MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sspmlem Structured version   Visualization version   GIF version

Theorem sspmlem 29674
Description: Lemma for sspm 29676 and others. (Contributed by NM, 1-Feb-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
sspmlem.y 𝑌 = (BaseSet‘𝑊)
sspmlem.h 𝐻 = (SubSp‘𝑈)
sspmlem.1 (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ (𝑥𝑌𝑦𝑌)) → (𝑥𝐹𝑦) = (𝑥𝐺𝑦))
sspmlem.2 (𝑊 ∈ NrmCVec → 𝐹:(𝑌 × 𝑌)⟶𝑅)
sspmlem.3 (𝑈 ∈ NrmCVec → 𝐺:((BaseSet‘𝑈) × (BaseSet‘𝑈))⟶𝑆)
Assertion
Ref Expression
sspmlem ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝐹 = (𝐺 ↾ (𝑌 × 𝑌)))
Distinct variable groups:   𝑥,𝑦,𝐹   𝑥,𝐺,𝑦   𝑥,𝐻,𝑦   𝑥,𝑈,𝑦   𝑥,𝑊,𝑦   𝑥,𝑌,𝑦
Allowed substitution hints:   𝑅(𝑥,𝑦)   𝑆(𝑥,𝑦)

Proof of Theorem sspmlem
StepHypRef Expression
1 sspmlem.1 . . . . 5 (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ (𝑥𝑌𝑦𝑌)) → (𝑥𝐹𝑦) = (𝑥𝐺𝑦))
2 ovres 7520 . . . . . 6 ((𝑥𝑌𝑦𝑌) → (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦) = (𝑥𝐺𝑦))
32adantl 482 . . . . 5 (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ (𝑥𝑌𝑦𝑌)) → (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦) = (𝑥𝐺𝑦))
41, 3eqtr4d 2779 . . . 4 (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ (𝑥𝑌𝑦𝑌)) → (𝑥𝐹𝑦) = (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦))
54ralrimivva 3197 . . 3 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → ∀𝑥𝑌𝑦𝑌 (𝑥𝐹𝑦) = (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦))
6 eqid 2736 . . 3 (𝑌 × 𝑌) = (𝑌 × 𝑌)
75, 6jctil 520 . 2 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → ((𝑌 × 𝑌) = (𝑌 × 𝑌) ∧ ∀𝑥𝑌𝑦𝑌 (𝑥𝐹𝑦) = (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦)))
8 sspmlem.h . . . . 5 𝐻 = (SubSp‘𝑈)
98sspnv 29668 . . . 4 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑊 ∈ NrmCVec)
10 sspmlem.2 . . . 4 (𝑊 ∈ NrmCVec → 𝐹:(𝑌 × 𝑌)⟶𝑅)
11 ffn 6668 . . . 4 (𝐹:(𝑌 × 𝑌)⟶𝑅𝐹 Fn (𝑌 × 𝑌))
129, 10, 113syl 18 . . 3 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝐹 Fn (𝑌 × 𝑌))
13 sspmlem.3 . . . . . 6 (𝑈 ∈ NrmCVec → 𝐺:((BaseSet‘𝑈) × (BaseSet‘𝑈))⟶𝑆)
1413ffnd 6669 . . . . 5 (𝑈 ∈ NrmCVec → 𝐺 Fn ((BaseSet‘𝑈) × (BaseSet‘𝑈)))
1514adantr 481 . . . 4 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝐺 Fn ((BaseSet‘𝑈) × (BaseSet‘𝑈)))
16 eqid 2736 . . . . . 6 (BaseSet‘𝑈) = (BaseSet‘𝑈)
17 sspmlem.y . . . . . 6 𝑌 = (BaseSet‘𝑊)
1816, 17, 8sspba 29669 . . . . 5 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑌 ⊆ (BaseSet‘𝑈))
19 xpss12 5648 . . . . 5 ((𝑌 ⊆ (BaseSet‘𝑈) ∧ 𝑌 ⊆ (BaseSet‘𝑈)) → (𝑌 × 𝑌) ⊆ ((BaseSet‘𝑈) × (BaseSet‘𝑈)))
2018, 18, 19syl2anc 584 . . . 4 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (𝑌 × 𝑌) ⊆ ((BaseSet‘𝑈) × (BaseSet‘𝑈)))
21 fnssres 6624 . . . 4 ((𝐺 Fn ((BaseSet‘𝑈) × (BaseSet‘𝑈)) ∧ (𝑌 × 𝑌) ⊆ ((BaseSet‘𝑈) × (BaseSet‘𝑈))) → (𝐺 ↾ (𝑌 × 𝑌)) Fn (𝑌 × 𝑌))
2215, 20, 21syl2anc 584 . . 3 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (𝐺 ↾ (𝑌 × 𝑌)) Fn (𝑌 × 𝑌))
23 eqfnov 7485 . . 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))))
2412, 22, 23syl2anc 584 . 2 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (𝐹 = (𝐺 ↾ (𝑌 × 𝑌)) ↔ ((𝑌 × 𝑌) = (𝑌 × 𝑌) ∧ ∀𝑥𝑌𝑦𝑌 (𝑥𝐹𝑦) = (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦))))
257, 24mpbird 256 1 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝐹 = (𝐺 ↾ (𝑌 × 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wral 3064  wss 3910   × cxp 5631  cres 5635   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7357  NrmCVeccnv 29526  BaseSetcba 29528  SubSpcss 29663
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-fo 6502  df-fv 6504  df-ov 7360  df-oprab 7361  df-1st 7921  df-2nd 7922  df-vc 29501  df-nv 29534  df-va 29537  df-ba 29538  df-sm 29539  df-nmcv 29542  df-ssp 29664
This theorem is referenced by:  sspm  29676  sspims  29681
  Copyright terms: Public domain W3C validator