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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 28761 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑊 ∈ NrmCVec) |
3 | eqid 2736 | . . . . . 6 ⊢ (BaseSet‘𝑊) = (BaseSet‘𝑊) | |
4 | sspz.q | . . . . . 6 ⊢ 𝑄 = (0vec‘𝑊) | |
5 | 3, 4 | nvzcl 28669 | . . . . 5 ⊢ (𝑊 ∈ NrmCVec → 𝑄 ∈ (BaseSet‘𝑊)) |
6 | 5, 5 | jca 515 | . . . 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 2736 | . . . 4 ⊢ ( −𝑣 ‘𝑈) = ( −𝑣 ‘𝑈) | |
9 | eqid 2736 | . . . 4 ⊢ ( −𝑣 ‘𝑊) = ( −𝑣 ‘𝑊) | |
10 | 3, 8, 9, 1 | sspmval 28768 | . . 3 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ (BaseSet‘𝑊) ∧ 𝑄 ∈ (BaseSet‘𝑊))) → (𝑄( −𝑣 ‘𝑊)𝑄) = (𝑄( −𝑣 ‘𝑈)𝑄)) |
11 | 7, 10 | mpdan 687 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (𝑄( −𝑣 ‘𝑊)𝑄) = (𝑄( −𝑣 ‘𝑈)𝑄)) |
12 | 3, 9, 4 | nvmid 28694 | . . 3 ⊢ ((𝑊 ∈ NrmCVec ∧ 𝑄 ∈ (BaseSet‘𝑊)) → (𝑄( −𝑣 ‘𝑊)𝑄) = 𝑄) |
13 | 2, 5, 12 | syl2anc2 588 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (𝑄( −𝑣 ‘𝑊)𝑄) = 𝑄) |
14 | eqid 2736 | . . . . 5 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈) | |
15 | 14, 3, 1 | sspba 28762 | . . . 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 3888 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑄 ∈ (BaseSet‘𝑈)) |
18 | sspz.z | . . . 4 ⊢ 𝑍 = (0vec‘𝑈) | |
19 | 14, 8, 18 | nvmid 28694 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑄 ∈ (BaseSet‘𝑈)) → (𝑄( −𝑣 ‘𝑈)𝑄) = 𝑍) |
20 | 17, 19 | syldan 594 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (𝑄( −𝑣 ‘𝑈)𝑄) = 𝑍) |
21 | 11, 13, 20 | 3eqtr3d 2779 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑄 = 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ‘cfv 6358 (class class class)co 7191 NrmCVeccnv 28619 BaseSetcba 28621 0veccn0v 28623 −𝑣 cnsb 28624 SubSpcss 28756 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-po 5453 df-so 5454 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-1st 7739 df-2nd 7740 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-pnf 10834 df-mnf 10835 df-ltxr 10837 df-sub 11029 df-neg 11030 df-grpo 28528 df-gid 28529 df-ginv 28530 df-gdiv 28531 df-ablo 28580 df-vc 28594 df-nv 28627 df-va 28630 df-ba 28631 df-sm 28632 df-0v 28633 df-vs 28634 df-nmcv 28635 df-ssp 28757 |
This theorem is referenced by: hhshsslem2 29303 |
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