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Theorem ipfval 20210
Description: The inner product operation as a function. (Contributed by Mario Carneiro, 14-Aug-2015.)
Hypotheses
Ref Expression
ipffval.1 𝑉 = (Base‘𝑊)
ipffval.2 , = (·𝑖𝑊)
ipffval.3 · = (·if𝑊)
Assertion
Ref Expression
ipfval ((𝑋𝑉𝑌𝑉) → (𝑋 · 𝑌) = (𝑋 , 𝑌))

Proof of Theorem ipfval
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq12 6801 . 2 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝑥 , 𝑦) = (𝑋 , 𝑌))
2 ipffval.1 . . 3 𝑉 = (Base‘𝑊)
3 ipffval.2 . . 3 , = (·𝑖𝑊)
4 ipffval.3 . . 3 · = (·if𝑊)
52, 3, 4ipffval 20209 . 2 · = (𝑥𝑉, 𝑦𝑉 ↦ (𝑥 , 𝑦))
6 ovex 6822 . 2 (𝑋 , 𝑌) ∈ V
71, 5, 6ovmpt2a 6937 1 ((𝑋𝑉𝑌𝑉) → (𝑋 · 𝑌) = (𝑋 , 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1630  wcel 2144  cfv 6031  (class class class)co 6792  Basecbs 16063  ·𝑖cip 16153  ·ifcipf 20186
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  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 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-1st 7314  df-2nd 7315  df-ipf 20188
This theorem is referenced by:  ipcn  23263  cnmpt1ip  23264  cnmpt2ip  23265
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