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Theorem ipffval 19912
 Description: The inner product operation as a function. (Contributed by Mario Carneiro, 12-Oct-2015.)
Hypotheses
Ref Expression
ipffval.1 𝑉 = (Base‘𝑊)
ipffval.2 , = (·𝑖𝑊)
ipffval.3 · = (·if𝑊)
Assertion
Ref Expression
ipffval · = (𝑥𝑉, 𝑦𝑉 ↦ (𝑥 , 𝑦))
Distinct variable groups:   𝑥,𝑦, ,   𝑥,𝑉,𝑦   𝑥,𝑊,𝑦
Allowed substitution hints:   · (𝑥,𝑦)

Proof of Theorem ipffval
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 ipffval.3 . 2 · = (·if𝑊)
2 fveq2 6148 . . . . . 6 (𝑔 = 𝑊 → (Base‘𝑔) = (Base‘𝑊))
3 ipffval.1 . . . . . 6 𝑉 = (Base‘𝑊)
42, 3syl6eqr 2673 . . . . 5 (𝑔 = 𝑊 → (Base‘𝑔) = 𝑉)
5 fveq2 6148 . . . . . . 7 (𝑔 = 𝑊 → (·𝑖𝑔) = (·𝑖𝑊))
6 ipffval.2 . . . . . . 7 , = (·𝑖𝑊)
75, 6syl6eqr 2673 . . . . . 6 (𝑔 = 𝑊 → (·𝑖𝑔) = , )
87oveqd 6621 . . . . 5 (𝑔 = 𝑊 → (𝑥(·𝑖𝑔)𝑦) = (𝑥 , 𝑦))
94, 4, 8mpt2eq123dv 6670 . . . 4 (𝑔 = 𝑊 → (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(·𝑖𝑔)𝑦)) = (𝑥𝑉, 𝑦𝑉 ↦ (𝑥 , 𝑦)))
10 df-ipf 19891 . . . 4 ·if = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(·𝑖𝑔)𝑦)))
11 df-ov 6607 . . . . . . . 8 (𝑥 , 𝑦) = ( , ‘⟨𝑥, 𝑦⟩)
12 fvrn0 6173 . . . . . . . 8 ( , ‘⟨𝑥, 𝑦⟩) ∈ (ran , ∪ {∅})
1311, 12eqeltri 2694 . . . . . . 7 (𝑥 , 𝑦) ∈ (ran , ∪ {∅})
1413rgen2w 2920 . . . . . 6 𝑥𝑉𝑦𝑉 (𝑥 , 𝑦) ∈ (ran , ∪ {∅})
15 eqid 2621 . . . . . . 7 (𝑥𝑉, 𝑦𝑉 ↦ (𝑥 , 𝑦)) = (𝑥𝑉, 𝑦𝑉 ↦ (𝑥 , 𝑦))
1615fmpt2 7182 . . . . . 6 (∀𝑥𝑉𝑦𝑉 (𝑥 , 𝑦) ∈ (ran , ∪ {∅}) ↔ (𝑥𝑉, 𝑦𝑉 ↦ (𝑥 , 𝑦)):(𝑉 × 𝑉)⟶(ran , ∪ {∅}))
1714, 16mpbi 220 . . . . 5 (𝑥𝑉, 𝑦𝑉 ↦ (𝑥 , 𝑦)):(𝑉 × 𝑉)⟶(ran , ∪ {∅})
18 fvex 6158 . . . . . . 7 (Base‘𝑊) ∈ V
193, 18eqeltri 2694 . . . . . 6 𝑉 ∈ V
2019, 19xpex 6915 . . . . 5 (𝑉 × 𝑉) ∈ V
21 fvex 6158 . . . . . . . 8 (·𝑖𝑊) ∈ V
226, 21eqeltri 2694 . . . . . . 7 , ∈ V
2322rnex 7047 . . . . . 6 ran , ∈ V
24 p0ex 4813 . . . . . 6 {∅} ∈ V
2523, 24unex 6909 . . . . 5 (ran , ∪ {∅}) ∈ V
26 fex2 7068 . . . . 5 (((𝑥𝑉, 𝑦𝑉 ↦ (𝑥 , 𝑦)):(𝑉 × 𝑉)⟶(ran , ∪ {∅}) ∧ (𝑉 × 𝑉) ∈ V ∧ (ran , ∪ {∅}) ∈ V) → (𝑥𝑉, 𝑦𝑉 ↦ (𝑥 , 𝑦)) ∈ V)
2717, 20, 25, 26mp3an 1421 . . . 4 (𝑥𝑉, 𝑦𝑉 ↦ (𝑥 , 𝑦)) ∈ V
289, 10, 27fvmpt 6239 . . 3 (𝑊 ∈ V → (·if𝑊) = (𝑥𝑉, 𝑦𝑉 ↦ (𝑥 , 𝑦)))
29 fvprc 6142 . . . . 5 𝑊 ∈ V → (·if𝑊) = ∅)
30 mpt20 6678 . . . . 5 (𝑥 ∈ ∅, 𝑦 ∈ ∅ ↦ (𝑥 , 𝑦)) = ∅
3129, 30syl6eqr 2673 . . . 4 𝑊 ∈ V → (·if𝑊) = (𝑥 ∈ ∅, 𝑦 ∈ ∅ ↦ (𝑥 , 𝑦)))
32 fvprc 6142 . . . . . 6 𝑊 ∈ V → (Base‘𝑊) = ∅)
333, 32syl5eq 2667 . . . . 5 𝑊 ∈ V → 𝑉 = ∅)
34 mpt2eq12 6668 . . . . 5 ((𝑉 = ∅ ∧ 𝑉 = ∅) → (𝑥𝑉, 𝑦𝑉 ↦ (𝑥 , 𝑦)) = (𝑥 ∈ ∅, 𝑦 ∈ ∅ ↦ (𝑥 , 𝑦)))
3533, 33, 34syl2anc 692 . . . 4 𝑊 ∈ V → (𝑥𝑉, 𝑦𝑉 ↦ (𝑥 , 𝑦)) = (𝑥 ∈ ∅, 𝑦 ∈ ∅ ↦ (𝑥 , 𝑦)))
3631, 35eqtr4d 2658 . . 3 𝑊 ∈ V → (·if𝑊) = (𝑥𝑉, 𝑦𝑉 ↦ (𝑥 , 𝑦)))
3728, 36pm2.61i 176 . 2 (·if𝑊) = (𝑥𝑉, 𝑦𝑉 ↦ (𝑥 , 𝑦))
381, 37eqtri 2643 1 · = (𝑥𝑉, 𝑦𝑉 ↦ (𝑥 , 𝑦))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1480   ∈ wcel 1987  ∀wral 2907  Vcvv 3186   ∪ cun 3553  ∅c0 3891  {csn 4148  ⟨cop 4154   × cxp 5072  ran crn 5075  ⟶wf 5843  ‘cfv 5847  (class class class)co 6604   ↦ cmpt2 6606  Basecbs 15781  ·𝑖cip 15867  ·ifcipf 19889 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-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902 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 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114  df-ipf 19891 This theorem is referenced by:  ipfval  19913  ipfeq  19914  ipffn  19915  phlipf  19916  phssip  19922
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