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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 2821 | . . . . 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 20771 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥(·𝑖‘𝑊)𝑦) ∈ 𝐾) |
6 | 5 | 3expb 1116 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑥(·𝑖‘𝑊)𝑦) ∈ 𝐾) |
7 | 6 | ralrimivva 3191 | . 2 ⊢ (𝑊 ∈ PreHil → ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥(·𝑖‘𝑊)𝑦) ∈ 𝐾) |
8 | ipffn.2 | . . . 4 ⊢ , = (·if‘𝑊) | |
9 | 3, 2, 8 | ipffval 20786 | . . 3 ⊢ , = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥(·𝑖‘𝑊)𝑦)) |
10 | 9 | fmpo 7760 | . 2 ⊢ (∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥(·𝑖‘𝑊)𝑦) ∈ 𝐾 ↔ , :(𝑉 × 𝑉)⟶𝐾) |
11 | 7, 10 | sylib 220 | 1 ⊢ (𝑊 ∈ PreHil → , :(𝑉 × 𝑉)⟶𝐾) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ∀wral 3138 × cxp 5547 ⟶wf 6345 ‘cfv 6349 (class class class)co 7150 Basecbs 16477 Scalarcsca 16562 ·𝑖cip 16564 PreHilcphl 20762 ·ifcipf 20763 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-sca 16575 df-vsca 16576 df-ip 16577 df-ghm 18350 df-lmhm 19788 df-sra 19938 df-rgmod 19939 df-phl 20764 df-ipf 20765 |
This theorem is referenced by: ipcn 23843 |
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