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Mirrors > Home > MPE Home > Th. List > thlle | Structured version Visualization version GIF version |
Description: Ordering on the Hilbert lattice of closed subspaces. (Contributed by Mario Carneiro, 25-Oct-2015.) |
Ref | Expression |
---|---|
thlval.k | ⊢ 𝐾 = (toHL‘𝑊) |
thlbas.c | ⊢ 𝐶 = (ClSubSp‘𝑊) |
thlle.i | ⊢ 𝐼 = (toInc‘𝐶) |
thlle.l | ⊢ ≤ = (le‘𝐼) |
Ref | Expression |
---|---|
thlle | ⊢ ≤ = (le‘𝐾) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | thlle.l | . . . 4 ⊢ ≤ = (le‘𝐼) | |
2 | pleid 16659 | . . . . 5 ⊢ le = Slot (le‘ndx) | |
3 | 10re 12105 | . . . . . . 7 ⊢ ;10 ∈ ℝ | |
4 | 1nn0 11901 | . . . . . . . 8 ⊢ 1 ∈ ℕ0 | |
5 | 0nn0 11900 | . . . . . . . 8 ⊢ 0 ∈ ℕ0 | |
6 | 1nn 11636 | . . . . . . . 8 ⊢ 1 ∈ ℕ | |
7 | 0lt1 11151 | . . . . . . . 8 ⊢ 0 < 1 | |
8 | 4, 5, 6, 7 | declt 12114 | . . . . . . 7 ⊢ ;10 < ;11 |
9 | 3, 8 | ltneii 10742 | . . . . . 6 ⊢ ;10 ≠ ;11 |
10 | plendx 16658 | . . . . . . 7 ⊢ (le‘ndx) = ;10 | |
11 | ocndx 16665 | . . . . . . 7 ⊢ (oc‘ndx) = ;11 | |
12 | 10, 11 | neeq12i 3053 | . . . . . 6 ⊢ ((le‘ndx) ≠ (oc‘ndx) ↔ ;10 ≠ ;11) |
13 | 9, 12 | mpbir 234 | . . . . 5 ⊢ (le‘ndx) ≠ (oc‘ndx) |
14 | 2, 13 | setsnid 16531 | . . . 4 ⊢ (le‘𝐼) = (le‘(𝐼 sSet 〈(oc‘ndx), (ocv‘𝑊)〉)) |
15 | 1, 14 | eqtri 2821 | . . 3 ⊢ ≤ = (le‘(𝐼 sSet 〈(oc‘ndx), (ocv‘𝑊)〉)) |
16 | thlval.k | . . . . 5 ⊢ 𝐾 = (toHL‘𝑊) | |
17 | thlbas.c | . . . . 5 ⊢ 𝐶 = (ClSubSp‘𝑊) | |
18 | thlle.i | . . . . 5 ⊢ 𝐼 = (toInc‘𝐶) | |
19 | eqid 2798 | . . . . 5 ⊢ (ocv‘𝑊) = (ocv‘𝑊) | |
20 | 16, 17, 18, 19 | thlval 20384 | . . . 4 ⊢ (𝑊 ∈ V → 𝐾 = (𝐼 sSet 〈(oc‘ndx), (ocv‘𝑊)〉)) |
21 | 20 | fveq2d 6649 | . . 3 ⊢ (𝑊 ∈ V → (le‘𝐾) = (le‘(𝐼 sSet 〈(oc‘ndx), (ocv‘𝑊)〉))) |
22 | 15, 21 | eqtr4id 2852 | . 2 ⊢ (𝑊 ∈ V → ≤ = (le‘𝐾)) |
23 | 2 | str0 16527 | . . 3 ⊢ ∅ = (le‘∅) |
24 | 17 | fvexi 6659 | . . . . . 6 ⊢ 𝐶 ∈ V |
25 | 18 | ipolerval 17758 | . . . . . 6 ⊢ (𝐶 ∈ V → {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐶 ∧ 𝑥 ⊆ 𝑦)} = (le‘𝐼)) |
26 | 24, 25 | ax-mp 5 | . . . . 5 ⊢ {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐶 ∧ 𝑥 ⊆ 𝑦)} = (le‘𝐼) |
27 | 1, 26 | eqtr4i 2824 | . . . 4 ⊢ ≤ = {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐶 ∧ 𝑥 ⊆ 𝑦)} |
28 | opabn0 5405 | . . . . . 6 ⊢ ({〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐶 ∧ 𝑥 ⊆ 𝑦)} ≠ ∅ ↔ ∃𝑥∃𝑦({𝑥, 𝑦} ⊆ 𝐶 ∧ 𝑥 ⊆ 𝑦)) | |
29 | vex 3444 | . . . . . . . . 9 ⊢ 𝑥 ∈ V | |
30 | vex 3444 | . . . . . . . . 9 ⊢ 𝑦 ∈ V | |
31 | 29, 30 | prss 4713 | . . . . . . . 8 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) ↔ {𝑥, 𝑦} ⊆ 𝐶) |
32 | elfvex 6678 | . . . . . . . . . 10 ⊢ (𝑥 ∈ (ClSubSp‘𝑊) → 𝑊 ∈ V) | |
33 | 32, 17 | eleq2s 2908 | . . . . . . . . 9 ⊢ (𝑥 ∈ 𝐶 → 𝑊 ∈ V) |
34 | 33 | ad2antrr 725 | . . . . . . . 8 ⊢ (((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ⊆ 𝑦) → 𝑊 ∈ V) |
35 | 31, 34 | sylanbr 585 | . . . . . . 7 ⊢ (({𝑥, 𝑦} ⊆ 𝐶 ∧ 𝑥 ⊆ 𝑦) → 𝑊 ∈ V) |
36 | 35 | exlimivv 1933 | . . . . . 6 ⊢ (∃𝑥∃𝑦({𝑥, 𝑦} ⊆ 𝐶 ∧ 𝑥 ⊆ 𝑦) → 𝑊 ∈ V) |
37 | 28, 36 | sylbi 220 | . . . . 5 ⊢ ({〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐶 ∧ 𝑥 ⊆ 𝑦)} ≠ ∅ → 𝑊 ∈ V) |
38 | 37 | necon1bi 3015 | . . . 4 ⊢ (¬ 𝑊 ∈ V → {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐶 ∧ 𝑥 ⊆ 𝑦)} = ∅) |
39 | 27, 38 | syl5eq 2845 | . . 3 ⊢ (¬ 𝑊 ∈ V → ≤ = ∅) |
40 | fvprc 6638 | . . . . 5 ⊢ (¬ 𝑊 ∈ V → (toHL‘𝑊) = ∅) | |
41 | 16, 40 | syl5eq 2845 | . . . 4 ⊢ (¬ 𝑊 ∈ V → 𝐾 = ∅) |
42 | 41 | fveq2d 6649 | . . 3 ⊢ (¬ 𝑊 ∈ V → (le‘𝐾) = (le‘∅)) |
43 | 23, 39, 42 | 3eqtr4a 2859 | . 2 ⊢ (¬ 𝑊 ∈ V → ≤ = (le‘𝐾)) |
44 | 22, 43 | pm2.61i 185 | 1 ⊢ ≤ = (le‘𝐾) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 399 = wceq 1538 ∃wex 1781 ∈ wcel 2111 ≠ wne 2987 Vcvv 3441 ⊆ wss 3881 ∅c0 4243 {cpr 4527 〈cop 4531 {copab 5092 ‘cfv 6324 (class class class)co 7135 0cc0 10526 1c1 10527 ;cdc 12086 ndxcnx 16472 sSet csts 16473 lecple 16564 occoc 16565 toInccipo 17753 ocvcocv 20349 ClSubSpccss 20350 toHLcthl 20351 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12886 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-tset 16576 df-ple 16577 df-ocomp 16578 df-ipo 17754 df-thl 20354 |
This theorem is referenced by: thlleval 20387 |
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