![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 17347 | . . . . 5 β’ le = Slot (leβndx) | |
3 | plendxnocndx 17364 | . . . . 5 β’ (leβndx) β (ocβndx) | |
4 | 2, 3 | setsnid 17177 | . . . 4 β’ (leβπΌ) = (leβ(πΌ sSet β¨(ocβndx), (ocvβπ)β©)) |
5 | 1, 4 | eqtri 2756 | . . 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 2728 | . . . . 5 β’ (ocvβπ) = (ocvβπ) | |
10 | 6, 7, 8, 9 | thlval 21626 | . . . 4 β’ (π β V β πΎ = (πΌ sSet β¨(ocβndx), (ocvβπ)β©)) |
11 | 10 | fveq2d 6901 | . . 3 β’ (π β V β (leβπΎ) = (leβ(πΌ sSet β¨(ocβndx), (ocvβπ)β©))) |
12 | 5, 11 | eqtr4id 2787 | . 2 β’ (π β V β β€ = (leβπΎ)) |
13 | 2 | str0 17157 | . . 3 β’ β = (leββ ) |
14 | 7 | fvexi 6911 | . . . . . 6 β’ πΆ β V |
15 | 8 | ipolerval 18523 | . . . . . 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 2759 | . . . 4 β’ β€ = {β¨π₯, π¦β© β£ ({π₯, π¦} β πΆ β§ π₯ β π¦)} |
18 | opabn0 5555 | . . . . . 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 3475 | . . . . . . . . 9 β’ π₯ β V | |
20 | vex 3475 | . . . . . . . . 9 β’ π¦ β V | |
21 | 19, 20 | prss 4824 | . . . . . . . 8 β’ ((π₯ β πΆ β§ π¦ β πΆ) β {π₯, π¦} β πΆ) |
22 | elfvex 6935 | . . . . . . . . . 10 β’ (π₯ β (ClSubSpβπ) β π β V) | |
23 | 22, 7 | eleq2s 2847 | . . . . . . . . 9 β’ (π₯ β πΆ β π β V) |
24 | 23 | ad2antrr 725 | . . . . . . . 8 β’ (((π₯ β πΆ β§ π¦ β πΆ) β§ π₯ β π¦) β π β V) |
25 | 21, 24 | sylanbr 581 | . . . . . . 7 β’ (({π₯, π¦} β πΆ β§ π₯ β π¦) β π β V) |
26 | 25 | exlimivv 1928 | . . . . . 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 2966 | . . . 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 2780 | . . 3 β’ (Β¬ π β V β β€ = β ) |
30 | fvprc 6889 | . . . . 5 β’ (Β¬ π β V β (toHLβπ) = β ) | |
31 | 6, 30 | eqtrid 2780 | . . . 4 β’ (Β¬ π β V β πΎ = β ) |
32 | 31 | fveq2d 6901 | . . 3 β’ (Β¬ π β V β (leβπΎ) = (leββ )) |
33 | 13, 29, 32 | 3eqtr4a 2794 | . 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 395 = wceq 1534 βwex 1774 β wcel 2099 β wne 2937 Vcvv 3471 β wss 3947 β c0 4323 {cpr 4631 β¨cop 4635 {copab 5210 βcfv 6548 (class class class)co 7420 sSet csts 17131 ndxcnx 17161 lecple 17239 occoc 17240 toInccipo 18518 ocvcocv 21591 ClSubSpccss 21592 toHLcthl 21593 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-er 8724 df-en 8964 df-dom 8965 df-sdom 8966 df-fin 8967 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-dec 12708 df-uz 12853 df-fz 13517 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-tset 17251 df-ple 17252 df-ocomp 17253 df-ipo 18519 df-thl 21596 |
This theorem is referenced by: thlleval 21631 |
Copyright terms: Public domain | W3C validator |