Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 | thlval.k | . . . . 5 ⊢ 𝐾 = (toHL‘𝑊) | |
2 | eqid 2821 | . . . . 5 ⊢ (ClSubSp‘𝑊) = (ClSubSp‘𝑊) | |
3 | eqid 2821 | . . . . 5 ⊢ (toInc‘(ClSubSp‘𝑊)) = (toInc‘(ClSubSp‘𝑊)) | |
4 | thloc.c | . . . . 5 ⊢ ⊥ = (ocv‘𝑊) | |
5 | 1, 2, 3, 4 | thlval 20839 | . . . 4 ⊢ (𝑊 ∈ V → 𝐾 = ((toInc‘(ClSubSp‘𝑊)) sSet 〈(oc‘ndx), ⊥ 〉)) |
6 | 5 | fveq2d 6674 | . . 3 ⊢ (𝑊 ∈ V → (oc‘𝐾) = (oc‘((toInc‘(ClSubSp‘𝑊)) sSet 〈(oc‘ndx), ⊥ 〉))) |
7 | fvex 6683 | . . . 4 ⊢ (toInc‘(ClSubSp‘𝑊)) ∈ V | |
8 | 4 | fvexi 6684 | . . . 4 ⊢ ⊥ ∈ V |
9 | ocid 16674 | . . . . 5 ⊢ oc = Slot (oc‘ndx) | |
10 | 9 | setsid 16538 | . . . 4 ⊢ (((toInc‘(ClSubSp‘𝑊)) ∈ V ∧ ⊥ ∈ V) → ⊥ = (oc‘((toInc‘(ClSubSp‘𝑊)) sSet 〈(oc‘ndx), ⊥ 〉))) |
11 | 7, 8, 10 | mp2an 690 | . . 3 ⊢ ⊥ = (oc‘((toInc‘(ClSubSp‘𝑊)) sSet 〈(oc‘ndx), ⊥ 〉)) |
12 | 6, 11 | syl6reqr 2875 | . 2 ⊢ (𝑊 ∈ V → ⊥ = (oc‘𝐾)) |
13 | 9 | str0 16535 | . . 3 ⊢ ∅ = (oc‘∅) |
14 | fvprc 6663 | . . . 4 ⊢ (¬ 𝑊 ∈ V → (ocv‘𝑊) = ∅) | |
15 | 4, 14 | syl5eq 2868 | . . 3 ⊢ (¬ 𝑊 ∈ V → ⊥ = ∅) |
16 | fvprc 6663 | . . . . 5 ⊢ (¬ 𝑊 ∈ V → (toHL‘𝑊) = ∅) | |
17 | 1, 16 | syl5eq 2868 | . . . 4 ⊢ (¬ 𝑊 ∈ V → 𝐾 = ∅) |
18 | 17 | fveq2d 6674 | . . 3 ⊢ (¬ 𝑊 ∈ V → (oc‘𝐾) = (oc‘∅)) |
19 | 13, 15, 18 | 3eqtr4a 2882 | . 2 ⊢ (¬ 𝑊 ∈ V → ⊥ = (oc‘𝐾)) |
20 | 12, 19 | pm2.61i 184 | 1 ⊢ ⊥ = (oc‘𝐾) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ∅c0 4291 〈cop 4573 ‘cfv 6355 (class class class)co 7156 ndxcnx 16480 sSet csts 16481 occoc 16573 toInccipo 17761 ocvcocv 20804 ClSubSpccss 20805 toHLcthl 20806 |
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 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 |
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 2654 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 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-ltxr 10680 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-dec 12100 df-ndx 16486 df-slot 16487 df-sets 16490 df-ocomp 16586 df-thl 20809 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |