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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 = (CSubSp‘𝑊) |
thlle.i | ⊢ 𝐼 = (toInc‘𝐶) |
thlle.l | ⊢ ≤ = (le‘𝐼) |
Ref | Expression |
---|---|
thlle | ⊢ ≤ = (le‘𝐾) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | thlval.k | . . . . 5 ⊢ 𝐾 = (toHL‘𝑊) | |
2 | thlbas.c | . . . . 5 ⊢ 𝐶 = (CSubSp‘𝑊) | |
3 | thlle.i | . . . . 5 ⊢ 𝐼 = (toInc‘𝐶) | |
4 | eqid 2610 | . . . . 5 ⊢ (ocv‘𝑊) = (ocv‘𝑊) | |
5 | 1, 2, 3, 4 | thlval 19858 | . . . 4 ⊢ (𝑊 ∈ V → 𝐾 = (𝐼 sSet 〈(oc‘ndx), (ocv‘𝑊)〉)) |
6 | 5 | fveq2d 6107 | . . 3 ⊢ (𝑊 ∈ V → (le‘𝐾) = (le‘(𝐼 sSet 〈(oc‘ndx), (ocv‘𝑊)〉))) |
7 | thlle.l | . . . 4 ⊢ ≤ = (le‘𝐼) | |
8 | pleid 15872 | . . . . 5 ⊢ le = Slot (le‘ndx) | |
9 | 10re 11393 | . . . . . . 7 ⊢ ;10 ∈ ℝ | |
10 | 1nn0 11185 | . . . . . . . 8 ⊢ 1 ∈ ℕ0 | |
11 | 0nn0 11184 | . . . . . . . 8 ⊢ 0 ∈ ℕ0 | |
12 | 1nn 10908 | . . . . . . . 8 ⊢ 1 ∈ ℕ | |
13 | 0lt1 10429 | . . . . . . . 8 ⊢ 0 < 1 | |
14 | 10, 11, 12, 13 | declt 11406 | . . . . . . 7 ⊢ ;10 < ;11 |
15 | 9, 14 | ltneii 10029 | . . . . . 6 ⊢ ;10 ≠ ;11 |
16 | plendx 15870 | . . . . . . 7 ⊢ (le‘ndx) = ;10 | |
17 | ocndx 15883 | . . . . . . 7 ⊢ (oc‘ndx) = ;11 | |
18 | 16, 17 | neeq12i 2848 | . . . . . 6 ⊢ ((le‘ndx) ≠ (oc‘ndx) ↔ ;10 ≠ ;11) |
19 | 15, 18 | mpbir 220 | . . . . 5 ⊢ (le‘ndx) ≠ (oc‘ndx) |
20 | 8, 19 | setsnid 15743 | . . . 4 ⊢ (le‘𝐼) = (le‘(𝐼 sSet 〈(oc‘ndx), (ocv‘𝑊)〉)) |
21 | 7, 20 | eqtri 2632 | . . 3 ⊢ ≤ = (le‘(𝐼 sSet 〈(oc‘ndx), (ocv‘𝑊)〉)) |
22 | 6, 21 | syl6reqr 2663 | . 2 ⊢ (𝑊 ∈ V → ≤ = (le‘𝐾)) |
23 | 8 | str0 15739 | . . 3 ⊢ ∅ = (le‘∅) |
24 | fvex 6113 | . . . . . . 7 ⊢ (CSubSp‘𝑊) ∈ V | |
25 | 2, 24 | eqeltri 2684 | . . . . . 6 ⊢ 𝐶 ∈ V |
26 | 3 | ipolerval 16979 | . . . . . 6 ⊢ (𝐶 ∈ V → {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐶 ∧ 𝑥 ⊆ 𝑦)} = (le‘𝐼)) |
27 | 25, 26 | ax-mp 5 | . . . . 5 ⊢ {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐶 ∧ 𝑥 ⊆ 𝑦)} = (le‘𝐼) |
28 | 7, 27 | eqtr4i 2635 | . . . 4 ⊢ ≤ = {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐶 ∧ 𝑥 ⊆ 𝑦)} |
29 | opabn0 4931 | . . . . . 6 ⊢ ({〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐶 ∧ 𝑥 ⊆ 𝑦)} ≠ ∅ ↔ ∃𝑥∃𝑦({𝑥, 𝑦} ⊆ 𝐶 ∧ 𝑥 ⊆ 𝑦)) | |
30 | vex 3176 | . . . . . . . . 9 ⊢ 𝑥 ∈ V | |
31 | vex 3176 | . . . . . . . . 9 ⊢ 𝑦 ∈ V | |
32 | 30, 31 | prss 4291 | . . . . . . . 8 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) ↔ {𝑥, 𝑦} ⊆ 𝐶) |
33 | elfvex 6131 | . . . . . . . . . 10 ⊢ (𝑥 ∈ (CSubSp‘𝑊) → 𝑊 ∈ V) | |
34 | 33, 2 | eleq2s 2706 | . . . . . . . . 9 ⊢ (𝑥 ∈ 𝐶 → 𝑊 ∈ V) |
35 | 34 | ad2antrr 758 | . . . . . . . 8 ⊢ (((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ⊆ 𝑦) → 𝑊 ∈ V) |
36 | 32, 35 | sylanbr 489 | . . . . . . 7 ⊢ (({𝑥, 𝑦} ⊆ 𝐶 ∧ 𝑥 ⊆ 𝑦) → 𝑊 ∈ V) |
37 | 36 | exlimivv 1847 | . . . . . 6 ⊢ (∃𝑥∃𝑦({𝑥, 𝑦} ⊆ 𝐶 ∧ 𝑥 ⊆ 𝑦) → 𝑊 ∈ V) |
38 | 29, 37 | sylbi 206 | . . . . 5 ⊢ ({〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐶 ∧ 𝑥 ⊆ 𝑦)} ≠ ∅ → 𝑊 ∈ V) |
39 | 38 | necon1bi 2810 | . . . 4 ⊢ (¬ 𝑊 ∈ V → {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐶 ∧ 𝑥 ⊆ 𝑦)} = ∅) |
40 | 28, 39 | syl5eq 2656 | . . 3 ⊢ (¬ 𝑊 ∈ V → ≤ = ∅) |
41 | fvprc 6097 | . . . . 5 ⊢ (¬ 𝑊 ∈ V → (toHL‘𝑊) = ∅) | |
42 | 1, 41 | syl5eq 2656 | . . . 4 ⊢ (¬ 𝑊 ∈ V → 𝐾 = ∅) |
43 | 42 | fveq2d 6107 | . . 3 ⊢ (¬ 𝑊 ∈ V → (le‘𝐾) = (le‘∅)) |
44 | 23, 40, 43 | 3eqtr4a 2670 | . 2 ⊢ (¬ 𝑊 ∈ V → ≤ = (le‘𝐾)) |
45 | 22, 44 | pm2.61i 175 | 1 ⊢ ≤ = (le‘𝐾) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 383 = wceq 1475 ∃wex 1695 ∈ wcel 1977 ≠ wne 2780 Vcvv 3173 ⊆ wss 3540 ∅c0 3874 {cpr 4127 〈cop 4131 {copab 4642 ‘cfv 5804 (class class class)co 6549 0cc0 9815 1c1 9816 ;cdc 11369 ndxcnx 15692 sSet csts 15693 lecple 15775 occoc 15776 toInccipo 16974 ocvcocv 19823 CSubSpccss 19824 toHLcthl 19825 |
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-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-int 4411 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-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 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-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-fz 12198 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-tset 15787 df-ple 15788 df-ocomp 15790 df-ipo 16975 df-thl 19828 |
This theorem is referenced by: thlleval 19861 |
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