![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > phlipf | Structured version Visualization version GIF version |
Description: The inner product operation is a function. (Contributed by Mario Carneiro, 14-Aug-2015.) |
Ref | Expression |
---|---|
ipffn.1 | ⊢ 𝑉 = (Base‘𝑊) |
ipffn.2 | ⊢ , = (·if‘𝑊) |
phlipf.s | ⊢ 𝑆 = (Scalar‘𝑊) |
phlipf.k | ⊢ 𝐾 = (Base‘𝑆) |
Ref | Expression |
---|---|
phlipf | ⊢ (𝑊 ∈ PreHil → , :(𝑉 × 𝑉)⟶𝐾) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | phlipf.s | . . . . 5 ⊢ 𝑆 = (Scalar‘𝑊) | |
2 | eqid 2825 | . . . . 5 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊) | |
3 | ipffn.1 | . . . . 5 ⊢ 𝑉 = (Base‘𝑊) | |
4 | phlipf.k | . . . . 5 ⊢ 𝐾 = (Base‘𝑆) | |
5 | 1, 2, 3, 4 | ipcl 20340 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥(·𝑖‘𝑊)𝑦) ∈ 𝐾) |
6 | 5 | 3expb 1155 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑥(·𝑖‘𝑊)𝑦) ∈ 𝐾) |
7 | 6 | ralrimivva 3180 | . 2 ⊢ (𝑊 ∈ PreHil → ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥(·𝑖‘𝑊)𝑦) ∈ 𝐾) |
8 | ipffn.2 | . . . 4 ⊢ , = (·if‘𝑊) | |
9 | 3, 2, 8 | ipffval 20355 | . . 3 ⊢ , = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥(·𝑖‘𝑊)𝑦)) |
10 | 9 | fmpt2 7500 | . 2 ⊢ (∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥(·𝑖‘𝑊)𝑦) ∈ 𝐾 ↔ , :(𝑉 × 𝑉)⟶𝐾) |
11 | 7, 10 | sylib 210 | 1 ⊢ (𝑊 ∈ PreHil → , :(𝑉 × 𝑉)⟶𝐾) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1658 ∈ wcel 2166 ∀wral 3117 × cxp 5340 ⟶wf 6119 ‘cfv 6123 (class class class)co 6905 Basecbs 16222 Scalarcsca 16308 ·𝑖cip 16310 PreHilcphl 20331 ·ifcipf 20332 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-1st 7428 df-2nd 7429 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-er 8009 df-en 8223 df-dom 8224 df-sdom 8225 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-nn 11351 df-2 11414 df-3 11415 df-4 11416 df-5 11417 df-6 11418 df-7 11419 df-8 11420 df-ndx 16225 df-slot 16226 df-base 16228 df-sets 16229 df-sca 16321 df-vsca 16322 df-ip 16323 df-ghm 18009 df-lmhm 19381 df-sra 19533 df-rgmod 19534 df-phl 20333 df-ipf 20334 |
This theorem is referenced by: ipcn 23414 |
Copyright terms: Public domain | W3C validator |