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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.) (Proof shortened by AV, 11-Nov-2024.) |
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 17294 | . . . . 5 ⊢ le = Slot (le‘ndx) | |
3 | plendxnocndx 17311 | . . . . 5 ⊢ (le‘ndx) ≠ (oc‘ndx) | |
4 | 2, 3 | setsnid 17124 | . . . 4 ⊢ (le‘𝐼) = (le‘(𝐼 sSet 〈(oc‘ndx), (ocv‘𝑊)〉)) |
5 | 1, 4 | eqtri 2759 | . . 3 ⊢ ≤ = (le‘(𝐼 sSet 〈(oc‘ndx), (ocv‘𝑊)〉)) |
6 | thlval.k | . . . . 5 ⊢ 𝐾 = (toHL‘𝑊) | |
7 | thlbas.c | . . . . 5 ⊢ 𝐶 = (ClSubSp‘𝑊) | |
8 | thlle.i | . . . . 5 ⊢ 𝐼 = (toInc‘𝐶) | |
9 | eqid 2731 | . . . . 5 ⊢ (ocv‘𝑊) = (ocv‘𝑊) | |
10 | 6, 7, 8, 9 | thlval 21181 | . . . 4 ⊢ (𝑊 ∈ V → 𝐾 = (𝐼 sSet 〈(oc‘ndx), (ocv‘𝑊)〉)) |
11 | 10 | fveq2d 6882 | . . 3 ⊢ (𝑊 ∈ V → (le‘𝐾) = (le‘(𝐼 sSet 〈(oc‘ndx), (ocv‘𝑊)〉))) |
12 | 5, 11 | eqtr4id 2790 | . 2 ⊢ (𝑊 ∈ V → ≤ = (le‘𝐾)) |
13 | 2 | str0 17104 | . . 3 ⊢ ∅ = (le‘∅) |
14 | 7 | fvexi 6892 | . . . . . 6 ⊢ 𝐶 ∈ V |
15 | 8 | ipolerval 18467 | . . . . . 6 ⊢ (𝐶 ∈ V → {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐶 ∧ 𝑥 ⊆ 𝑦)} = (le‘𝐼)) |
16 | 14, 15 | ax-mp 5 | . . . . 5 ⊢ {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐶 ∧ 𝑥 ⊆ 𝑦)} = (le‘𝐼) |
17 | 1, 16 | eqtr4i 2762 | . . . 4 ⊢ ≤ = {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐶 ∧ 𝑥 ⊆ 𝑦)} |
18 | opabn0 5546 | . . . . . 6 ⊢ ({〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐶 ∧ 𝑥 ⊆ 𝑦)} ≠ ∅ ↔ ∃𝑥∃𝑦({𝑥, 𝑦} ⊆ 𝐶 ∧ 𝑥 ⊆ 𝑦)) | |
19 | vex 3477 | . . . . . . . . 9 ⊢ 𝑥 ∈ V | |
20 | vex 3477 | . . . . . . . . 9 ⊢ 𝑦 ∈ V | |
21 | 19, 20 | prss 4816 | . . . . . . . 8 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) ↔ {𝑥, 𝑦} ⊆ 𝐶) |
22 | elfvex 6916 | . . . . . . . . . 10 ⊢ (𝑥 ∈ (ClSubSp‘𝑊) → 𝑊 ∈ V) | |
23 | 22, 7 | eleq2s 2850 | . . . . . . . . 9 ⊢ (𝑥 ∈ 𝐶 → 𝑊 ∈ V) |
24 | 23 | ad2antrr 724 | . . . . . . . 8 ⊢ (((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ⊆ 𝑦) → 𝑊 ∈ V) |
25 | 21, 24 | sylanbr 582 | . . . . . . 7 ⊢ (({𝑥, 𝑦} ⊆ 𝐶 ∧ 𝑥 ⊆ 𝑦) → 𝑊 ∈ V) |
26 | 25 | exlimivv 1935 | . . . . . 6 ⊢ (∃𝑥∃𝑦({𝑥, 𝑦} ⊆ 𝐶 ∧ 𝑥 ⊆ 𝑦) → 𝑊 ∈ V) |
27 | 18, 26 | sylbi 216 | . . . . 5 ⊢ ({〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐶 ∧ 𝑥 ⊆ 𝑦)} ≠ ∅ → 𝑊 ∈ V) |
28 | 27 | necon1bi 2968 | . . . 4 ⊢ (¬ 𝑊 ∈ V → {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐶 ∧ 𝑥 ⊆ 𝑦)} = ∅) |
29 | 17, 28 | eqtrid 2783 | . . 3 ⊢ (¬ 𝑊 ∈ V → ≤ = ∅) |
30 | fvprc 6870 | . . . . 5 ⊢ (¬ 𝑊 ∈ V → (toHL‘𝑊) = ∅) | |
31 | 6, 30 | eqtrid 2783 | . . . 4 ⊢ (¬ 𝑊 ∈ V → 𝐾 = ∅) |
32 | 31 | fveq2d 6882 | . . 3 ⊢ (¬ 𝑊 ∈ V → (le‘𝐾) = (le‘∅)) |
33 | 13, 29, 32 | 3eqtr4a 2797 | . 2 ⊢ (¬ 𝑊 ∈ V → ≤ = (le‘𝐾)) |
34 | 12, 33 | pm2.61i 182 | 1 ⊢ ≤ = (le‘𝐾) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 396 = wceq 1541 ∃wex 1781 ∈ wcel 2106 ≠ wne 2939 Vcvv 3473 ⊆ wss 3944 ∅c0 4318 {cpr 4624 〈cop 4628 {copab 5203 ‘cfv 6532 (class class class)co 7393 sSet csts 17078 ndxcnx 17108 lecple 17186 occoc 17187 toInccipo 18462 ocvcocv 21146 ClSubSpccss 21147 toHLcthl 21148 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-cnex 11148 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168 ax-pre-mulgt0 11169 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-om 7839 df-1st 7957 df-2nd 7958 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-1o 8448 df-er 8686 df-en 8923 df-dom 8924 df-sdom 8925 df-fin 8926 df-pnf 11232 df-mnf 11233 df-xr 11234 df-ltxr 11235 df-le 11236 df-sub 11428 df-neg 11429 df-nn 12195 df-2 12257 df-3 12258 df-4 12259 df-5 12260 df-6 12261 df-7 12262 df-8 12263 df-9 12264 df-n0 12455 df-z 12541 df-dec 12660 df-uz 12805 df-fz 13467 df-struct 17062 df-sets 17079 df-slot 17097 df-ndx 17109 df-base 17127 df-tset 17198 df-ple 17199 df-ocomp 17200 df-ipo 18463 df-thl 21151 |
This theorem is referenced by: thlleval 21186 |
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