![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 30755 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑊 ∈ NrmCVec) |
3 | eqid 2735 | . . . . . 6 ⊢ (BaseSet‘𝑊) = (BaseSet‘𝑊) | |
4 | sspz.q | . . . . . 6 ⊢ 𝑄 = (0vec‘𝑊) | |
5 | 3, 4 | nvzcl 30663 | . . . . 5 ⊢ (𝑊 ∈ NrmCVec → 𝑄 ∈ (BaseSet‘𝑊)) |
6 | 5, 5 | jca 511 | . . . 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 2735 | . . . 4 ⊢ ( −𝑣 ‘𝑈) = ( −𝑣 ‘𝑈) | |
9 | eqid 2735 | . . . 4 ⊢ ( −𝑣 ‘𝑊) = ( −𝑣 ‘𝑊) | |
10 | 3, 8, 9, 1 | sspmval 30762 | . . 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 30688 | . . 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 2735 | . . . . 5 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈) | |
15 | 14, 3, 1 | sspba 30756 | . . . 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 3996 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑄 ∈ (BaseSet‘𝑈)) |
18 | sspz.z | . . . 4 ⊢ 𝑍 = (0vec‘𝑈) | |
19 | 14, 8, 18 | nvmid 30688 | . . 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 2783 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑄 = 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ‘cfv 6563 (class class class)co 7431 NrmCVeccnv 30613 BaseSetcba 30615 0veccn0v 30617 −𝑣 cnsb 30618 SubSpcss 30750 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-ltxr 11298 df-sub 11492 df-neg 11493 df-grpo 30522 df-gid 30523 df-ginv 30524 df-gdiv 30525 df-ablo 30574 df-vc 30588 df-nv 30621 df-va 30624 df-ba 30625 df-sm 30626 df-0v 30627 df-vs 30628 df-nmcv 30629 df-ssp 30751 |
This theorem is referenced by: hhshsslem2 31297 |
Copyright terms: Public domain | W3C validator |