Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > hlhilipval | Structured version Visualization version GIF version |
Description: Value of inner product operation for the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.) |
Ref | Expression |
---|---|
hlhilip.h | ⊢ 𝐻 = (LHyp‘𝐾) |
hlhilip.l | ⊢ 𝐿 = ((DVecH‘𝐾)‘𝑊) |
hlhilip.v | ⊢ 𝑉 = (Base‘𝐿) |
hlhilip.s | ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) |
hlhilip.u | ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) |
hlhilip.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
hlhilip.i | ⊢ , = (·𝑖‘𝑈) |
hlhilip.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
hlhilip.y | ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
Ref | Expression |
---|---|
hlhilipval | ⊢ (𝜑 → (𝑋 , 𝑌) = ((𝑆‘𝑌)‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlhilip.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | hlhilip.l | . . . . 5 ⊢ 𝐿 = ((DVecH‘𝐾)‘𝑊) | |
3 | hlhilip.v | . . . . 5 ⊢ 𝑉 = (Base‘𝐿) | |
4 | hlhilip.s | . . . . 5 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) | |
5 | hlhilip.u | . . . . 5 ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) | |
6 | hlhilip.k | . . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
7 | eqid 2820 | . . . . 5 ⊢ (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ ((𝑆‘𝑦)‘𝑥)) = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ ((𝑆‘𝑦)‘𝑥)) | |
8 | 1, 2, 3, 4, 5, 6, 7 | hlhilip 39111 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ ((𝑆‘𝑦)‘𝑥)) = (·𝑖‘𝑈)) |
9 | hlhilip.i | . . . 4 ⊢ , = (·𝑖‘𝑈) | |
10 | 8, 9 | syl6reqr 2874 | . . 3 ⊢ (𝜑 → , = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ ((𝑆‘𝑦)‘𝑥))) |
11 | 10 | oveqd 7154 | . 2 ⊢ (𝜑 → (𝑋 , 𝑌) = (𝑋(𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ ((𝑆‘𝑦)‘𝑥))𝑌)) |
12 | hlhilip.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
13 | hlhilip.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
14 | fveq2 6651 | . . . 4 ⊢ (𝑥 = 𝑋 → ((𝑆‘𝑦)‘𝑥) = ((𝑆‘𝑦)‘𝑋)) | |
15 | fveq2 6651 | . . . . 5 ⊢ (𝑦 = 𝑌 → (𝑆‘𝑦) = (𝑆‘𝑌)) | |
16 | 15 | fveq1d 6653 | . . . 4 ⊢ (𝑦 = 𝑌 → ((𝑆‘𝑦)‘𝑋) = ((𝑆‘𝑌)‘𝑋)) |
17 | fvex 6664 | . . . 4 ⊢ ((𝑆‘𝑌)‘𝑋) ∈ V | |
18 | 14, 16, 7, 17 | ovmpo 7291 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋(𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ ((𝑆‘𝑦)‘𝑥))𝑌) = ((𝑆‘𝑌)‘𝑋)) |
19 | 12, 13, 18 | syl2anc 586 | . 2 ⊢ (𝜑 → (𝑋(𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ ((𝑆‘𝑦)‘𝑥))𝑌) = ((𝑆‘𝑌)‘𝑋)) |
20 | 11, 19 | eqtrd 2855 | 1 ⊢ (𝜑 → (𝑋 , 𝑌) = ((𝑆‘𝑌)‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ‘cfv 6336 (class class class)co 7137 ∈ cmpo 7139 Basecbs 16461 ·𝑖cip 16548 HLchlt 36513 LHypclh 37147 DVecHcdvh 38241 HDMapchdma 38955 HLHilchlh 39095 |
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-rep 5171 ax-sep 5184 ax-nul 5191 ax-pow 5247 ax-pr 5311 ax-un 7442 ax-cnex 10574 ax-resscn 10575 ax-1cn 10576 ax-icn 10577 ax-addcl 10578 ax-addrcl 10579 ax-mulcl 10580 ax-mulrcl 10581 ax-mulcom 10582 ax-addass 10583 ax-mulass 10584 ax-distr 10585 ax-i2m1 10586 ax-1ne0 10587 ax-1rid 10588 ax-rnegex 10589 ax-rrecex 10590 ax-cnre 10591 ax-pre-lttri 10592 ax-pre-lttrn 10593 ax-pre-ltadd 10594 ax-pre-mulgt0 10595 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 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-nel 3119 df-ral 3138 df-rex 3139 df-reu 3140 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-pss 3937 df-nul 4275 df-if 4449 df-pw 4522 df-sn 4549 df-pr 4551 df-tp 4553 df-op 4555 df-uni 4820 df-int 4858 df-iun 4902 df-br 5048 df-opab 5110 df-mpt 5128 df-tr 5154 df-id 5441 df-eprel 5446 df-po 5455 df-so 5456 df-fr 5495 df-we 5497 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-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6295 df-fun 6338 df-fn 6339 df-f 6340 df-f1 6341 df-fo 6342 df-f1o 6343 df-fv 6344 df-riota 7095 df-ov 7140 df-oprab 7141 df-mpo 7142 df-om 7562 df-1st 7670 df-2nd 7671 df-wrecs 7928 df-recs 7989 df-rdg 8027 df-1o 8083 df-oadd 8087 df-er 8270 df-en 8491 df-dom 8492 df-sdom 8493 df-fin 8494 df-pnf 10658 df-mnf 10659 df-xr 10660 df-ltxr 10661 df-le 10662 df-sub 10853 df-neg 10854 df-nn 11620 df-2 11682 df-3 11683 df-4 11684 df-5 11685 df-6 11686 df-7 11687 df-8 11688 df-n0 11880 df-z 11964 df-uz 12226 df-fz 12878 df-struct 16463 df-ndx 16464 df-slot 16465 df-base 16467 df-plusg 16556 df-sca 16559 df-vsca 16560 df-ip 16561 df-hlhil 39096 |
This theorem is referenced by: hlhilocv 39120 hlhilphllem 39122 |
Copyright terms: Public domain | W3C validator |