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Mirrors > Home > MPE Home > Th. List > sspm | Structured version Visualization version GIF version |
Description: Vector subtraction on a subspace is a restriction of vector subtraction on the parent space. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.) |
Ref | Expression |
---|---|
sspm.y | ⊢ 𝑌 = (BaseSet‘𝑊) |
sspm.m | ⊢ 𝑀 = ( −𝑣 ‘𝑈) |
sspm.l | ⊢ 𝐿 = ( −𝑣 ‘𝑊) |
sspm.h | ⊢ 𝐻 = (SubSp‘𝑈) |
Ref | Expression |
---|---|
sspm | ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝐿 = (𝑀 ↾ (𝑌 × 𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sspm.y | . 2 ⊢ 𝑌 = (BaseSet‘𝑊) | |
2 | sspm.h | . 2 ⊢ 𝐻 = (SubSp‘𝑈) | |
3 | sspm.m | . . 3 ⊢ 𝑀 = ( −𝑣 ‘𝑈) | |
4 | sspm.l | . . 3 ⊢ 𝐿 = ( −𝑣 ‘𝑊) | |
5 | 1, 3, 4, 2 | sspmval 28504 | . 2 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → (𝑥𝐿𝑦) = (𝑥𝑀𝑦)) |
6 | 1, 4 | nvmf 28416 | . 2 ⊢ (𝑊 ∈ NrmCVec → 𝐿:(𝑌 × 𝑌)⟶𝑌) |
7 | eqid 2821 | . . 3 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈) | |
8 | 7, 3 | nvmf 28416 | . 2 ⊢ (𝑈 ∈ NrmCVec → 𝑀:((BaseSet‘𝑈) × (BaseSet‘𝑈))⟶(BaseSet‘𝑈)) |
9 | 1, 2, 5, 6, 8 | sspmlem 28503 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝐿 = (𝑀 ↾ (𝑌 × 𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 × cxp 5547 ↾ cres 5551 ‘cfv 6349 NrmCVeccnv 28355 BaseSetcba 28357 −𝑣 cnsb 28360 SubSpcss 28492 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-po 5468 df-so 5469 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-ltxr 10674 df-sub 10866 df-neg 10867 df-grpo 28264 df-gid 28265 df-ginv 28266 df-gdiv 28267 df-ablo 28316 df-vc 28330 df-nv 28363 df-va 28366 df-ba 28367 df-sm 28368 df-0v 28369 df-vs 28370 df-nmcv 28371 df-ssp 28493 |
This theorem is referenced by: hhssvs 29043 |
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