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Theorem sspmlem 27571
Description: Lemma for sspm 27573 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 6797 . . . . . 6 ((𝑥𝑌𝑦𝑌) → (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦) = (𝑥𝐺𝑦))
32adantl 482 . . . . 5 (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ (𝑥𝑌𝑦𝑌)) → (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦) = (𝑥𝐺𝑦))
41, 3eqtr4d 2658 . . . 4 (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ (𝑥𝑌𝑦𝑌)) → (𝑥𝐹𝑦) = (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦))
54ralrimivva 2970 . . 3 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → ∀𝑥𝑌𝑦𝑌 (𝑥𝐹𝑦) = (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦))
6 eqid 2621 . . 3 (𝑌 × 𝑌) = (𝑌 × 𝑌)
75, 6jctil 560 . 2 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → ((𝑌 × 𝑌) = (𝑌 × 𝑌) ∧ ∀𝑥𝑌𝑦𝑌 (𝑥𝐹𝑦) = (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦)))
8 sspmlem.h . . . . 5 𝐻 = (SubSp‘𝑈)
98sspnv 27565 . . . 4 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑊 ∈ NrmCVec)
10 sspmlem.2 . . . 4 (𝑊 ∈ NrmCVec → 𝐹:(𝑌 × 𝑌)⟶𝑅)
11 ffn 6043 . . . 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)
14 ffn 6043 . . . . . 6 (𝐺:((BaseSet‘𝑈) × (BaseSet‘𝑈))⟶𝑆𝐺 Fn ((BaseSet‘𝑈) × (BaseSet‘𝑈)))
1513, 14syl 17 . . . . 5 (𝑈 ∈ NrmCVec → 𝐺 Fn ((BaseSet‘𝑈) × (BaseSet‘𝑈)))
1615adantr 481 . . . 4 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝐺 Fn ((BaseSet‘𝑈) × (BaseSet‘𝑈)))
17 eqid 2621 . . . . . 6 (BaseSet‘𝑈) = (BaseSet‘𝑈)
18 sspmlem.y . . . . . 6 𝑌 = (BaseSet‘𝑊)
1917, 18, 8sspba 27566 . . . . 5 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑌 ⊆ (BaseSet‘𝑈))
20 xpss12 5223 . . . . 5 ((𝑌 ⊆ (BaseSet‘𝑈) ∧ 𝑌 ⊆ (BaseSet‘𝑈)) → (𝑌 × 𝑌) ⊆ ((BaseSet‘𝑈) × (BaseSet‘𝑈)))
2119, 19, 20syl2anc 693 . . . 4 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (𝑌 × 𝑌) ⊆ ((BaseSet‘𝑈) × (BaseSet‘𝑈)))
22 fnssres 6002 . . . 4 ((𝐺 Fn ((BaseSet‘𝑈) × (BaseSet‘𝑈)) ∧ (𝑌 × 𝑌) ⊆ ((BaseSet‘𝑈) × (BaseSet‘𝑈))) → (𝐺 ↾ (𝑌 × 𝑌)) Fn (𝑌 × 𝑌))
2316, 21, 22syl2anc 693 . . 3 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (𝐺 ↾ (𝑌 × 𝑌)) Fn (𝑌 × 𝑌))
24 eqfnov 6763 . . 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))))
2512, 23, 24syl2anc 693 . 2 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (𝐹 = (𝐺 ↾ (𝑌 × 𝑌)) ↔ ((𝑌 × 𝑌) = (𝑌 × 𝑌) ∧ ∀𝑥𝑌𝑦𝑌 (𝑥𝐹𝑦) = (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦))))
267, 25mpbird 247 1 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝐹 = (𝐺 ↾ (𝑌 × 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1482  wcel 1989  wral 2911  wss 3572   × cxp 5110  cres 5114   Fn wfn 5881  wf 5882  cfv 5886  (class class class)co 6647  NrmCVeccnv 27423  BaseSetcba 27425  SubSpcss 27560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-fo 5892  df-fv 5894  df-ov 6650  df-oprab 6651  df-1st 7165  df-2nd 7166  df-vc 27398  df-nv 27431  df-va 27434  df-ba 27435  df-sm 27436  df-nmcv 27439  df-ssp 27561
This theorem is referenced by:  sspm  27573  sspims  27578
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