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Mirrors > Home > MPE Home > Th. List > mrccss | Structured version Visualization version GIF version |
Description: The Moore closure corresponding to the system of closed subspaces is the double orthocomplement operation. (Contributed by Mario Carneiro, 13-Oct-2015.) |
Ref | Expression |
---|---|
mrccss.v | ⊢ 𝑉 = (Base‘𝑊) |
mrccss.o | ⊢ ⊥ = (ocv‘𝑊) |
mrccss.c | ⊢ 𝐶 = (ClSubSp‘𝑊) |
mrccss.f | ⊢ 𝐹 = (mrCls‘𝐶) |
Ref | Expression |
---|---|
mrccss | ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) → (𝐹‘𝑆) = ( ⊥ ‘( ⊥ ‘𝑆))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mrccss.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑊) | |
2 | mrccss.c | . . . . 5 ⊢ 𝐶 = (ClSubSp‘𝑊) | |
3 | 1, 2 | cssmre 20829 | . . . 4 ⊢ (𝑊 ∈ PreHil → 𝐶 ∈ (Moore‘𝑉)) |
4 | 3 | adantr 483 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) → 𝐶 ∈ (Moore‘𝑉)) |
5 | mrccss.o | . . . 4 ⊢ ⊥ = (ocv‘𝑊) | |
6 | 1, 5 | ocvocv 20807 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) → 𝑆 ⊆ ( ⊥ ‘( ⊥ ‘𝑆))) |
7 | 1, 5 | ocvss 20806 | . . . . 5 ⊢ ( ⊥ ‘𝑆) ⊆ 𝑉 |
8 | 7 | a1i 11 | . . . 4 ⊢ (𝑆 ⊆ 𝑉 → ( ⊥ ‘𝑆) ⊆ 𝑉) |
9 | 1, 2, 5 | ocvcss 20823 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ ( ⊥ ‘𝑆) ⊆ 𝑉) → ( ⊥ ‘( ⊥ ‘𝑆)) ∈ 𝐶) |
10 | 8, 9 | sylan2 594 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) → ( ⊥ ‘( ⊥ ‘𝑆)) ∈ 𝐶) |
11 | mrccss.f | . . . 4 ⊢ 𝐹 = (mrCls‘𝐶) | |
12 | 11 | mrcsscl 16883 | . . 3 ⊢ ((𝐶 ∈ (Moore‘𝑉) ∧ 𝑆 ⊆ ( ⊥ ‘( ⊥ ‘𝑆)) ∧ ( ⊥ ‘( ⊥ ‘𝑆)) ∈ 𝐶) → (𝐹‘𝑆) ⊆ ( ⊥ ‘( ⊥ ‘𝑆))) |
13 | 4, 6, 10, 12 | syl3anc 1366 | . 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) → (𝐹‘𝑆) ⊆ ( ⊥ ‘( ⊥ ‘𝑆))) |
14 | 11 | mrcssid 16880 | . . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑉) ∧ 𝑆 ⊆ 𝑉) → 𝑆 ⊆ (𝐹‘𝑆)) |
15 | 3, 14 | sylan 582 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) → 𝑆 ⊆ (𝐹‘𝑆)) |
16 | 5 | ocv2ss 20809 | . . . 4 ⊢ (𝑆 ⊆ (𝐹‘𝑆) → ( ⊥ ‘(𝐹‘𝑆)) ⊆ ( ⊥ ‘𝑆)) |
17 | 5 | ocv2ss 20809 | . . . 4 ⊢ (( ⊥ ‘(𝐹‘𝑆)) ⊆ ( ⊥ ‘𝑆) → ( ⊥ ‘( ⊥ ‘𝑆)) ⊆ ( ⊥ ‘( ⊥ ‘(𝐹‘𝑆)))) |
18 | 15, 16, 17 | 3syl 18 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) → ( ⊥ ‘( ⊥ ‘𝑆)) ⊆ ( ⊥ ‘( ⊥ ‘(𝐹‘𝑆)))) |
19 | 11 | mrccl 16874 | . . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑉) ∧ 𝑆 ⊆ 𝑉) → (𝐹‘𝑆) ∈ 𝐶) |
20 | 3, 19 | sylan 582 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) → (𝐹‘𝑆) ∈ 𝐶) |
21 | 5, 2 | cssi 20820 | . . . 4 ⊢ ((𝐹‘𝑆) ∈ 𝐶 → (𝐹‘𝑆) = ( ⊥ ‘( ⊥ ‘(𝐹‘𝑆)))) |
22 | 20, 21 | syl 17 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) → (𝐹‘𝑆) = ( ⊥ ‘( ⊥ ‘(𝐹‘𝑆)))) |
23 | 18, 22 | sseqtrrd 4006 | . 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) → ( ⊥ ‘( ⊥ ‘𝑆)) ⊆ (𝐹‘𝑆)) |
24 | 13, 23 | eqssd 3982 | 1 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) → (𝐹‘𝑆) = ( ⊥ ‘( ⊥ ‘𝑆))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1531 ∈ wcel 2108 ⊆ wss 3934 ‘cfv 6348 Basecbs 16475 Moorecmre 16845 mrClscmrc 16846 PreHilcphl 20760 ocvcocv 20796 ClSubSpccss 20797 |
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 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-om 7573 df-tpos 7884 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-er 8281 df-map 8400 df-en 8502 df-dom 8503 df-sdom 8504 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-nn 11631 df-2 11692 df-3 11693 df-4 11694 df-5 11695 df-6 11696 df-7 11697 df-8 11698 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-plusg 16570 df-mulr 16571 df-sca 16573 df-vsca 16574 df-ip 16575 df-0g 16707 df-mre 16849 df-mrc 16850 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-mhm 17948 df-grp 18098 df-ghm 18348 df-mgp 19232 df-ur 19244 df-ring 19291 df-oppr 19365 df-rnghom 19459 df-staf 19608 df-srng 19609 df-lmod 19628 df-lmhm 19786 df-lvec 19867 df-sra 19936 df-rgmod 19937 df-phl 20762 df-ocv 20799 df-css 20800 |
This theorem is referenced by: (None) |
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