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Theorem sspg 27471
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 2621 . . . . . . . . . . 11 (BaseSet‘𝑈) = (BaseSet‘𝑈)
2 sspg.g . . . . . . . . . . 11 𝐺 = ( +𝑣𝑈)
31, 2nvgf 27361 . . . . . . . . . 10 (𝑈 ∈ NrmCVec → 𝐺:((BaseSet‘𝑈) × (BaseSet‘𝑈))⟶(BaseSet‘𝑈))
4 ffun 6015 . . . . . . . . . 10 (𝐺:((BaseSet‘𝑈) × (BaseSet‘𝑈))⟶(BaseSet‘𝑈) → Fun 𝐺)
53, 4syl 17 . . . . . . . . 9 (𝑈 ∈ NrmCVec → Fun 𝐺)
6 funres 5897 . . . . . . . . 9 (Fun 𝐺 → Fun (𝐺 ↾ (𝑌 × 𝑌)))
75, 6syl 17 . . . . . . . 8 (𝑈 ∈ NrmCVec → Fun (𝐺 ↾ (𝑌 × 𝑌)))
87adantr 481 . . . . . . 7 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → Fun (𝐺 ↾ (𝑌 × 𝑌)))
9 sspg.h . . . . . . . . . 10 𝐻 = (SubSp‘𝑈)
109sspnv 27469 . . . . . . . . 9 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑊 ∈ NrmCVec)
11 sspg.y . . . . . . . . . 10 𝑌 = (BaseSet‘𝑊)
12 sspg.f . . . . . . . . . 10 𝐹 = ( +𝑣𝑊)
1311, 12nvgf 27361 . . . . . . . . 9 (𝑊 ∈ NrmCVec → 𝐹:(𝑌 × 𝑌)⟶𝑌)
1410, 13syl 17 . . . . . . . 8 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝐹:(𝑌 × 𝑌)⟶𝑌)
15 ffn 6012 . . . . . . . 8 (𝐹:(𝑌 × 𝑌)⟶𝑌𝐹 Fn (𝑌 × 𝑌))
1614, 15syl 17 . . . . . . 7 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝐹 Fn (𝑌 × 𝑌))
17 fnresdm 5968 . . . . . . . . 9 (𝐹 Fn (𝑌 × 𝑌) → (𝐹 ↾ (𝑌 × 𝑌)) = 𝐹)
1816, 17syl 17 . . . . . . . 8 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (𝐹 ↾ (𝑌 × 𝑌)) = 𝐹)
19 eqid 2621 . . . . . . . . . . . 12 ( ·𝑠OLD𝑈) = ( ·𝑠OLD𝑈)
20 eqid 2621 . . . . . . . . . . . 12 ( ·𝑠OLD𝑊) = ( ·𝑠OLD𝑊)
21 eqid 2621 . . . . . . . . . . . 12 (normCV𝑈) = (normCV𝑈)
22 eqid 2621 . . . . . . . . . . . 12 (normCV𝑊) = (normCV𝑊)
232, 12, 19, 20, 21, 22, 9isssp 27467 . . . . . . . . . . 11 (𝑈 ∈ NrmCVec → (𝑊𝐻 ↔ (𝑊 ∈ NrmCVec ∧ (𝐹𝐺 ∧ ( ·𝑠OLD𝑊) ⊆ ( ·𝑠OLD𝑈) ∧ (normCV𝑊) ⊆ (normCV𝑈)))))
2423simplbda 653 . . . . . . . . . 10 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (𝐹𝐺 ∧ ( ·𝑠OLD𝑊) ⊆ ( ·𝑠OLD𝑈) ∧ (normCV𝑊) ⊆ (normCV𝑈)))
2524simp1d 1071 . . . . . . . . 9 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝐹𝐺)
26 ssres 5393 . . . . . . . . 9 (𝐹𝐺 → (𝐹 ↾ (𝑌 × 𝑌)) ⊆ (𝐺 ↾ (𝑌 × 𝑌)))
2725, 26syl 17 . . . . . . . 8 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (𝐹 ↾ (𝑌 × 𝑌)) ⊆ (𝐺 ↾ (𝑌 × 𝑌)))
2818, 27eqsstr3d 3625 . . . . . . 7 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝐹 ⊆ (𝐺 ↾ (𝑌 × 𝑌)))
298, 16, 283jca 1240 . . . . . 6 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (Fun (𝐺 ↾ (𝑌 × 𝑌)) ∧ 𝐹 Fn (𝑌 × 𝑌) ∧ 𝐹 ⊆ (𝐺 ↾ (𝑌 × 𝑌))))
30 oprssov 6768 . . . . . 6 (((Fun (𝐺 ↾ (𝑌 × 𝑌)) ∧ 𝐹 Fn (𝑌 × 𝑌) ∧ 𝐹 ⊆ (𝐺 ↾ (𝑌 × 𝑌))) ∧ (𝑥𝑌𝑦𝑌)) → (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦) = (𝑥𝐹𝑦))
3129, 30sylan 488 . . . . 5 (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ (𝑥𝑌𝑦𝑌)) → (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦) = (𝑥𝐹𝑦))
3231eqcomd 2627 . . . 4 (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ (𝑥𝑌𝑦𝑌)) → (𝑥𝐹𝑦) = (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦))
3332ralrimivva 2967 . . 3 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → ∀𝑥𝑌𝑦𝑌 (𝑥𝐹𝑦) = (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦))
34 eqid 2621 . . 3 (𝑌 × 𝑌) = (𝑌 × 𝑌)
3533, 34jctil 559 . 2 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → ((𝑌 × 𝑌) = (𝑌 × 𝑌) ∧ ∀𝑥𝑌𝑦𝑌 (𝑥𝐹𝑦) = (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦)))
36 ffn 6012 . . . . . 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 481 . . . 4 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝐺 Fn ((BaseSet‘𝑈) × (BaseSet‘𝑈)))
391, 11, 9sspba 27470 . . . . 5 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑌 ⊆ (BaseSet‘𝑈))
40 xpss12 5196 . . . . 5 ((𝑌 ⊆ (BaseSet‘𝑈) ∧ 𝑌 ⊆ (BaseSet‘𝑈)) → (𝑌 × 𝑌) ⊆ ((BaseSet‘𝑈) × (BaseSet‘𝑈)))
4139, 39, 40syl2anc 692 . . . 4 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (𝑌 × 𝑌) ⊆ ((BaseSet‘𝑈) × (BaseSet‘𝑈)))
42 fnssres 5972 . . . 4 ((𝐺 Fn ((BaseSet‘𝑈) × (BaseSet‘𝑈)) ∧ (𝑌 × 𝑌) ⊆ ((BaseSet‘𝑈) × (BaseSet‘𝑈))) → (𝐺 ↾ (𝑌 × 𝑌)) Fn (𝑌 × 𝑌))
4338, 41, 42syl2anc 692 . . 3 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (𝐺 ↾ (𝑌 × 𝑌)) Fn (𝑌 × 𝑌))
44 eqfnov 6731 . . 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 692 . 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 384  w3a 1036   = wceq 1480  wcel 1987  wral 2908  wss 3560   × cxp 5082  cres 5086  Fun wfun 5851   Fn wfn 5852  wf 5853  cfv 5857  (class class class)co 6615  NrmCVeccnv 27327   +𝑣 cpv 27328  BaseSetcba 27329   ·𝑠OLD cns 27330  normCVcnmcv 27333  SubSpcss 27464
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-oprab 6619  df-1st 7128  df-2nd 7129  df-grpo 27235  df-ablo 27287  df-vc 27302  df-nv 27335  df-va 27338  df-ba 27339  df-sm 27340  df-0v 27341  df-nmcv 27343  df-ssp 27465
This theorem is referenced by:  sspgval  27472
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