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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 4353 | . . . . 5 ⊢ ∅ ⊆ 𝑈 | |
2 | lss0v.x | . . . . . 6 ⊢ 𝑋 = (𝑊 ↾s 𝑈) | |
3 | eqid 2824 | . . . . . 6 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊) | |
4 | eqid 2824 | . . . . . 6 ⊢ (LSpan‘𝑋) = (LSpan‘𝑋) | |
5 | lss0v.l | . . . . . 6 ⊢ 𝐿 = (LSubSp‘𝑊) | |
6 | 2, 3, 4, 5 | lsslsp 19790 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿 ∧ ∅ ⊆ 𝑈) → ((LSpan‘𝑊)‘∅) = ((LSpan‘𝑋)‘∅)) |
7 | 1, 6 | mp3an3 1446 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿) → ((LSpan‘𝑊)‘∅) = ((LSpan‘𝑋)‘∅)) |
8 | lss0v.o | . . . . . 6 ⊢ 0 = (0g‘𝑊) | |
9 | 8, 3 | lsp0 19784 | . . . . 5 ⊢ (𝑊 ∈ LMod → ((LSpan‘𝑊)‘∅) = { 0 }) |
10 | 9 | adantr 483 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿) → ((LSpan‘𝑊)‘∅) = { 0 }) |
11 | 2, 5 | lsslmod 19735 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿) → 𝑋 ∈ LMod) |
12 | lss0v.z | . . . . . 6 ⊢ 𝑍 = (0g‘𝑋) | |
13 | 12, 4 | lsp0 19784 | . . . . 5 ⊢ (𝑋 ∈ LMod → ((LSpan‘𝑋)‘∅) = {𝑍}) |
14 | 11, 13 | syl 17 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿) → ((LSpan‘𝑋)‘∅) = {𝑍}) |
15 | 7, 10, 14 | 3eqtr3rd 2868 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿) → {𝑍} = { 0 }) |
16 | 15 | unieqd 4855 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿) → ∪ {𝑍} = ∪ { 0 }) |
17 | 12 | fvexi 6687 | . . 3 ⊢ 𝑍 ∈ V |
18 | 17 | unisn 4861 | . 2 ⊢ ∪ {𝑍} = 𝑍 |
19 | 8 | fvexi 6687 | . . 3 ⊢ 0 ∈ V |
20 | 19 | unisn 4861 | . 2 ⊢ ∪ { 0 } = 0 |
21 | 16, 18, 20 | 3eqtr3g 2882 | 1 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿) → 𝑍 = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ⊆ wss 3939 ∅c0 4294 {csn 4570 ∪ cuni 4841 ‘cfv 6358 (class class class)co 7159 ↾s cress 16487 0gc0g 16716 LModclmod 19637 LSubSpclss 19706 LSpanclspn 19746 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-ress 16494 df-plusg 16581 df-sca 16584 df-vsca 16585 df-0g 16718 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-grp 18109 df-minusg 18110 df-sbg 18111 df-subg 18279 df-mgp 19243 df-ur 19255 df-ring 19302 df-lmod 19639 df-lss 19707 df-lsp 19747 |
This theorem is referenced by: phlssphl 20806 lcd0v 38751 |
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