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Theorem sspg 30414
Description: Vector addition on a subspace is a restriction of vector addition on the parent space. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
sspg.y 𝑌 = (BaseSet‘𝑊)
sspg.g 𝐺 = ( +𝑣𝑈)
sspg.f 𝐹 = ( +𝑣𝑊)
sspg.h 𝐻 = (SubSp‘𝑈)
Assertion
Ref Expression
sspg ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝐹 = (𝐺 ↾ (𝑌 × 𝑌)))

Proof of Theorem sspg
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2731 . . . . . . . . . . 11 (BaseSet‘𝑈) = (BaseSet‘𝑈)
2 sspg.g . . . . . . . . . . 11 𝐺 = ( +𝑣𝑈)
31, 2nvgf 30304 . . . . . . . . . 10 (𝑈 ∈ NrmCVec → 𝐺:((BaseSet‘𝑈) × (BaseSet‘𝑈))⟶(BaseSet‘𝑈))
43ffund 6721 . . . . . . . . 9 (𝑈 ∈ NrmCVec → Fun 𝐺)
54funresd 6591 . . . . . . . 8 (𝑈 ∈ NrmCVec → Fun (𝐺 ↾ (𝑌 × 𝑌)))
65adantr 480 . . . . . . 7 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → Fun (𝐺 ↾ (𝑌 × 𝑌)))
7 sspg.h . . . . . . . . . 10 𝐻 = (SubSp‘𝑈)
87sspnv 30412 . . . . . . . . 9 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑊 ∈ NrmCVec)
9 sspg.y . . . . . . . . . 10 𝑌 = (BaseSet‘𝑊)
10 sspg.f . . . . . . . . . 10 𝐹 = ( +𝑣𝑊)
119, 10nvgf 30304 . . . . . . . . 9 (𝑊 ∈ NrmCVec → 𝐹:(𝑌 × 𝑌)⟶𝑌)
128, 11syl 17 . . . . . . . 8 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝐹:(𝑌 × 𝑌)⟶𝑌)
1312ffnd 6718 . . . . . . 7 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝐹 Fn (𝑌 × 𝑌))
14 fnresdm 6669 . . . . . . . . 9 (𝐹 Fn (𝑌 × 𝑌) → (𝐹 ↾ (𝑌 × 𝑌)) = 𝐹)
1513, 14syl 17 . . . . . . . 8 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (𝐹 ↾ (𝑌 × 𝑌)) = 𝐹)
16 eqid 2731 . . . . . . . . . . . 12 ( ·𝑠OLD𝑈) = ( ·𝑠OLD𝑈)
17 eqid 2731 . . . . . . . . . . . 12 ( ·𝑠OLD𝑊) = ( ·𝑠OLD𝑊)
18 eqid 2731 . . . . . . . . . . . 12 (normCV𝑈) = (normCV𝑈)
19 eqid 2731 . . . . . . . . . . . 12 (normCV𝑊) = (normCV𝑊)
202, 10, 16, 17, 18, 19, 7isssp 30410 . . . . . . . . . . 11 (𝑈 ∈ NrmCVec → (𝑊𝐻 ↔ (𝑊 ∈ NrmCVec ∧ (𝐹𝐺 ∧ ( ·𝑠OLD𝑊) ⊆ ( ·𝑠OLD𝑈) ∧ (normCV𝑊) ⊆ (normCV𝑈)))))
2120simplbda 499 . . . . . . . . . 10 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (𝐹𝐺 ∧ ( ·𝑠OLD𝑊) ⊆ ( ·𝑠OLD𝑈) ∧ (normCV𝑊) ⊆ (normCV𝑈)))
2221simp1d 1141 . . . . . . . . 9 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝐹𝐺)
23 ssres 6008 . . . . . . . . 9 (𝐹𝐺 → (𝐹 ↾ (𝑌 × 𝑌)) ⊆ (𝐺 ↾ (𝑌 × 𝑌)))
2422, 23syl 17 . . . . . . . 8 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (𝐹 ↾ (𝑌 × 𝑌)) ⊆ (𝐺 ↾ (𝑌 × 𝑌)))
2515, 24eqsstrrd 4021 . . . . . . 7 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝐹 ⊆ (𝐺 ↾ (𝑌 × 𝑌)))
266, 13, 253jca 1127 . . . . . 6 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (Fun (𝐺 ↾ (𝑌 × 𝑌)) ∧ 𝐹 Fn (𝑌 × 𝑌) ∧ 𝐹 ⊆ (𝐺 ↾ (𝑌 × 𝑌))))
27 oprssov 7580 . . . . . 6 (((Fun (𝐺 ↾ (𝑌 × 𝑌)) ∧ 𝐹 Fn (𝑌 × 𝑌) ∧ 𝐹 ⊆ (𝐺 ↾ (𝑌 × 𝑌))) ∧ (𝑥𝑌𝑦𝑌)) → (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦) = (𝑥𝐹𝑦))
2826, 27sylan 579 . . . . 5 (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ (𝑥𝑌𝑦𝑌)) → (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦) = (𝑥𝐹𝑦))
2928eqcomd 2737 . . . 4 (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ (𝑥𝑌𝑦𝑌)) → (𝑥𝐹𝑦) = (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦))
3029ralrimivva 3199 . . 3 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → ∀𝑥𝑌𝑦𝑌 (𝑥𝐹𝑦) = (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦))
31 eqid 2731 . . 3 (𝑌 × 𝑌) = (𝑌 × 𝑌)
3230, 31jctil 519 . 2 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → ((𝑌 × 𝑌) = (𝑌 × 𝑌) ∧ ∀𝑥𝑌𝑦𝑌 (𝑥𝐹𝑦) = (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦)))
333ffnd 6718 . . . . 5 (𝑈 ∈ NrmCVec → 𝐺 Fn ((BaseSet‘𝑈) × (BaseSet‘𝑈)))
3433adantr 480 . . . 4 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝐺 Fn ((BaseSet‘𝑈) × (BaseSet‘𝑈)))
351, 9, 7sspba 30413 . . . . 5 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑌 ⊆ (BaseSet‘𝑈))
36 xpss12 5691 . . . . 5 ((𝑌 ⊆ (BaseSet‘𝑈) ∧ 𝑌 ⊆ (BaseSet‘𝑈)) → (𝑌 × 𝑌) ⊆ ((BaseSet‘𝑈) × (BaseSet‘𝑈)))
3735, 35, 36syl2anc 583 . . . 4 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (𝑌 × 𝑌) ⊆ ((BaseSet‘𝑈) × (BaseSet‘𝑈)))
38 fnssres 6673 . . . 4 ((𝐺 Fn ((BaseSet‘𝑈) × (BaseSet‘𝑈)) ∧ (𝑌 × 𝑌) ⊆ ((BaseSet‘𝑈) × (BaseSet‘𝑈))) → (𝐺 ↾ (𝑌 × 𝑌)) Fn (𝑌 × 𝑌))
3934, 37, 38syl2anc 583 . . 3 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (𝐺 ↾ (𝑌 × 𝑌)) Fn (𝑌 × 𝑌))
40 eqfnov 7541 . . 3 ((𝐹 Fn (𝑌 × 𝑌) ∧ (𝐺 ↾ (𝑌 × 𝑌)) Fn (𝑌 × 𝑌)) → (𝐹 = (𝐺 ↾ (𝑌 × 𝑌)) ↔ ((𝑌 × 𝑌) = (𝑌 × 𝑌) ∧ ∀𝑥𝑌𝑦𝑌 (𝑥𝐹𝑦) = (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦))))
4113, 39, 40syl2anc 583 . 2 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (𝐹 = (𝐺 ↾ (𝑌 × 𝑌)) ↔ ((𝑌 × 𝑌) = (𝑌 × 𝑌) ∧ ∀𝑥𝑌𝑦𝑌 (𝑥𝐹𝑦) = (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦))))
4232, 41mpbird 257 1 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝐹 = (𝐺 ↾ (𝑌 × 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1086   = wceq 1540  wcel 2105  wral 3060  wss 3948   × cxp 5674  cres 5678  Fun wfun 6537   Fn wfn 6538  wf 6539  cfv 6543  (class class class)co 7412  NrmCVeccnv 30270   +𝑣 cpv 30271  BaseSetcba 30272   ·𝑠OLD cns 30273  normCVcnmcv 30276  SubSpcss 30407
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7415  df-oprab 7416  df-1st 7979  df-2nd 7980  df-grpo 30179  df-ablo 30231  df-vc 30245  df-nv 30278  df-va 30281  df-ba 30282  df-sm 30283  df-0v 30284  df-nmcv 30286  df-ssp 30408
This theorem is referenced by:  sspgval  30415
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