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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 17317 | . . . . 5 β’ le = Slot (leβndx) | |
3 | plendxnocndx 17334 | . . . . 5 β’ (leβndx) β (ocβndx) | |
4 | 2, 3 | setsnid 17147 | . . . 4 β’ (leβπΌ) = (leβ(πΌ sSet β¨(ocβndx), (ocvβπ)β©)) |
5 | 1, 4 | eqtri 2752 | . . 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 2724 | . . . . 5 β’ (ocvβπ) = (ocvβπ) | |
10 | 6, 7, 8, 9 | thlval 21577 | . . . 4 β’ (π β V β πΎ = (πΌ sSet β¨(ocβndx), (ocvβπ)β©)) |
11 | 10 | fveq2d 6886 | . . 3 β’ (π β V β (leβπΎ) = (leβ(πΌ sSet β¨(ocβndx), (ocvβπ)β©))) |
12 | 5, 11 | eqtr4id 2783 | . 2 β’ (π β V β β€ = (leβπΎ)) |
13 | 2 | str0 17127 | . . 3 β’ β = (leββ ) |
14 | 7 | fvexi 6896 | . . . . . 6 β’ πΆ β V |
15 | 8 | ipolerval 18493 | . . . . . 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 2755 | . . . 4 β’ β€ = {β¨π₯, π¦β© β£ ({π₯, π¦} β πΆ β§ π₯ β π¦)} |
18 | opabn0 5544 | . . . . . 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 3470 | . . . . . . . . 9 β’ π₯ β V | |
20 | vex 3470 | . . . . . . . . 9 β’ π¦ β V | |
21 | 19, 20 | prss 4816 | . . . . . . . 8 β’ ((π₯ β πΆ β§ π¦ β πΆ) β {π₯, π¦} β πΆ) |
22 | elfvex 6920 | . . . . . . . . . 10 β’ (π₯ β (ClSubSpβπ) β π β V) | |
23 | 22, 7 | eleq2s 2843 | . . . . . . . . 9 β’ (π₯ β πΆ β π β V) |
24 | 23 | ad2antrr 723 | . . . . . . . 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 1927 | . . . . . 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 2961 | . . . 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 2776 | . . 3 β’ (Β¬ π β V β β€ = β ) |
30 | fvprc 6874 | . . . . 5 β’ (Β¬ π β V β (toHLβπ) = β ) | |
31 | 6, 30 | eqtrid 2776 | . . . 4 β’ (Β¬ π β V β πΎ = β ) |
32 | 31 | fveq2d 6886 | . . 3 β’ (Β¬ π β V β (leβπΎ) = (leββ )) |
33 | 13, 29, 32 | 3eqtr4a 2790 | . 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 1533 βwex 1773 β wcel 2098 β wne 2932 Vcvv 3466 β wss 3941 β c0 4315 {cpr 4623 β¨cop 4627 {copab 5201 βcfv 6534 (class class class)co 7402 sSet csts 17101 ndxcnx 17131 lecple 17209 occoc 17210 toInccipo 18488 ocvcocv 21542 ClSubSpccss 21543 toHLcthl 21544 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-z 12558 df-dec 12677 df-uz 12822 df-fz 13486 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-tset 17221 df-ple 17222 df-ocomp 17223 df-ipo 18489 df-thl 21547 |
This theorem is referenced by: thlleval 21582 |
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