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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 2610 | . . . . 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 19797 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥(·𝑖‘𝑊)𝑦) ∈ 𝐾) |
6 | 5 | 3expb 1258 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑥(·𝑖‘𝑊)𝑦) ∈ 𝐾) |
7 | 6 | ralrimivva 2954 | . 2 ⊢ (𝑊 ∈ PreHil → ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥(·𝑖‘𝑊)𝑦) ∈ 𝐾) |
8 | ipffn.2 | . . . 4 ⊢ , = (·if‘𝑊) | |
9 | 3, 2, 8 | ipffval 19812 | . . 3 ⊢ , = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥(·𝑖‘𝑊)𝑦)) |
10 | 9 | fmpt2 7126 | . 2 ⊢ (∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥(·𝑖‘𝑊)𝑦) ∈ 𝐾 ↔ , :(𝑉 × 𝑉)⟶𝐾) |
11 | 7, 10 | sylib 207 | 1 ⊢ (𝑊 ∈ PreHil → , :(𝑉 × 𝑉)⟶𝐾) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ∀wral 2896 × cxp 5036 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 Scalarcsca 15771 ·𝑖cip 15773 PreHilcphl 19788 ·ifcipf 19789 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-sca 15784 df-vsca 15785 df-ip 15786 df-ghm 17481 df-lmhm 18843 df-sra 18993 df-rgmod 18994 df-phl 19790 df-ipf 19791 |
This theorem is referenced by: ipcn 22853 |
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