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Theorem ipffval 20203
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 6330 . . . . . 6 (𝑔 = 𝑊 → (Base‘𝑔) = (Base‘𝑊))
3 ipffval.1 . . . . . 6 𝑉 = (Base‘𝑊)
42, 3syl6eqr 2823 . . . . 5 (𝑔 = 𝑊 → (Base‘𝑔) = 𝑉)
5 fveq2 6330 . . . . . . 7 (𝑔 = 𝑊 → (·𝑖𝑔) = (·𝑖𝑊))
6 ipffval.2 . . . . . . 7 , = (·𝑖𝑊)
75, 6syl6eqr 2823 . . . . . 6 (𝑔 = 𝑊 → (·𝑖𝑔) = , )
87oveqd 6808 . . . . 5 (𝑔 = 𝑊 → (𝑥(·𝑖𝑔)𝑦) = (𝑥 , 𝑦))
94, 4, 8mpt2eq123dv 6862 . . . 4 (𝑔 = 𝑊 → (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(·𝑖𝑔)𝑦)) = (𝑥𝑉, 𝑦𝑉 ↦ (𝑥 , 𝑦)))
10 df-ipf 20182 . . . 4 ·if = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(·𝑖𝑔)𝑦)))
11 df-ov 6794 . . . . . . . 8 (𝑥 , 𝑦) = ( , ‘⟨𝑥, 𝑦⟩)
12 fvrn0 6355 . . . . . . . 8 ( , ‘⟨𝑥, 𝑦⟩) ∈ (ran , ∪ {∅})
1311, 12eqeltri 2846 . . . . . . 7 (𝑥 , 𝑦) ∈ (ran , ∪ {∅})
1413rgen2w 3074 . . . . . 6 𝑥𝑉𝑦𝑉 (𝑥 , 𝑦) ∈ (ran , ∪ {∅})
15 eqid 2771 . . . . . . 7 (𝑥𝑉, 𝑦𝑉 ↦ (𝑥 , 𝑦)) = (𝑥𝑉, 𝑦𝑉 ↦ (𝑥 , 𝑦))
1615fmpt2 7385 . . . . . 6 (∀𝑥𝑉𝑦𝑉 (𝑥 , 𝑦) ∈ (ran , ∪ {∅}) ↔ (𝑥𝑉, 𝑦𝑉 ↦ (𝑥 , 𝑦)):(𝑉 × 𝑉)⟶(ran , ∪ {∅}))
1714, 16mpbi 220 . . . . 5 (𝑥𝑉, 𝑦𝑉 ↦ (𝑥 , 𝑦)):(𝑉 × 𝑉)⟶(ran , ∪ {∅})
183fvexi 6341 . . . . . 6 𝑉 ∈ V
1918, 18xpex 7107 . . . . 5 (𝑉 × 𝑉) ∈ V
206fvexi 6341 . . . . . . 7 , ∈ V
2120rnex 7245 . . . . . 6 ran , ∈ V
22 p0ex 4984 . . . . . 6 {∅} ∈ V
2321, 22unex 7101 . . . . 5 (ran , ∪ {∅}) ∈ V
24 fex2 7266 . . . . 5 (((𝑥𝑉, 𝑦𝑉 ↦ (𝑥 , 𝑦)):(𝑉 × 𝑉)⟶(ran , ∪ {∅}) ∧ (𝑉 × 𝑉) ∈ V ∧ (ran , ∪ {∅}) ∈ V) → (𝑥𝑉, 𝑦𝑉 ↦ (𝑥 , 𝑦)) ∈ V)
2517, 19, 23, 24mp3an 1572 . . . 4 (𝑥𝑉, 𝑦𝑉 ↦ (𝑥 , 𝑦)) ∈ V
269, 10, 25fvmpt 6422 . . 3 (𝑊 ∈ V → (·if𝑊) = (𝑥𝑉, 𝑦𝑉 ↦ (𝑥 , 𝑦)))
27 fvprc 6324 . . . . 5 𝑊 ∈ V → (·if𝑊) = ∅)
28 mpt20 6870 . . . . 5 (𝑥 ∈ ∅, 𝑦 ∈ ∅ ↦ (𝑥 , 𝑦)) = ∅
2927, 28syl6eqr 2823 . . . 4 𝑊 ∈ V → (·if𝑊) = (𝑥 ∈ ∅, 𝑦 ∈ ∅ ↦ (𝑥 , 𝑦)))
30 fvprc 6324 . . . . . 6 𝑊 ∈ V → (Base‘𝑊) = ∅)
313, 30syl5eq 2817 . . . . 5 𝑊 ∈ V → 𝑉 = ∅)
32 mpt2eq12 6860 . . . . 5 ((𝑉 = ∅ ∧ 𝑉 = ∅) → (𝑥𝑉, 𝑦𝑉 ↦ (𝑥 , 𝑦)) = (𝑥 ∈ ∅, 𝑦 ∈ ∅ ↦ (𝑥 , 𝑦)))
3331, 31, 32syl2anc 573 . . . 4 𝑊 ∈ V → (𝑥𝑉, 𝑦𝑉 ↦ (𝑥 , 𝑦)) = (𝑥 ∈ ∅, 𝑦 ∈ ∅ ↦ (𝑥 , 𝑦)))
3429, 33eqtr4d 2808 . . 3 𝑊 ∈ V → (·if𝑊) = (𝑥𝑉, 𝑦𝑉 ↦ (𝑥 , 𝑦)))
3526, 34pm2.61i 176 . 2 (·if𝑊) = (𝑥𝑉, 𝑦𝑉 ↦ (𝑥 , 𝑦))
361, 35eqtri 2793 1 · = (𝑥𝑉, 𝑦𝑉 ↦ (𝑥 , 𝑦))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1631  wcel 2145  wral 3061  Vcvv 3351  cun 3721  c0 4063  {csn 4316  cop 4322   × cxp 5247  ran crn 5250  wf 6025  cfv 6029  (class class class)co 6791  cmpt2 6793  Basecbs 16057  ·𝑖cip 16147  ·ifcipf 20180
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7094
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5992  df-fun 6031  df-fn 6032  df-f 6033  df-fv 6037  df-ov 6794  df-oprab 6795  df-mpt2 6796  df-1st 7313  df-2nd 7314  df-ipf 20182
This theorem is referenced by:  ipfval  20204  ipfeq  20205  ipffn  20206  phlipf  20207  phssip  20213
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