![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > ipfval | Structured version Visualization version GIF version |
Description: The inner product operation as a function. (Contributed by Mario Carneiro, 14-Aug-2015.) |
Ref | Expression |
---|---|
ipffval.1 | ⊢ 𝑉 = (Base‘𝑊) |
ipffval.2 | ⊢ , = (·𝑖‘𝑊) |
ipffval.3 | ⊢ · = (·if‘𝑊) |
Ref | Expression |
---|---|
ipfval | ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 · 𝑌) = (𝑋 , 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq12 6985 | . 2 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑥 , 𝑦) = (𝑋 , 𝑌)) | |
2 | ipffval.1 | . . 3 ⊢ 𝑉 = (Base‘𝑊) | |
3 | ipffval.2 | . . 3 ⊢ , = (·𝑖‘𝑊) | |
4 | ipffval.3 | . . 3 ⊢ · = (·if‘𝑊) | |
5 | 2, 3, 4 | ipffval 20494 | . 2 ⊢ · = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦)) |
6 | ovex 7008 | . 2 ⊢ (𝑋 , 𝑌) ∈ V | |
7 | 1, 5, 6 | ovmpoa 7121 | 1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 · 𝑌) = (𝑋 , 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1507 ∈ wcel 2050 ‘cfv 6188 (class class class)co 6976 Basecbs 16339 ·𝑖cip 16426 ·ifcipf 20471 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2751 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2760 df-cleq 2772 df-clel 2847 df-nfc 2919 df-ne 2969 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3418 df-sbc 3683 df-csb 3788 df-dif 3833 df-un 3835 df-in 3837 df-ss 3844 df-nul 4180 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-op 4448 df-uni 4713 df-iun 4794 df-br 4930 df-opab 4992 df-mpt 5009 df-id 5312 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-fv 6196 df-ov 6979 df-oprab 6980 df-mpo 6981 df-1st 7501 df-2nd 7502 df-ipf 20473 |
This theorem is referenced by: ipcn 23552 cnmpt1ip 23553 cnmpt2ip 23554 |
Copyright terms: Public domain | W3C validator |