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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 28505 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑊 ∈ NrmCVec) |
3 | eqid 2823 | . . . . . 6 ⊢ (BaseSet‘𝑊) = (BaseSet‘𝑊) | |
4 | sspz.q | . . . . . 6 ⊢ 𝑄 = (0vec‘𝑊) | |
5 | 3, 4 | nvzcl 28413 | . . . . 5 ⊢ (𝑊 ∈ NrmCVec → 𝑄 ∈ (BaseSet‘𝑊)) |
6 | 5, 5 | jca 514 | . . . 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 2823 | . . . 4 ⊢ ( −𝑣 ‘𝑈) = ( −𝑣 ‘𝑈) | |
9 | eqid 2823 | . . . 4 ⊢ ( −𝑣 ‘𝑊) = ( −𝑣 ‘𝑊) | |
10 | 3, 8, 9, 1 | sspmval 28512 | . . 3 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ (BaseSet‘𝑊) ∧ 𝑄 ∈ (BaseSet‘𝑊))) → (𝑄( −𝑣 ‘𝑊)𝑄) = (𝑄( −𝑣 ‘𝑈)𝑄)) |
11 | 7, 10 | mpdan 685 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (𝑄( −𝑣 ‘𝑊)𝑄) = (𝑄( −𝑣 ‘𝑈)𝑄)) |
12 | 3, 9, 4 | nvmid 28438 | . . 3 ⊢ ((𝑊 ∈ NrmCVec ∧ 𝑄 ∈ (BaseSet‘𝑊)) → (𝑄( −𝑣 ‘𝑊)𝑄) = 𝑄) |
13 | 2, 5, 12 | syl2anc2 587 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (𝑄( −𝑣 ‘𝑊)𝑄) = 𝑄) |
14 | eqid 2823 | . . . . 5 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈) | |
15 | 14, 3, 1 | sspba 28506 | . . . 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 3970 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑄 ∈ (BaseSet‘𝑈)) |
18 | sspz.z | . . . 4 ⊢ 𝑍 = (0vec‘𝑈) | |
19 | 14, 8, 18 | nvmid 28438 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑄 ∈ (BaseSet‘𝑈)) → (𝑄( −𝑣 ‘𝑈)𝑄) = 𝑍) |
20 | 17, 19 | syldan 593 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (𝑄( −𝑣 ‘𝑈)𝑄) = 𝑍) |
21 | 11, 13, 20 | 3eqtr3d 2866 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑄 = 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ‘cfv 6357 (class class class)co 7158 NrmCVeccnv 28363 BaseSetcba 28365 0veccn0v 28367 −𝑣 cnsb 28368 SubSpcss 28500 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-po 5476 df-so 5477 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-1st 7691 df-2nd 7692 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-ltxr 10682 df-sub 10874 df-neg 10875 df-grpo 28272 df-gid 28273 df-ginv 28274 df-gdiv 28275 df-ablo 28324 df-vc 28338 df-nv 28371 df-va 28374 df-ba 28375 df-sm 28376 df-0v 28377 df-vs 28378 df-nmcv 28379 df-ssp 28501 |
This theorem is referenced by: hhshsslem2 29047 |
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