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Mirrors > Home > MPE Home > Th. List > ipfeq | Structured version Visualization version GIF version |
Description: If the inner product operation is already a function, the functionalization of it is equal to the original operation. (Contributed by Mario Carneiro, 14-Aug-2015.) |
Ref | Expression |
---|---|
ipffval.1 | ⊢ 𝑉 = (Base‘𝑊) |
ipffval.2 | ⊢ , = (·𝑖‘𝑊) |
ipffval.3 | ⊢ · = (·if‘𝑊) |
Ref | Expression |
---|---|
ipfeq | ⊢ ( , Fn (𝑉 × 𝑉) → · = , ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ipffval.1 | . . 3 ⊢ 𝑉 = (Base‘𝑊) | |
2 | ipffval.2 | . . 3 ⊢ , = (·𝑖‘𝑊) | |
3 | ipffval.3 | . . 3 ⊢ · = (·if‘𝑊) | |
4 | 1, 2, 3 | ipffval 21037 | . 2 ⊢ · = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦)) |
5 | fnov 7483 | . . 3 ⊢ ( , Fn (𝑉 × 𝑉) ↔ , = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦))) | |
6 | 5 | biimpi 215 | . 2 ⊢ ( , Fn (𝑉 × 𝑉) → , = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦))) |
7 | 4, 6 | eqtr4id 2795 | 1 ⊢ ( , Fn (𝑉 × 𝑉) → · = , ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 × cxp 5629 Fn wfn 6488 ‘cfv 6493 (class class class)co 7353 ∈ cmpo 7355 Basecbs 17075 ·𝑖cip 17130 ·ifcipf 21014 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-fv 6501 df-ov 7356 df-oprab 7357 df-mpo 7358 df-1st 7917 df-2nd 7918 df-ipf 21016 |
This theorem is referenced by: (None) |
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