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Mirrors > Home > MPE Home > Th. List > sspz | Structured version Visualization version GIF version |
Description: The zero vector of a subspace is the same as the parent's. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.) |
Ref | Expression |
---|---|
sspz.z | ⊢ 𝑍 = (0vec‘𝑈) |
sspz.q | ⊢ 𝑄 = (0vec‘𝑊) |
sspz.h | ⊢ 𝐻 = (SubSp‘𝑈) |
Ref | Expression |
---|---|
sspz | ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑄 = 𝑍) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sspz.h | . . . . 5 ⊢ 𝐻 = (SubSp‘𝑈) | |
2 | 1 | sspnv 28430 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑊 ∈ NrmCVec) |
3 | eqid 2818 | . . . . . 6 ⊢ (BaseSet‘𝑊) = (BaseSet‘𝑊) | |
4 | sspz.q | . . . . . 6 ⊢ 𝑄 = (0vec‘𝑊) | |
5 | 3, 4 | nvzcl 28338 | . . . . 5 ⊢ (𝑊 ∈ NrmCVec → 𝑄 ∈ (BaseSet‘𝑊)) |
6 | 5, 5 | jca 512 | . . . 4 ⊢ (𝑊 ∈ NrmCVec → (𝑄 ∈ (BaseSet‘𝑊) ∧ 𝑄 ∈ (BaseSet‘𝑊))) |
7 | 2, 6 | syl 17 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (𝑄 ∈ (BaseSet‘𝑊) ∧ 𝑄 ∈ (BaseSet‘𝑊))) |
8 | eqid 2818 | . . . 4 ⊢ ( −𝑣 ‘𝑈) = ( −𝑣 ‘𝑈) | |
9 | eqid 2818 | . . . 4 ⊢ ( −𝑣 ‘𝑊) = ( −𝑣 ‘𝑊) | |
10 | 3, 8, 9, 1 | sspmval 28437 | . . 3 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ (BaseSet‘𝑊) ∧ 𝑄 ∈ (BaseSet‘𝑊))) → (𝑄( −𝑣 ‘𝑊)𝑄) = (𝑄( −𝑣 ‘𝑈)𝑄)) |
11 | 7, 10 | mpdan 683 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (𝑄( −𝑣 ‘𝑊)𝑄) = (𝑄( −𝑣 ‘𝑈)𝑄)) |
12 | 3, 9, 4 | nvmid 28363 | . . 3 ⊢ ((𝑊 ∈ NrmCVec ∧ 𝑄 ∈ (BaseSet‘𝑊)) → (𝑄( −𝑣 ‘𝑊)𝑄) = 𝑄) |
13 | 2, 5, 12 | syl2anc2 585 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (𝑄( −𝑣 ‘𝑊)𝑄) = 𝑄) |
14 | eqid 2818 | . . . . 5 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈) | |
15 | 14, 3, 1 | sspba 28431 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (BaseSet‘𝑊) ⊆ (BaseSet‘𝑈)) |
16 | 2, 5 | syl 17 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑄 ∈ (BaseSet‘𝑊)) |
17 | 15, 16 | sseldd 3965 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑄 ∈ (BaseSet‘𝑈)) |
18 | sspz.z | . . . 4 ⊢ 𝑍 = (0vec‘𝑈) | |
19 | 14, 8, 18 | nvmid 28363 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑄 ∈ (BaseSet‘𝑈)) → (𝑄( −𝑣 ‘𝑈)𝑄) = 𝑍) |
20 | 17, 19 | syldan 591 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (𝑄( −𝑣 ‘𝑈)𝑄) = 𝑍) |
21 | 11, 13, 20 | 3eqtr3d 2861 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑄 = 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ‘cfv 6348 (class class class)co 7145 NrmCVeccnv 28288 BaseSetcba 28290 0veccn0v 28292 −𝑣 cnsb 28293 SubSpcss 28425 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-1st 7678 df-2nd 7679 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-ltxr 10668 df-sub 10860 df-neg 10861 df-grpo 28197 df-gid 28198 df-ginv 28199 df-gdiv 28200 df-ablo 28249 df-vc 28263 df-nv 28296 df-va 28299 df-ba 28300 df-sm 28301 df-0v 28302 df-vs 28303 df-nmcv 28304 df-ssp 28426 |
This theorem is referenced by: hhshsslem2 28972 |
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