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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 6651 | . . . . . 6 ⊢ (𝑔 = 𝑊 → (Base‘𝑔) = (Base‘𝑊)) | |
3 | ipffval.1 | . . . . . 6 ⊢ 𝑉 = (Base‘𝑊) | |
4 | 2, 3 | syl6eqr 2873 | . . . . 5 ⊢ (𝑔 = 𝑊 → (Base‘𝑔) = 𝑉) |
5 | fveq2 6651 | . . . . . . 7 ⊢ (𝑔 = 𝑊 → (·𝑖‘𝑔) = (·𝑖‘𝑊)) | |
6 | ipffval.2 | . . . . . . 7 ⊢ , = (·𝑖‘𝑊) | |
7 | 5, 6 | syl6eqr 2873 | . . . . . 6 ⊢ (𝑔 = 𝑊 → (·𝑖‘𝑔) = , ) |
8 | 7 | oveqd 7154 | . . . . 5 ⊢ (𝑔 = 𝑊 → (𝑥(·𝑖‘𝑔)𝑦) = (𝑥 , 𝑦)) |
9 | 4, 4, 8 | mpoeq123dv 7210 | . . . 4 ⊢ (𝑔 = 𝑊 → (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(·𝑖‘𝑔)𝑦)) = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦))) |
10 | df-ipf 20749 | . . . 4 ⊢ ·if = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(·𝑖‘𝑔)𝑦))) | |
11 | 3 | fvexi 6665 | . . . . 5 ⊢ 𝑉 ∈ V |
12 | 6 | fvexi 6665 | . . . . . . 7 ⊢ , ∈ V |
13 | 12 | rnex 7598 | . . . . . 6 ⊢ ran , ∈ V |
14 | p0ex 5266 | . . . . . 6 ⊢ {∅} ∈ V | |
15 | 13, 14 | unex 7450 | . . . . 5 ⊢ (ran , ∪ {∅}) ∈ V |
16 | df-ov 7140 | . . . . . . 7 ⊢ (𝑥 , 𝑦) = ( , ‘〈𝑥, 𝑦〉) | |
17 | fvrn0 6679 | . . . . . . 7 ⊢ ( , ‘〈𝑥, 𝑦〉) ∈ (ran , ∪ {∅}) | |
18 | 16, 17 | eqeltri 2907 | . . . . . 6 ⊢ (𝑥 , 𝑦) ∈ (ran , ∪ {∅}) |
19 | 18 | rgen2w 3146 | . . . . 5 ⊢ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥 , 𝑦) ∈ (ran , ∪ {∅}) |
20 | 11, 11, 15, 19 | mpoexw 7757 | . . . 4 ⊢ (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦)) ∈ V |
21 | 9, 10, 20 | fvmpt 6749 | . . 3 ⊢ (𝑊 ∈ V → (·if‘𝑊) = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦))) |
22 | fvprc 6644 | . . . 4 ⊢ (¬ 𝑊 ∈ V → (·if‘𝑊) = ∅) | |
23 | fvprc 6644 | . . . . . . 7 ⊢ (¬ 𝑊 ∈ V → (Base‘𝑊) = ∅) | |
24 | 3, 23 | syl5eq 2867 | . . . . . 6 ⊢ (¬ 𝑊 ∈ V → 𝑉 = ∅) |
25 | 24 | olcd 870 | . . . . 5 ⊢ (¬ 𝑊 ∈ V → (𝑉 = ∅ ∨ 𝑉 = ∅)) |
26 | 0mpo0 7218 | . . . . 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 2858 | . . 3 ⊢ (¬ 𝑊 ∈ V → (·if‘𝑊) = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦))) |
29 | 21, 28 | pm2.61i 184 | . 2 ⊢ (·if‘𝑊) = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦)) |
30 | 1, 29 | eqtri 2843 | 1 ⊢ · = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∨ wo 843 = wceq 1537 ∈ wcel 2114 Vcvv 3481 ∪ cun 3917 ∅c0 4274 {csn 4548 〈cop 4554 ran crn 5537 ‘cfv 6336 (class class class)co 7137 ∈ cmpo 7139 Basecbs 16461 ·𝑖cip 16548 ·ifcipf 20747 |
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 2792 ax-sep 5184 ax-nul 5191 ax-pow 5247 ax-pr 5311 ax-un 7442 |
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 2653 df-clab 2799 df-cleq 2813 df-clel 2891 df-nfc 2959 df-ne 3012 df-ral 3138 df-rex 3139 df-rab 3142 df-v 3483 df-sbc 3759 df-csb 3867 df-dif 3922 df-un 3924 df-in 3926 df-ss 3935 df-nul 4275 df-if 4449 df-pw 4522 df-sn 4549 df-pr 4551 df-op 4555 df-uni 4820 df-iun 4902 df-br 5048 df-opab 5110 df-mpt 5128 df-id 5441 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6295 df-fun 6338 df-fn 6339 df-f 6340 df-fv 6344 df-ov 7140 df-oprab 7141 df-mpo 7142 df-1st 7670 df-2nd 7671 df-ipf 20749 |
This theorem is referenced by: ipfval 20771 ipfeq 20772 ipffn 20773 phlipf 20774 phssip 20780 |
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