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Mirrors > Home > MPE Home > Th. List > ipffval | Structured version Visualization version GIF version |
Description: The inner product operation as a function. (Contributed by Mario Carneiro, 12-Oct-2015.) (Proof shortened by AV, 2-Mar-2024.) |
Ref | Expression |
---|---|
ipffval.1 | ⊢ 𝑉 = (Base‘𝑊) |
ipffval.2 | ⊢ , = (·𝑖‘𝑊) |
ipffval.3 | ⊢ · = (·if‘𝑊) |
Ref | Expression |
---|---|
ipffval | ⊢ · = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ipffval.3 | . 2 ⊢ · = (·if‘𝑊) | |
2 | fveq2 6842 | . . . . . 6 ⊢ (𝑔 = 𝑊 → (Base‘𝑔) = (Base‘𝑊)) | |
3 | ipffval.1 | . . . . . 6 ⊢ 𝑉 = (Base‘𝑊) | |
4 | 2, 3 | eqtr4di 2794 | . . . . 5 ⊢ (𝑔 = 𝑊 → (Base‘𝑔) = 𝑉) |
5 | fveq2 6842 | . . . . . . 7 ⊢ (𝑔 = 𝑊 → (·𝑖‘𝑔) = (·𝑖‘𝑊)) | |
6 | ipffval.2 | . . . . . . 7 ⊢ , = (·𝑖‘𝑊) | |
7 | 5, 6 | eqtr4di 2794 | . . . . . 6 ⊢ (𝑔 = 𝑊 → (·𝑖‘𝑔) = , ) |
8 | 7 | oveqd 7374 | . . . . 5 ⊢ (𝑔 = 𝑊 → (𝑥(·𝑖‘𝑔)𝑦) = (𝑥 , 𝑦)) |
9 | 4, 4, 8 | mpoeq123dv 7432 | . . . 4 ⊢ (𝑔 = 𝑊 → (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(·𝑖‘𝑔)𝑦)) = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦))) |
10 | df-ipf 21031 | . . . 4 ⊢ ·if = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(·𝑖‘𝑔)𝑦))) | |
11 | 3 | fvexi 6856 | . . . . 5 ⊢ 𝑉 ∈ V |
12 | 6 | fvexi 6856 | . . . . . . 7 ⊢ , ∈ V |
13 | 12 | rnex 7849 | . . . . . 6 ⊢ ran , ∈ V |
14 | p0ex 5339 | . . . . . 6 ⊢ {∅} ∈ V | |
15 | 13, 14 | unex 7680 | . . . . 5 ⊢ (ran , ∪ {∅}) ∈ V |
16 | df-ov 7360 | . . . . . . 7 ⊢ (𝑥 , 𝑦) = ( , ‘〈𝑥, 𝑦〉) | |
17 | fvrn0 6872 | . . . . . . 7 ⊢ ( , ‘〈𝑥, 𝑦〉) ∈ (ran , ∪ {∅}) | |
18 | 16, 17 | eqeltri 2834 | . . . . . 6 ⊢ (𝑥 , 𝑦) ∈ (ran , ∪ {∅}) |
19 | 18 | rgen2w 3069 | . . . . 5 ⊢ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥 , 𝑦) ∈ (ran , ∪ {∅}) |
20 | 11, 11, 15, 19 | mpoexw 8011 | . . . 4 ⊢ (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦)) ∈ V |
21 | 9, 10, 20 | fvmpt 6948 | . . 3 ⊢ (𝑊 ∈ V → (·if‘𝑊) = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦))) |
22 | fvprc 6834 | . . . 4 ⊢ (¬ 𝑊 ∈ V → (·if‘𝑊) = ∅) | |
23 | fvprc 6834 | . . . . . . 7 ⊢ (¬ 𝑊 ∈ V → (Base‘𝑊) = ∅) | |
24 | 3, 23 | eqtrid 2788 | . . . . . 6 ⊢ (¬ 𝑊 ∈ V → 𝑉 = ∅) |
25 | 24 | olcd 872 | . . . . 5 ⊢ (¬ 𝑊 ∈ V → (𝑉 = ∅ ∨ 𝑉 = ∅)) |
26 | 0mpo0 7440 | . . . . 5 ⊢ ((𝑉 = ∅ ∨ 𝑉 = ∅) → (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦)) = ∅) | |
27 | 25, 26 | syl 17 | . . . 4 ⊢ (¬ 𝑊 ∈ V → (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦)) = ∅) |
28 | 22, 27 | eqtr4d 2779 | . . 3 ⊢ (¬ 𝑊 ∈ V → (·if‘𝑊) = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦))) |
29 | 21, 28 | pm2.61i 182 | . 2 ⊢ (·if‘𝑊) = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦)) |
30 | 1, 29 | eqtri 2764 | 1 ⊢ · = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∨ wo 845 = wceq 1541 ∈ wcel 2106 Vcvv 3445 ∪ cun 3908 ∅c0 4282 {csn 4586 〈cop 4592 ran crn 5634 ‘cfv 6496 (class class class)co 7357 ∈ cmpo 7359 Basecbs 17083 ·𝑖cip 17138 ·ifcipf 21029 |
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 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 |
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 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-fv 6504 df-ov 7360 df-oprab 7361 df-mpo 7362 df-1st 7921 df-2nd 7922 df-ipf 21031 |
This theorem is referenced by: ipfval 21053 ipfeq 21054 ipffn 21055 phlipf 21056 phssip 21062 |
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