Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 | fnov 7284 | . . 3 ⊢ ( , Fn (𝑉 × 𝑉) ↔ , = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦))) | |
2 | 1 | biimpi 218 | . 2 ⊢ ( , Fn (𝑉 × 𝑉) → , = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦))) |
3 | ipffval.1 | . . 3 ⊢ 𝑉 = (Base‘𝑊) | |
4 | ipffval.2 | . . 3 ⊢ , = (·𝑖‘𝑊) | |
5 | ipffval.3 | . . 3 ⊢ · = (·if‘𝑊) | |
6 | 3, 4, 5 | ipffval 20794 | . 2 ⊢ · = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦)) |
7 | 2, 6 | syl6reqr 2877 | 1 ⊢ ( , Fn (𝑉 × 𝑉) → · = , ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 × cxp 5555 Fn wfn 6352 ‘cfv 6357 (class class class)co 7158 ∈ cmpo 7160 Basecbs 16485 ·𝑖cip 16572 ·ifcipf 20771 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-1st 7691 df-2nd 7692 df-ipf 20773 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |