 11, 13 | syl 17 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿) → ((LSpan‘𝑋)‘∅) = {𝑍}) |
15 | 7, 10, 14 | 3eqtr3rd 2665 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿) → {𝑍} = { 0 }) |
16 | 15 | unieqd 4446 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿) → ∪ {𝑍} = ∪ { 0 }) |
17 | fvex 6201 | . . . 4 ⊢ (0g‘𝑋) ∈ V | |
18 | 12, 17 | eqeltri 2697 | . . 3 ⊢ 𝑍 ∈ V |
19 | 18 | unisn 4451 | . 2 ⊢ ∪ {𝑍} = 𝑍 |
20 | fvex 6201 | . . . 4 ⊢ (0g‘𝑊) ∈ V | |
21 | 8, 20 | eqeltri 2697 | . . 3 ⊢ 0 ∈ V |
22 | 21 | unisn 4451 | . 2 ⊢ ∪ { 0 } = 0 |
23 | 16, 19, 22 | 3eqtr3g 2679 | 1 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿) → 𝑍 = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ⊆ wss 3574 ∅c0 3915 {csn 4177 ∪ cuni 4436 ‘cfv 5888 (class class class)co 6650 ↾s cress 15858 0gc0g 16100 LModclmod 18863 LSubSpclss 18932 LSpanclspn 18971 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-sca 15957 df-vsca 15958 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 df-sbg 17427 df-subg 17591 df-mgp 18490 df-ur 18502 df-ring 18549 df-lmod 18865 df-lss 18933 df-lsp 18972 |
This theorem is referenced by: lcd0v 36900 |
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