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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 | β’ πΎ = (toHLβπ) |
thlbas.c | β’ πΆ = (ClSubSpβπ) |
thlle.i | β’ πΌ = (toIncβπΆ) |
thlle.l | β’ β€ = (leβπΌ) |
Ref | Expression |
---|---|
thlle | β’ β€ = (leβπΎ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | thlle.l | . . . 4 β’ β€ = (leβπΌ) | |
2 | pleid 17308 | . . . . 5 β’ le = Slot (leβndx) | |
3 | plendxnocndx 17325 | . . . . 5 β’ (leβndx) β (ocβndx) | |
4 | 2, 3 | setsnid 17138 | . . . 4 β’ (leβπΌ) = (leβ(πΌ sSet β¨(ocβndx), (ocvβπ)β©)) |
5 | 1, 4 | eqtri 2760 | . . 3 β’ β€ = (leβ(πΌ sSet β¨(ocβndx), (ocvβπ)β©)) |
6 | thlval.k | . . . . 5 β’ πΎ = (toHLβπ) | |
7 | thlbas.c | . . . . 5 β’ πΆ = (ClSubSpβπ) | |
8 | thlle.i | . . . . 5 β’ πΌ = (toIncβπΆ) | |
9 | eqid 2732 | . . . . 5 β’ (ocvβπ) = (ocvβπ) | |
10 | 6, 7, 8, 9 | thlval 21239 | . . . 4 β’ (π β V β πΎ = (πΌ sSet β¨(ocβndx), (ocvβπ)β©)) |
11 | 10 | fveq2d 6892 | . . 3 β’ (π β V β (leβπΎ) = (leβ(πΌ sSet β¨(ocβndx), (ocvβπ)β©))) |
12 | 5, 11 | eqtr4id 2791 | . 2 β’ (π β V β β€ = (leβπΎ)) |
13 | 2 | str0 17118 | . . 3 β’ β = (leββ ) |
14 | 7 | fvexi 6902 | . . . . . 6 β’ πΆ β V |
15 | 8 | ipolerval 18481 | . . . . . 6 β’ (πΆ β V β {β¨π₯, π¦β© β£ ({π₯, π¦} β πΆ β§ π₯ β π¦)} = (leβπΌ)) |
16 | 14, 15 | ax-mp 5 | . . . . 5 β’ {β¨π₯, π¦β© β£ ({π₯, π¦} β πΆ β§ π₯ β π¦)} = (leβπΌ) |
17 | 1, 16 | eqtr4i 2763 | . . . 4 β’ β€ = {β¨π₯, π¦β© β£ ({π₯, π¦} β πΆ β§ π₯ β π¦)} |
18 | opabn0 5552 | . . . . . 6 β’ ({β¨π₯, π¦β© β£ ({π₯, π¦} β πΆ β§ π₯ β π¦)} β β β βπ₯βπ¦({π₯, π¦} β πΆ β§ π₯ β π¦)) | |
19 | vex 3478 | . . . . . . . . 9 β’ π₯ β V | |
20 | vex 3478 | . . . . . . . . 9 β’ π¦ β V | |
21 | 19, 20 | prss 4822 | . . . . . . . 8 β’ ((π₯ β πΆ β§ π¦ β πΆ) β {π₯, π¦} β πΆ) |
22 | elfvex 6926 | . . . . . . . . . 10 β’ (π₯ β (ClSubSpβπ) β π β V) | |
23 | 22, 7 | eleq2s 2851 | . . . . . . . . 9 β’ (π₯ β πΆ β π β V) |
24 | 23 | ad2antrr 724 | . . . . . . . 8 β’ (((π₯ β πΆ β§ π¦ β πΆ) β§ π₯ β π¦) β π β V) |
25 | 21, 24 | sylanbr 582 | . . . . . . 7 β’ (({π₯, π¦} β πΆ β§ π₯ β π¦) β π β V) |
26 | 25 | exlimivv 1935 | . . . . . 6 β’ (βπ₯βπ¦({π₯, π¦} β πΆ β§ π₯ β π¦) β π β V) |
27 | 18, 26 | sylbi 216 | . . . . 5 β’ ({β¨π₯, π¦β© β£ ({π₯, π¦} β πΆ β§ π₯ β π¦)} β β β π β V) |
28 | 27 | necon1bi 2969 | . . . 4 β’ (Β¬ π β V β {β¨π₯, π¦β© β£ ({π₯, π¦} β πΆ β§ π₯ β π¦)} = β ) |
29 | 17, 28 | eqtrid 2784 | . . 3 β’ (Β¬ π β V β β€ = β ) |
30 | fvprc 6880 | . . . . 5 β’ (Β¬ π β V β (toHLβπ) = β ) | |
31 | 6, 30 | eqtrid 2784 | . . . 4 β’ (Β¬ π β V β πΎ = β ) |
32 | 31 | fveq2d 6892 | . . 3 β’ (Β¬ π β V β (leβπΎ) = (leββ )) |
33 | 13, 29, 32 | 3eqtr4a 2798 | . 2 β’ (Β¬ π β V β β€ = (leβπΎ)) |
34 | 12, 33 | pm2.61i 182 | 1 β’ β€ = (leβπΎ) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β§ wa 396 = wceq 1541 βwex 1781 β wcel 2106 β wne 2940 Vcvv 3474 β wss 3947 β c0 4321 {cpr 4629 β¨cop 4633 {copab 5209 βcfv 6540 (class class class)co 7405 sSet csts 17092 ndxcnx 17122 lecple 17200 occoc 17201 toInccipo 18476 ocvcocv 21204 ClSubSpccss 21205 toHLcthl 21206 |
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 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-fz 13481 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-tset 17212 df-ple 17213 df-ocomp 17214 df-ipo 18477 df-thl 21209 |
This theorem is referenced by: thlleval 21244 |
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