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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 2622 | . . . . 5 ⊢ (ocv‘𝑊) = (ocv‘𝑊) | |
5 | 1, 2, 3, 4 | thlval 20039 | . . . 4 ⊢ (𝑊 ∈ V → 𝐾 = (𝐼 sSet 〈(oc‘ndx), (ocv‘𝑊)〉)) |
6 | 5 | fveq2d 6195 | . . 3 ⊢ (𝑊 ∈ V → (le‘𝐾) = (le‘(𝐼 sSet 〈(oc‘ndx), (ocv‘𝑊)〉))) |
7 | thlle.l | . . . 4 ⊢ ≤ = (le‘𝐼) | |
8 | pleid 16049 | . . . . 5 ⊢ le = Slot (le‘ndx) | |
9 | 10re 11517 | . . . . . . 7 ⊢ ;10 ∈ ℝ | |
10 | 1nn0 11308 | . . . . . . . 8 ⊢ 1 ∈ ℕ0 | |
11 | 0nn0 11307 | . . . . . . . 8 ⊢ 0 ∈ ℕ0 | |
12 | 1nn 11031 | . . . . . . . 8 ⊢ 1 ∈ ℕ | |
13 | 0lt1 10550 | . . . . . . . 8 ⊢ 0 < 1 | |
14 | 10, 11, 12, 13 | declt 11530 | . . . . . . 7 ⊢ ;10 < ;11 |
15 | 9, 14 | ltneii 10150 | . . . . . 6 ⊢ ;10 ≠ ;11 |
16 | plendx 16047 | . . . . . . 7 ⊢ (le‘ndx) = ;10 | |
17 | ocndx 16060 | . . . . . . 7 ⊢ (oc‘ndx) = ;11 | |
18 | 16, 17 | neeq12i 2860 | . . . . . 6 ⊢ ((le‘ndx) ≠ (oc‘ndx) ↔ ;10 ≠ ;11) |
19 | 15, 18 | mpbir 221 | . . . . 5 ⊢ (le‘ndx) ≠ (oc‘ndx) |
20 | 8, 19 | setsnid 15915 | . . . 4 ⊢ (le‘𝐼) = (le‘(𝐼 sSet 〈(oc‘ndx), (ocv‘𝑊)〉)) |
21 | 7, 20 | eqtri 2644 | . . 3 ⊢ ≤ = (le‘(𝐼 sSet 〈(oc‘ndx), (ocv‘𝑊)〉)) |
22 | 6, 21 | syl6reqr 2675 | . 2 ⊢ (𝑊 ∈ V → ≤ = (le‘𝐾)) |
23 | 8 | str0 15911 | . . 3 ⊢ ∅ = (le‘∅) |
24 | fvex 6201 | . . . . . . 7 ⊢ (CSubSp‘𝑊) ∈ V | |
25 | 2, 24 | eqeltri 2697 | . . . . . 6 ⊢ 𝐶 ∈ V |
26 | 3 | ipolerval 17156 | . . . . . 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 2647 | . . . 4 ⊢ ≤ = {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐶 ∧ 𝑥 ⊆ 𝑦)} |
29 | opabn0 5006 | . . . . . 6 ⊢ ({〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐶 ∧ 𝑥 ⊆ 𝑦)} ≠ ∅ ↔ ∃𝑥∃𝑦({𝑥, 𝑦} ⊆ 𝐶 ∧ 𝑥 ⊆ 𝑦)) | |
30 | vex 3203 | . . . . . . . . 9 ⊢ 𝑥 ∈ V | |
31 | vex 3203 | . . . . . . . . 9 ⊢ 𝑦 ∈ V | |
32 | 30, 31 | prss 4351 | . . . . . . . 8 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) ↔ {𝑥, 𝑦} ⊆ 𝐶) |
33 | elfvex 6221 | . . . . . . . . . 10 ⊢ (𝑥 ∈ (CSubSp‘𝑊) → 𝑊 ∈ V) | |
34 | 33, 2 | eleq2s 2719 | . . . . . . . . 9 ⊢ (𝑥 ∈ 𝐶 → 𝑊 ∈ V) |
35 | 34 | ad2antrr 762 | . . . . . . . 8 ⊢ (((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ⊆ 𝑦) → 𝑊 ∈ V) |
36 | 32, 35 | sylanbr 490 | . . . . . . 7 ⊢ (({𝑥, 𝑦} ⊆ 𝐶 ∧ 𝑥 ⊆ 𝑦) → 𝑊 ∈ V) |
37 | 36 | exlimivv 1860 | . . . . . 6 ⊢ (∃𝑥∃𝑦({𝑥, 𝑦} ⊆ 𝐶 ∧ 𝑥 ⊆ 𝑦) → 𝑊 ∈ V) |
38 | 29, 37 | sylbi 207 | . . . . 5 ⊢ ({〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐶 ∧ 𝑥 ⊆ 𝑦)} ≠ ∅ → 𝑊 ∈ V) |
39 | 38 | necon1bi 2822 | . . . 4 ⊢ (¬ 𝑊 ∈ V → {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐶 ∧ 𝑥 ⊆ 𝑦)} = ∅) |
40 | 28, 39 | syl5eq 2668 | . . 3 ⊢ (¬ 𝑊 ∈ V → ≤ = ∅) |
41 | fvprc 6185 | . . . . 5 ⊢ (¬ 𝑊 ∈ V → (toHL‘𝑊) = ∅) | |
42 | 1, 41 | syl5eq 2668 | . . . 4 ⊢ (¬ 𝑊 ∈ V → 𝐾 = ∅) |
43 | 42 | fveq2d 6195 | . . 3 ⊢ (¬ 𝑊 ∈ V → (le‘𝐾) = (le‘∅)) |
44 | 23, 40, 43 | 3eqtr4a 2682 | . 2 ⊢ (¬ 𝑊 ∈ V → ≤ = (le‘𝐾)) |
45 | 22, 44 | pm2.61i 176 | 1 ⊢ ≤ = (le‘𝐾) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 384 = wceq 1483 ∃wex 1704 ∈ wcel 1990 ≠ wne 2794 Vcvv 3200 ⊆ wss 3574 ∅c0 3915 {cpr 4179 〈cop 4183 {copab 4712 ‘cfv 5888 (class class class)co 6650 0cc0 9936 1c1 9937 ;cdc 11493 ndxcnx 15854 sSet csts 15855 lecple 15948 occoc 15949 toInccipo 17151 ocvcocv 20004 CSubSpccss 20005 toHLcthl 20006 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-fz 12327 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-tset 15960 df-ple 15961 df-ocomp 15963 df-ipo 17152 df-thl 20009 |
This theorem is referenced by: thlleval 20042 |
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