![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > thlleOLD | Structured version Visualization version GIF version |
Description: Obsolete version of thlle 21733 as of 11-Nov-2024. Ordering on the Hilbert lattice of closed subspaces. (Contributed by Mario Carneiro, 25-Oct-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
thlval.k | ⊢ 𝐾 = (toHL‘𝑊) |
thlbas.c | ⊢ 𝐶 = (ClSubSp‘𝑊) |
thlle.i | ⊢ 𝐼 = (toInc‘𝐶) |
thlle.l | ⊢ ≤ = (le‘𝐼) |
Ref | Expression |
---|---|
thlleOLD | ⊢ ≤ = (le‘𝐾) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | thlle.l | . . . 4 ⊢ ≤ = (le‘𝐼) | |
2 | pleid 17412 | . . . . 5 ⊢ le = Slot (le‘ndx) | |
3 | 10re 12749 | . . . . . . 7 ⊢ ;10 ∈ ℝ | |
4 | 1nn0 12539 | . . . . . . . 8 ⊢ 1 ∈ ℕ0 | |
5 | 0nn0 12538 | . . . . . . . 8 ⊢ 0 ∈ ℕ0 | |
6 | 1nn 12274 | . . . . . . . 8 ⊢ 1 ∈ ℕ | |
7 | 0lt1 11782 | . . . . . . . 8 ⊢ 0 < 1 | |
8 | 4, 5, 6, 7 | declt 12758 | . . . . . . 7 ⊢ ;10 < ;11 |
9 | 3, 8 | ltneii 11371 | . . . . . 6 ⊢ ;10 ≠ ;11 |
10 | plendx 17411 | . . . . . . 7 ⊢ (le‘ndx) = ;10 | |
11 | ocndx 17426 | . . . . . . 7 ⊢ (oc‘ndx) = ;11 | |
12 | 10, 11 | neeq12i 3004 | . . . . . 6 ⊢ ((le‘ndx) ≠ (oc‘ndx) ↔ ;10 ≠ ;11) |
13 | 9, 12 | mpbir 231 | . . . . 5 ⊢ (le‘ndx) ≠ (oc‘ndx) |
14 | 2, 13 | setsnid 17242 | . . . 4 ⊢ (le‘𝐼) = (le‘(𝐼 sSet 〈(oc‘ndx), (ocv‘𝑊)〉)) |
15 | 1, 14 | eqtri 2762 | . . 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 2734 | . . . . 5 ⊢ (ocv‘𝑊) = (ocv‘𝑊) | |
20 | 16, 17, 18, 19 | thlval 21730 | . . . 4 ⊢ (𝑊 ∈ V → 𝐾 = (𝐼 sSet 〈(oc‘ndx), (ocv‘𝑊)〉)) |
21 | 20 | fveq2d 6910 | . . 3 ⊢ (𝑊 ∈ V → (le‘𝐾) = (le‘(𝐼 sSet 〈(oc‘ndx), (ocv‘𝑊)〉))) |
22 | 15, 21 | eqtr4id 2793 | . 2 ⊢ (𝑊 ∈ V → ≤ = (le‘𝐾)) |
23 | 2 | str0 17222 | . . 3 ⊢ ∅ = (le‘∅) |
24 | 17 | fvexi 6920 | . . . . . 6 ⊢ 𝐶 ∈ V |
25 | 18 | ipolerval 18589 | . . . . . 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 2765 | . . . 4 ⊢ ≤ = {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐶 ∧ 𝑥 ⊆ 𝑦)} |
28 | opabn0 5562 | . . . . . 6 ⊢ ({〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐶 ∧ 𝑥 ⊆ 𝑦)} ≠ ∅ ↔ ∃𝑥∃𝑦({𝑥, 𝑦} ⊆ 𝐶 ∧ 𝑥 ⊆ 𝑦)) | |
29 | vex 3481 | . . . . . . . . 9 ⊢ 𝑥 ∈ V | |
30 | vex 3481 | . . . . . . . . 9 ⊢ 𝑦 ∈ V | |
31 | 29, 30 | prss 4824 | . . . . . . . 8 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) ↔ {𝑥, 𝑦} ⊆ 𝐶) |
32 | elfvex 6944 | . . . . . . . . . 10 ⊢ (𝑥 ∈ (ClSubSp‘𝑊) → 𝑊 ∈ V) | |
33 | 32, 17 | eleq2s 2856 | . . . . . . . . 9 ⊢ (𝑥 ∈ 𝐶 → 𝑊 ∈ V) |
34 | 33 | ad2antrr 726 | . . . . . . . 8 ⊢ (((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ⊆ 𝑦) → 𝑊 ∈ V) |
35 | 31, 34 | sylanbr 582 | . . . . . . 7 ⊢ (({𝑥, 𝑦} ⊆ 𝐶 ∧ 𝑥 ⊆ 𝑦) → 𝑊 ∈ V) |
36 | 35 | exlimivv 1929 | . . . . . 6 ⊢ (∃𝑥∃𝑦({𝑥, 𝑦} ⊆ 𝐶 ∧ 𝑥 ⊆ 𝑦) → 𝑊 ∈ V) |
37 | 28, 36 | sylbi 217 | . . . . 5 ⊢ ({〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐶 ∧ 𝑥 ⊆ 𝑦)} ≠ ∅ → 𝑊 ∈ V) |
38 | 37 | necon1bi 2966 | . . . 4 ⊢ (¬ 𝑊 ∈ V → {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐶 ∧ 𝑥 ⊆ 𝑦)} = ∅) |
39 | 27, 38 | eqtrid 2786 | . . 3 ⊢ (¬ 𝑊 ∈ V → ≤ = ∅) |
40 | fvprc 6898 | . . . . 5 ⊢ (¬ 𝑊 ∈ V → (toHL‘𝑊) = ∅) | |
41 | 16, 40 | eqtrid 2786 | . . . 4 ⊢ (¬ 𝑊 ∈ V → 𝐾 = ∅) |
42 | 41 | fveq2d 6910 | . . 3 ⊢ (¬ 𝑊 ∈ V → (le‘𝐾) = (le‘∅)) |
43 | 23, 39, 42 | 3eqtr4a 2800 | . 2 ⊢ (¬ 𝑊 ∈ V → ≤ = (le‘𝐾)) |
44 | 22, 43 | pm2.61i 182 | 1 ⊢ ≤ = (le‘𝐾) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 395 = wceq 1536 ∃wex 1775 ∈ wcel 2105 ≠ wne 2937 Vcvv 3477 ⊆ wss 3962 ∅c0 4338 {cpr 4632 〈cop 4636 {copab 5209 ‘cfv 6562 (class class class)co 7430 0cc0 11152 1c1 11153 ;cdc 12730 sSet csts 17196 ndxcnx 17226 lecple 17304 occoc 17305 toInccipo 18584 ocvcocv 21695 ClSubSpccss 21696 toHLcthl 21697 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-z 12611 df-dec 12731 df-uz 12876 df-fz 13544 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-tset 17316 df-ple 17317 df-ocomp 17318 df-ipo 18585 df-thl 21700 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |