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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 6670 | . . . . . 6 ⊢ (𝑔 = 𝑊 → (Base‘𝑔) = (Base‘𝑊)) | |
3 | ipffval.1 | . . . . . 6 ⊢ 𝑉 = (Base‘𝑊) | |
4 | 2, 3 | syl6eqr 2874 | . . . . 5 ⊢ (𝑔 = 𝑊 → (Base‘𝑔) = 𝑉) |
5 | fveq2 6670 | . . . . . . 7 ⊢ (𝑔 = 𝑊 → (·𝑖‘𝑔) = (·𝑖‘𝑊)) | |
6 | ipffval.2 | . . . . . . 7 ⊢ , = (·𝑖‘𝑊) | |
7 | 5, 6 | syl6eqr 2874 | . . . . . 6 ⊢ (𝑔 = 𝑊 → (·𝑖‘𝑔) = , ) |
8 | 7 | oveqd 7173 | . . . . 5 ⊢ (𝑔 = 𝑊 → (𝑥(·𝑖‘𝑔)𝑦) = (𝑥 , 𝑦)) |
9 | 4, 4, 8 | mpoeq123dv 7229 | . . . 4 ⊢ (𝑔 = 𝑊 → (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(·𝑖‘𝑔)𝑦)) = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦))) |
10 | df-ipf 20771 | . . . 4 ⊢ ·if = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(·𝑖‘𝑔)𝑦))) | |
11 | 3 | fvexi 6684 | . . . . 5 ⊢ 𝑉 ∈ V |
12 | 6 | fvexi 6684 | . . . . . . 7 ⊢ , ∈ V |
13 | 12 | rnex 7617 | . . . . . 6 ⊢ ran , ∈ V |
14 | p0ex 5285 | . . . . . 6 ⊢ {∅} ∈ V | |
15 | 13, 14 | unex 7469 | . . . . 5 ⊢ (ran , ∪ {∅}) ∈ V |
16 | df-ov 7159 | . . . . . . 7 ⊢ (𝑥 , 𝑦) = ( , ‘〈𝑥, 𝑦〉) | |
17 | fvrn0 6698 | . . . . . . 7 ⊢ ( , ‘〈𝑥, 𝑦〉) ∈ (ran , ∪ {∅}) | |
18 | 16, 17 | eqeltri 2909 | . . . . . 6 ⊢ (𝑥 , 𝑦) ∈ (ran , ∪ {∅}) |
19 | 18 | rgen2w 3151 | . . . . 5 ⊢ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥 , 𝑦) ∈ (ran , ∪ {∅}) |
20 | 11, 11, 15, 19 | mpoexw 7776 | . . . 4 ⊢ (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦)) ∈ V |
21 | 9, 10, 20 | fvmpt 6768 | . . 3 ⊢ (𝑊 ∈ V → (·if‘𝑊) = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦))) |
22 | fvprc 6663 | . . . 4 ⊢ (¬ 𝑊 ∈ V → (·if‘𝑊) = ∅) | |
23 | fvprc 6663 | . . . . . . 7 ⊢ (¬ 𝑊 ∈ V → (Base‘𝑊) = ∅) | |
24 | 3, 23 | syl5eq 2868 | . . . . . 6 ⊢ (¬ 𝑊 ∈ V → 𝑉 = ∅) |
25 | 24 | olcd 870 | . . . . 5 ⊢ (¬ 𝑊 ∈ V → (𝑉 = ∅ ∨ 𝑉 = ∅)) |
26 | 0mpo0 7237 | . . . . 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 2859 | . . 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 2844 | 1 ⊢ · = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∨ wo 843 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ∪ cun 3934 ∅c0 4291 {csn 4567 〈cop 4573 ran crn 5556 ‘cfv 6355 (class class class)co 7156 ∈ cmpo 7158 Basecbs 16483 ·𝑖cip 16570 ·ifcipf 20769 |
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 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 df-ipf 20771 |
This theorem is referenced by: ipfval 20793 ipfeq 20794 ipffn 20795 phlipf 20796 phssip 20802 |
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