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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 | thlval.k | . . . . 5 ⊢ 𝐾 = (toHL‘𝑊) | |
2 | thlbas.c | . . . . 5 ⊢ 𝐶 = (ClSubSp‘𝑊) | |
3 | thlle.i | . . . . 5 ⊢ 𝐼 = (toInc‘𝐶) | |
4 | eqid 2826 | . . . . 5 ⊢ (ocv‘𝑊) = (ocv‘𝑊) | |
5 | 1, 2, 3, 4 | thlval 20403 | . . . 4 ⊢ (𝑊 ∈ V → 𝐾 = (𝐼 sSet 〈(oc‘ndx), (ocv‘𝑊)〉)) |
6 | 5 | fveq2d 6438 | . . 3 ⊢ (𝑊 ∈ V → (le‘𝐾) = (le‘(𝐼 sSet 〈(oc‘ndx), (ocv‘𝑊)〉))) |
7 | thlle.l | . . . 4 ⊢ ≤ = (le‘𝐼) | |
8 | pleid 16408 | . . . . 5 ⊢ le = Slot (le‘ndx) | |
9 | 10re 11841 | . . . . . . 7 ⊢ ;10 ∈ ℝ | |
10 | 1nn0 11637 | . . . . . . . 8 ⊢ 1 ∈ ℕ0 | |
11 | 0nn0 11636 | . . . . . . . 8 ⊢ 0 ∈ ℕ0 | |
12 | 1nn 11364 | . . . . . . . 8 ⊢ 1 ∈ ℕ | |
13 | 0lt1 10875 | . . . . . . . 8 ⊢ 0 < 1 | |
14 | 10, 11, 12, 13 | declt 11851 | . . . . . . 7 ⊢ ;10 < ;11 |
15 | 9, 14 | ltneii 10470 | . . . . . 6 ⊢ ;10 ≠ ;11 |
16 | plendx 16407 | . . . . . . 7 ⊢ (le‘ndx) = ;10 | |
17 | ocndx 16414 | . . . . . . 7 ⊢ (oc‘ndx) = ;11 | |
18 | 16, 17 | neeq12i 3066 | . . . . . 6 ⊢ ((le‘ndx) ≠ (oc‘ndx) ↔ ;10 ≠ ;11) |
19 | 15, 18 | mpbir 223 | . . . . 5 ⊢ (le‘ndx) ≠ (oc‘ndx) |
20 | 8, 19 | setsnid 16279 | . . . 4 ⊢ (le‘𝐼) = (le‘(𝐼 sSet 〈(oc‘ndx), (ocv‘𝑊)〉)) |
21 | 7, 20 | eqtri 2850 | . . 3 ⊢ ≤ = (le‘(𝐼 sSet 〈(oc‘ndx), (ocv‘𝑊)〉)) |
22 | 6, 21 | syl6reqr 2881 | . 2 ⊢ (𝑊 ∈ V → ≤ = (le‘𝐾)) |
23 | 8 | str0 16275 | . . 3 ⊢ ∅ = (le‘∅) |
24 | 2 | fvexi 6448 | . . . . . 6 ⊢ 𝐶 ∈ V |
25 | 3 | ipolerval 17510 | . . . . . 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 | 7, 26 | eqtr4i 2853 | . . . 4 ⊢ ≤ = {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐶 ∧ 𝑥 ⊆ 𝑦)} |
28 | opabn0 5233 | . . . . . 6 ⊢ ({〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐶 ∧ 𝑥 ⊆ 𝑦)} ≠ ∅ ↔ ∃𝑥∃𝑦({𝑥, 𝑦} ⊆ 𝐶 ∧ 𝑥 ⊆ 𝑦)) | |
29 | vex 3418 | . . . . . . . . 9 ⊢ 𝑥 ∈ V | |
30 | vex 3418 | . . . . . . . . 9 ⊢ 𝑦 ∈ V | |
31 | 29, 30 | prss 4570 | . . . . . . . 8 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) ↔ {𝑥, 𝑦} ⊆ 𝐶) |
32 | elfvex 6468 | . . . . . . . . . 10 ⊢ (𝑥 ∈ (ClSubSp‘𝑊) → 𝑊 ∈ V) | |
33 | 32, 2 | eleq2s 2925 | . . . . . . . . 9 ⊢ (𝑥 ∈ 𝐶 → 𝑊 ∈ V) |
34 | 33 | ad2antrr 719 | . . . . . . . 8 ⊢ (((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ⊆ 𝑦) → 𝑊 ∈ V) |
35 | 31, 34 | sylanbr 579 | . . . . . . 7 ⊢ (({𝑥, 𝑦} ⊆ 𝐶 ∧ 𝑥 ⊆ 𝑦) → 𝑊 ∈ V) |
36 | 35 | exlimivv 2033 | . . . . . 6 ⊢ (∃𝑥∃𝑦({𝑥, 𝑦} ⊆ 𝐶 ∧ 𝑥 ⊆ 𝑦) → 𝑊 ∈ V) |
37 | 28, 36 | sylbi 209 | . . . . 5 ⊢ ({〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐶 ∧ 𝑥 ⊆ 𝑦)} ≠ ∅ → 𝑊 ∈ V) |
38 | 37 | necon1bi 3028 | . . . 4 ⊢ (¬ 𝑊 ∈ V → {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐶 ∧ 𝑥 ⊆ 𝑦)} = ∅) |
39 | 27, 38 | syl5eq 2874 | . . 3 ⊢ (¬ 𝑊 ∈ V → ≤ = ∅) |
40 | fvprc 6427 | . . . . 5 ⊢ (¬ 𝑊 ∈ V → (toHL‘𝑊) = ∅) | |
41 | 1, 40 | syl5eq 2874 | . . . 4 ⊢ (¬ 𝑊 ∈ V → 𝐾 = ∅) |
42 | 41 | fveq2d 6438 | . . 3 ⊢ (¬ 𝑊 ∈ V → (le‘𝐾) = (le‘∅)) |
43 | 23, 39, 42 | 3eqtr4a 2888 | . 2 ⊢ (¬ 𝑊 ∈ V → ≤ = (le‘𝐾)) |
44 | 22, 43 | pm2.61i 177 | 1 ⊢ ≤ = (le‘𝐾) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 386 = wceq 1658 ∃wex 1880 ∈ wcel 2166 ≠ wne 3000 Vcvv 3415 ⊆ wss 3799 ∅c0 4145 {cpr 4400 〈cop 4404 {copab 4936 ‘cfv 6124 (class class class)co 6906 0cc0 10253 1c1 10254 ;cdc 11822 ndxcnx 16220 sSet csts 16221 lecple 16313 occoc 16314 toInccipo 17505 ocvcocv 20368 ClSubSpccss 20369 toHLcthl 20370 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-cnex 10309 ax-resscn 10310 ax-1cn 10311 ax-icn 10312 ax-addcl 10313 ax-addrcl 10314 ax-mulcl 10315 ax-mulrcl 10316 ax-mulcom 10317 ax-addass 10318 ax-mulass 10319 ax-distr 10320 ax-i2m1 10321 ax-1ne0 10322 ax-1rid 10323 ax-rnegex 10324 ax-rrecex 10325 ax-cnre 10326 ax-pre-lttri 10327 ax-pre-lttrn 10328 ax-pre-ltadd 10329 ax-pre-mulgt0 10330 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-nel 3104 df-ral 3123 df-rex 3124 df-reu 3125 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-pss 3815 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4660 df-int 4699 df-iun 4743 df-br 4875 df-opab 4937 df-mpt 4954 df-tr 4977 df-id 5251 df-eprel 5256 df-po 5264 df-so 5265 df-fr 5302 df-we 5304 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-pred 5921 df-ord 5967 df-on 5968 df-lim 5969 df-suc 5970 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-riota 6867 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-om 7328 df-1st 7429 df-2nd 7430 df-wrecs 7673 df-recs 7735 df-rdg 7773 df-1o 7827 df-oadd 7831 df-er 8010 df-en 8224 df-dom 8225 df-sdom 8226 df-fin 8227 df-pnf 10394 df-mnf 10395 df-xr 10396 df-ltxr 10397 df-le 10398 df-sub 10588 df-neg 10589 df-nn 11352 df-2 11415 df-3 11416 df-4 11417 df-5 11418 df-6 11419 df-7 11420 df-8 11421 df-9 11422 df-n0 11620 df-z 11706 df-dec 11823 df-uz 11970 df-fz 12621 df-struct 16225 df-ndx 16226 df-slot 16227 df-base 16229 df-sets 16230 df-tset 16325 df-ple 16326 df-ocomp 16327 df-ipo 17506 df-thl 20373 |
This theorem is referenced by: thlleval 20406 |
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