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Mirrors > Home > MPE Home > Th. List > thloc | Structured version Visualization version GIF version |
Description: Orthocomplement on the Hilbert lattice of closed subspaces. (Contributed by Mario Carneiro, 25-Oct-2015.) |
Ref | Expression |
---|---|
thlval.k | ⊢ 𝐾 = (toHL‘𝑊) |
thloc.c | ⊢ ⊥ = (ocv‘𝑊) |
Ref | Expression |
---|---|
thloc | ⊢ ⊥ = (oc‘𝐾) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvex 6708 | . . . 4 ⊢ (toInc‘(ClSubSp‘𝑊)) ∈ V | |
2 | thloc.c | . . . . 5 ⊢ ⊥ = (ocv‘𝑊) | |
3 | 2 | fvexi 6709 | . . . 4 ⊢ ⊥ ∈ V |
4 | ocid 16858 | . . . . 5 ⊢ oc = Slot (oc‘ndx) | |
5 | 4 | setsid 16719 | . . . 4 ⊢ (((toInc‘(ClSubSp‘𝑊)) ∈ V ∧ ⊥ ∈ V) → ⊥ = (oc‘((toInc‘(ClSubSp‘𝑊)) sSet 〈(oc‘ndx), ⊥ 〉))) |
6 | 1, 3, 5 | mp2an 692 | . . 3 ⊢ ⊥ = (oc‘((toInc‘(ClSubSp‘𝑊)) sSet 〈(oc‘ndx), ⊥ 〉)) |
7 | thlval.k | . . . . 5 ⊢ 𝐾 = (toHL‘𝑊) | |
8 | eqid 2736 | . . . . 5 ⊢ (ClSubSp‘𝑊) = (ClSubSp‘𝑊) | |
9 | eqid 2736 | . . . . 5 ⊢ (toInc‘(ClSubSp‘𝑊)) = (toInc‘(ClSubSp‘𝑊)) | |
10 | 7, 8, 9, 2 | thlval 20611 | . . . 4 ⊢ (𝑊 ∈ V → 𝐾 = ((toInc‘(ClSubSp‘𝑊)) sSet 〈(oc‘ndx), ⊥ 〉)) |
11 | 10 | fveq2d 6699 | . . 3 ⊢ (𝑊 ∈ V → (oc‘𝐾) = (oc‘((toInc‘(ClSubSp‘𝑊)) sSet 〈(oc‘ndx), ⊥ 〉))) |
12 | 6, 11 | eqtr4id 2790 | . 2 ⊢ (𝑊 ∈ V → ⊥ = (oc‘𝐾)) |
13 | 4 | str0 16717 | . . 3 ⊢ ∅ = (oc‘∅) |
14 | fvprc 6687 | . . . 4 ⊢ (¬ 𝑊 ∈ V → (ocv‘𝑊) = ∅) | |
15 | 2, 14 | syl5eq 2783 | . . 3 ⊢ (¬ 𝑊 ∈ V → ⊥ = ∅) |
16 | fvprc 6687 | . . . . 5 ⊢ (¬ 𝑊 ∈ V → (toHL‘𝑊) = ∅) | |
17 | 7, 16 | syl5eq 2783 | . . . 4 ⊢ (¬ 𝑊 ∈ V → 𝐾 = ∅) |
18 | 17 | fveq2d 6699 | . . 3 ⊢ (¬ 𝑊 ∈ V → (oc‘𝐾) = (oc‘∅)) |
19 | 13, 15, 18 | 3eqtr4a 2797 | . 2 ⊢ (¬ 𝑊 ∈ V → ⊥ = (oc‘𝐾)) |
20 | 12, 19 | pm2.61i 185 | 1 ⊢ ⊥ = (oc‘𝐾) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1543 ∈ wcel 2112 Vcvv 3398 ∅c0 4223 〈cop 4533 ‘cfv 6358 (class class class)co 7191 ndxcnx 16663 sSet csts 16664 occoc 16757 toInccipo 17987 ocvcocv 20576 ClSubSpccss 20577 toHLcthl 20578 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 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 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-pnf 10834 df-mnf 10835 df-ltxr 10837 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-5 11861 df-6 11862 df-7 11863 df-8 11864 df-9 11865 df-n0 12056 df-dec 12259 df-ndx 16669 df-slot 16670 df-sets 16673 df-ocomp 16770 df-thl 20581 |
This theorem is referenced by: (None) |
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