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Theorem sspmlem 28619
 Description: Lemma for sspm 28621 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 7315 . . . . . 6 ((𝑥𝑌𝑦𝑌) → (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦) = (𝑥𝐺𝑦))
32adantl 485 . . . . 5 (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ (𝑥𝑌𝑦𝑌)) → (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦) = (𝑥𝐺𝑦))
41, 3eqtr4d 2796 . . . 4 (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ (𝑥𝑌𝑦𝑌)) → (𝑥𝐹𝑦) = (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦))
54ralrimivva 3120 . . 3 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → ∀𝑥𝑌𝑦𝑌 (𝑥𝐹𝑦) = (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦))
6 eqid 2758 . . 3 (𝑌 × 𝑌) = (𝑌 × 𝑌)
75, 6jctil 523 . 2 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → ((𝑌 × 𝑌) = (𝑌 × 𝑌) ∧ ∀𝑥𝑌𝑦𝑌 (𝑥𝐹𝑦) = (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦)))
8 sspmlem.h . . . . 5 𝐻 = (SubSp‘𝑈)
98sspnv 28613 . . . 4 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑊 ∈ NrmCVec)
10 sspmlem.2 . . . 4 (𝑊 ∈ NrmCVec → 𝐹:(𝑌 × 𝑌)⟶𝑅)
11 ffn 6502 . . . 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 6503 . . . . 5 (𝑈 ∈ NrmCVec → 𝐺 Fn ((BaseSet‘𝑈) × (BaseSet‘𝑈)))
1514adantr 484 . . . 4 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝐺 Fn ((BaseSet‘𝑈) × (BaseSet‘𝑈)))
16 eqid 2758 . . . . . 6 (BaseSet‘𝑈) = (BaseSet‘𝑈)
17 sspmlem.y . . . . . 6 𝑌 = (BaseSet‘𝑊)
1816, 17, 8sspba 28614 . . . . 5 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑌 ⊆ (BaseSet‘𝑈))
19 xpss12 5542 . . . . 5 ((𝑌 ⊆ (BaseSet‘𝑈) ∧ 𝑌 ⊆ (BaseSet‘𝑈)) → (𝑌 × 𝑌) ⊆ ((BaseSet‘𝑈) × (BaseSet‘𝑈)))
2018, 18, 19syl2anc 587 . . . 4 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (𝑌 × 𝑌) ⊆ ((BaseSet‘𝑈) × (BaseSet‘𝑈)))
21 fnssres 6457 . . . 4 ((𝐺 Fn ((BaseSet‘𝑈) × (BaseSet‘𝑈)) ∧ (𝑌 × 𝑌) ⊆ ((BaseSet‘𝑈) × (BaseSet‘𝑈))) → (𝐺 ↾ (𝑌 × 𝑌)) Fn (𝑌 × 𝑌))
2215, 20, 21syl2anc 587 . . 3 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (𝐺 ↾ (𝑌 × 𝑌)) Fn (𝑌 × 𝑌))
23 eqfnov 7280 . . 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 587 . 2 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (𝐹 = (𝐺 ↾ (𝑌 × 𝑌)) ↔ ((𝑌 × 𝑌) = (𝑌 × 𝑌) ∧ ∀𝑥𝑌𝑦𝑌 (𝑥𝐹𝑦) = (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦))))
257, 24mpbird 260 1 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝐹 = (𝐺 ↾ (𝑌 × 𝑌)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3070   ⊆ wss 3860   × cxp 5525   ↾ cres 5529   Fn wfn 6334  ⟶wf 6335  ‘cfv 6339  (class class class)co 7155  NrmCVeccnv 28471  BaseSetcba 28473  SubSpcss 28608 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5172  ax-nul 5179  ax-pow 5237  ax-pr 5301  ax-un 7464 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4888  df-br 5036  df-opab 5098  df-mpt 5116  df-id 5433  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-iota 6298  df-fun 6341  df-fn 6342  df-f 6343  df-fo 6345  df-fv 6347  df-ov 7158  df-oprab 7159  df-1st 7698  df-2nd 7699  df-vc 28446  df-nv 28479  df-va 28482  df-ba 28483  df-sm 28484  df-nmcv 28487  df-ssp 28609 This theorem is referenced by:  sspm  28621  sspims  28626
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