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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 6659 | . 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 19993 | . 2 ⊢ · = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦)) |
6 | ovex 6678 | . 2 ⊢ (𝑋 , 𝑌) ∈ V | |
7 | 1, 5, 6 | ovmpt2a 6791 | 1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 · 𝑌) = (𝑋 , 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 ·𝑖cip 15946 ·ifcipf 19970 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-ipf 19972 |
This theorem is referenced by: ipcn 23045 cnmpt1ip 23046 cnmpt2ip 23047 |
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