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Theorem ipffval 20426
 Description: The inner product operation as a function. (Contributed by Mario Carneiro, 12-Oct-2015.) (Proof shortened by AV, 2-Mar-2024.)
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 6663 . . . . . 6 (𝑔 = 𝑊 → (Base‘𝑔) = (Base‘𝑊))
3 ipffval.1 . . . . . 6 𝑉 = (Base‘𝑊)
42, 3eqtr4di 2811 . . . . 5 (𝑔 = 𝑊 → (Base‘𝑔) = 𝑉)
5 fveq2 6663 . . . . . . 7 (𝑔 = 𝑊 → (·𝑖𝑔) = (·𝑖𝑊))
6 ipffval.2 . . . . . . 7 , = (·𝑖𝑊)
75, 6eqtr4di 2811 . . . . . 6 (𝑔 = 𝑊 → (·𝑖𝑔) = , )
87oveqd 7173 . . . . 5 (𝑔 = 𝑊 → (𝑥(·𝑖𝑔)𝑦) = (𝑥 , 𝑦))
94, 4, 8mpoeq123dv 7229 . . . 4 (𝑔 = 𝑊 → (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(·𝑖𝑔)𝑦)) = (𝑥𝑉, 𝑦𝑉 ↦ (𝑥 , 𝑦)))
10 df-ipf 20405 . . . 4 ·if = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(·𝑖𝑔)𝑦)))
113fvexi 6677 . . . . 5 𝑉 ∈ V
126fvexi 6677 . . . . . . 7 , ∈ V
1312rnex 7628 . . . . . 6 ran , ∈ V
14 p0ex 5257 . . . . . 6 {∅} ∈ V
1513, 14unex 7473 . . . . 5 (ran , ∪ {∅}) ∈ V
16 df-ov 7159 . . . . . . 7 (𝑥 , 𝑦) = ( , ‘⟨𝑥, 𝑦⟩)
17 fvrn0 6691 . . . . . . 7 ( , ‘⟨𝑥, 𝑦⟩) ∈ (ran , ∪ {∅})
1816, 17eqeltri 2848 . . . . . 6 (𝑥 , 𝑦) ∈ (ran , ∪ {∅})
1918rgen2w 3083 . . . . 5 𝑥𝑉𝑦𝑉 (𝑥 , 𝑦) ∈ (ran , ∪ {∅})
2011, 11, 15, 19mpoexw 7787 . . . 4 (𝑥𝑉, 𝑦𝑉 ↦ (𝑥 , 𝑦)) ∈ V
219, 10, 20fvmpt 6764 . . 3 (𝑊 ∈ V → (·if𝑊) = (𝑥𝑉, 𝑦𝑉 ↦ (𝑥 , 𝑦)))
22 fvprc 6655 . . . 4 𝑊 ∈ V → (·if𝑊) = ∅)
23 fvprc 6655 . . . . . . 7 𝑊 ∈ V → (Base‘𝑊) = ∅)
243, 23syl5eq 2805 . . . . . 6 𝑊 ∈ V → 𝑉 = ∅)
2524olcd 871 . . . . 5 𝑊 ∈ V → (𝑉 = ∅ ∨ 𝑉 = ∅))
26 0mpo0 7237 . . . . 5 ((𝑉 = ∅ ∨ 𝑉 = ∅) → (𝑥𝑉, 𝑦𝑉 ↦ (𝑥 , 𝑦)) = ∅)
2725, 26syl 17 . . . 4 𝑊 ∈ V → (𝑥𝑉, 𝑦𝑉 ↦ (𝑥 , 𝑦)) = ∅)
2822, 27eqtr4d 2796 . . 3 𝑊 ∈ V → (·if𝑊) = (𝑥𝑉, 𝑦𝑉 ↦ (𝑥 , 𝑦)))
2921, 28pm2.61i 185 . 2 (·if𝑊) = (𝑥𝑉, 𝑦𝑉 ↦ (𝑥 , 𝑦))
301, 29eqtri 2781 1 · = (𝑥𝑉, 𝑦𝑉 ↦ (𝑥 , 𝑦))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∨ wo 844   = wceq 1538   ∈ wcel 2111  Vcvv 3409   ∪ cun 3858  ∅c0 4227  {csn 4525  ⟨cop 4531  ran crn 5529  ‘cfv 6340  (class class class)co 7156   ∈ cmpo 7158  Basecbs 16554  ·𝑖cip 16641  ·ifcipf 20403 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 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465 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 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-fv 6348  df-ov 7159  df-oprab 7160  df-mpo 7161  df-1st 7699  df-2nd 7700  df-ipf 20405 This theorem is referenced by:  ipfval  20427  ipfeq  20428  ipffn  20429  phlipf  20430  phssip  20436
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