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Theorem sspg 27923
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 2771 . . . . . . . . . . 11 (BaseSet‘𝑈) = (BaseSet‘𝑈)
2 sspg.g . . . . . . . . . . 11 𝐺 = ( +𝑣𝑈)
31, 2nvgf 27813 . . . . . . . . . 10 (𝑈 ∈ NrmCVec → 𝐺:((BaseSet‘𝑈) × (BaseSet‘𝑈))⟶(BaseSet‘𝑈))
4 ffun 6188 . . . . . . . . . 10 (𝐺:((BaseSet‘𝑈) × (BaseSet‘𝑈))⟶(BaseSet‘𝑈) → Fun 𝐺)
53, 4syl 17 . . . . . . . . 9 (𝑈 ∈ NrmCVec → Fun 𝐺)
6 funres 6072 . . . . . . . . 9 (Fun 𝐺 → Fun (𝐺 ↾ (𝑌 × 𝑌)))
75, 6syl 17 . . . . . . . 8 (𝑈 ∈ NrmCVec → Fun (𝐺 ↾ (𝑌 × 𝑌)))
87adantr 466 . . . . . . 7 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → Fun (𝐺 ↾ (𝑌 × 𝑌)))
9 sspg.h . . . . . . . . . 10 𝐻 = (SubSp‘𝑈)
109sspnv 27921 . . . . . . . . 9 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑊 ∈ NrmCVec)
11 sspg.y . . . . . . . . . 10 𝑌 = (BaseSet‘𝑊)
12 sspg.f . . . . . . . . . 10 𝐹 = ( +𝑣𝑊)
1311, 12nvgf 27813 . . . . . . . . 9 (𝑊 ∈ NrmCVec → 𝐹:(𝑌 × 𝑌)⟶𝑌)
1410, 13syl 17 . . . . . . . 8 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝐹:(𝑌 × 𝑌)⟶𝑌)
15 ffn 6185 . . . . . . . 8 (𝐹:(𝑌 × 𝑌)⟶𝑌𝐹 Fn (𝑌 × 𝑌))
1614, 15syl 17 . . . . . . 7 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝐹 Fn (𝑌 × 𝑌))
17 fnresdm 6140 . . . . . . . . 9 (𝐹 Fn (𝑌 × 𝑌) → (𝐹 ↾ (𝑌 × 𝑌)) = 𝐹)
1816, 17syl 17 . . . . . . . 8 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (𝐹 ↾ (𝑌 × 𝑌)) = 𝐹)
19 eqid 2771 . . . . . . . . . . . 12 ( ·𝑠OLD𝑈) = ( ·𝑠OLD𝑈)
20 eqid 2771 . . . . . . . . . . . 12 ( ·𝑠OLD𝑊) = ( ·𝑠OLD𝑊)
21 eqid 2771 . . . . . . . . . . . 12 (normCV𝑈) = (normCV𝑈)
22 eqid 2771 . . . . . . . . . . . 12 (normCV𝑊) = (normCV𝑊)
232, 12, 19, 20, 21, 22, 9isssp 27919 . . . . . . . . . . 11 (𝑈 ∈ NrmCVec → (𝑊𝐻 ↔ (𝑊 ∈ NrmCVec ∧ (𝐹𝐺 ∧ ( ·𝑠OLD𝑊) ⊆ ( ·𝑠OLD𝑈) ∧ (normCV𝑊) ⊆ (normCV𝑈)))))
2423simplbda 487 . . . . . . . . . 10 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (𝐹𝐺 ∧ ( ·𝑠OLD𝑊) ⊆ ( ·𝑠OLD𝑈) ∧ (normCV𝑊) ⊆ (normCV𝑈)))
2524simp1d 1136 . . . . . . . . 9 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝐹𝐺)
26 ssres 5565 . . . . . . . . 9 (𝐹𝐺 → (𝐹 ↾ (𝑌 × 𝑌)) ⊆ (𝐺 ↾ (𝑌 × 𝑌)))
2725, 26syl 17 . . . . . . . 8 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (𝐹 ↾ (𝑌 × 𝑌)) ⊆ (𝐺 ↾ (𝑌 × 𝑌)))
2818, 27eqsstr3d 3789 . . . . . . 7 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝐹 ⊆ (𝐺 ↾ (𝑌 × 𝑌)))
298, 16, 283jca 1122 . . . . . 6 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (Fun (𝐺 ↾ (𝑌 × 𝑌)) ∧ 𝐹 Fn (𝑌 × 𝑌) ∧ 𝐹 ⊆ (𝐺 ↾ (𝑌 × 𝑌))))
30 oprssov 6950 . . . . . 6 (((Fun (𝐺 ↾ (𝑌 × 𝑌)) ∧ 𝐹 Fn (𝑌 × 𝑌) ∧ 𝐹 ⊆ (𝐺 ↾ (𝑌 × 𝑌))) ∧ (𝑥𝑌𝑦𝑌)) → (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦) = (𝑥𝐹𝑦))
3129, 30sylan 569 . . . . 5 (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ (𝑥𝑌𝑦𝑌)) → (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦) = (𝑥𝐹𝑦))
3231eqcomd 2777 . . . 4 (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ (𝑥𝑌𝑦𝑌)) → (𝑥𝐹𝑦) = (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦))
3332ralrimivva 3120 . . 3 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → ∀𝑥𝑌𝑦𝑌 (𝑥𝐹𝑦) = (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦))
34 eqid 2771 . . 3 (𝑌 × 𝑌) = (𝑌 × 𝑌)
3533, 34jctil 509 . 2 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → ((𝑌 × 𝑌) = (𝑌 × 𝑌) ∧ ∀𝑥𝑌𝑦𝑌 (𝑥𝐹𝑦) = (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦)))
36 ffn 6185 . . . . . 6 (𝐺:((BaseSet‘𝑈) × (BaseSet‘𝑈))⟶(BaseSet‘𝑈) → 𝐺 Fn ((BaseSet‘𝑈) × (BaseSet‘𝑈)))
373, 36syl 17 . . . . 5 (𝑈 ∈ NrmCVec → 𝐺 Fn ((BaseSet‘𝑈) × (BaseSet‘𝑈)))
3837adantr 466 . . . 4 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝐺 Fn ((BaseSet‘𝑈) × (BaseSet‘𝑈)))
391, 11, 9sspba 27922 . . . . 5 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑌 ⊆ (BaseSet‘𝑈))
40 xpss12 5264 . . . . 5 ((𝑌 ⊆ (BaseSet‘𝑈) ∧ 𝑌 ⊆ (BaseSet‘𝑈)) → (𝑌 × 𝑌) ⊆ ((BaseSet‘𝑈) × (BaseSet‘𝑈)))
4139, 39, 40syl2anc 573 . . . 4 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (𝑌 × 𝑌) ⊆ ((BaseSet‘𝑈) × (BaseSet‘𝑈)))
42 fnssres 6144 . . . 4 ((𝐺 Fn ((BaseSet‘𝑈) × (BaseSet‘𝑈)) ∧ (𝑌 × 𝑌) ⊆ ((BaseSet‘𝑈) × (BaseSet‘𝑈))) → (𝐺 ↾ (𝑌 × 𝑌)) Fn (𝑌 × 𝑌))
4338, 41, 42syl2anc 573 . . 3 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (𝐺 ↾ (𝑌 × 𝑌)) Fn (𝑌 × 𝑌))
44 eqfnov 6913 . . 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))))
4516, 43, 44syl2anc 573 . 2 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (𝐹 = (𝐺 ↾ (𝑌 × 𝑌)) ↔ ((𝑌 × 𝑌) = (𝑌 × 𝑌) ∧ ∀𝑥𝑌𝑦𝑌 (𝑥𝐹𝑦) = (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦))))
4635, 45mpbird 247 1 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝐹 = (𝐺 ↾ (𝑌 × 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  wral 3061  wss 3723   × cxp 5247  cres 5251  Fun wfun 6025   Fn wfn 6026  wf 6027  cfv 6031  (class class class)co 6793  NrmCVeccnv 27779   +𝑣 cpv 27780  BaseSetcba 27781   ·𝑠OLD cns 27782  normCVcnmcv 27785  SubSpcss 27916
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-1st 7315  df-2nd 7316  df-grpo 27687  df-ablo 27739  df-vc 27754  df-nv 27787  df-va 27790  df-ba 27791  df-sm 27792  df-0v 27793  df-nmcv 27795  df-ssp 27917
This theorem is referenced by:  sspgval  27924
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