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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 20400 | . . . 4 ⊢ (𝑊 ∈ PreHil → 𝐶 ∈ (Moore‘𝑉)) |
4 | 3 | adantr 474 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) → 𝐶 ∈ (Moore‘𝑉)) |
5 | mrccss.o | . . . 4 ⊢ ⊥ = (ocv‘𝑊) | |
6 | 1, 5 | ocvocv 20378 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) → 𝑆 ⊆ ( ⊥ ‘( ⊥ ‘𝑆))) |
7 | 1, 5 | ocvss 20377 | . . . . 5 ⊢ ( ⊥ ‘𝑆) ⊆ 𝑉 |
8 | 7 | a1i 11 | . . . 4 ⊢ (𝑆 ⊆ 𝑉 → ( ⊥ ‘𝑆) ⊆ 𝑉) |
9 | 1, 2, 5 | ocvcss 20394 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ ( ⊥ ‘𝑆) ⊆ 𝑉) → ( ⊥ ‘( ⊥ ‘𝑆)) ∈ 𝐶) |
10 | 8, 9 | sylan2 588 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) → ( ⊥ ‘( ⊥ ‘𝑆)) ∈ 𝐶) |
11 | mrccss.f | . . . 4 ⊢ 𝐹 = (mrCls‘𝐶) | |
12 | 11 | mrcsscl 16633 | . . 3 ⊢ ((𝐶 ∈ (Moore‘𝑉) ∧ 𝑆 ⊆ ( ⊥ ‘( ⊥ ‘𝑆)) ∧ ( ⊥ ‘( ⊥ ‘𝑆)) ∈ 𝐶) → (𝐹‘𝑆) ⊆ ( ⊥ ‘( ⊥ ‘𝑆))) |
13 | 4, 6, 10, 12 | syl3anc 1496 | . 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) → (𝐹‘𝑆) ⊆ ( ⊥ ‘( ⊥ ‘𝑆))) |
14 | 11 | mrcssid 16630 | . . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑉) ∧ 𝑆 ⊆ 𝑉) → 𝑆 ⊆ (𝐹‘𝑆)) |
15 | 3, 14 | sylan 577 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) → 𝑆 ⊆ (𝐹‘𝑆)) |
16 | 5 | ocv2ss 20380 | . . . 4 ⊢ (𝑆 ⊆ (𝐹‘𝑆) → ( ⊥ ‘(𝐹‘𝑆)) ⊆ ( ⊥ ‘𝑆)) |
17 | 5 | ocv2ss 20380 | . . . 4 ⊢ (( ⊥ ‘(𝐹‘𝑆)) ⊆ ( ⊥ ‘𝑆) → ( ⊥ ‘( ⊥ ‘𝑆)) ⊆ ( ⊥ ‘( ⊥ ‘(𝐹‘𝑆)))) |
18 | 15, 16, 17 | 3syl 18 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) → ( ⊥ ‘( ⊥ ‘𝑆)) ⊆ ( ⊥ ‘( ⊥ ‘(𝐹‘𝑆)))) |
19 | 11 | mrccl 16624 | . . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑉) ∧ 𝑆 ⊆ 𝑉) → (𝐹‘𝑆) ∈ 𝐶) |
20 | 3, 19 | sylan 577 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) → (𝐹‘𝑆) ∈ 𝐶) |
21 | 5, 2 | cssi 20391 | . . . 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 | sseqtr4d 3867 | . 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) → ( ⊥ ‘( ⊥ ‘𝑆)) ⊆ (𝐹‘𝑆)) |
24 | 13, 23 | eqssd 3844 | 1 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) → (𝐹‘𝑆) = ( ⊥ ‘( ⊥ ‘𝑆))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1658 ∈ wcel 2166 ⊆ wss 3798 ‘cfv 6123 Basecbs 16222 Moorecmre 16595 mrClscmrc 16596 PreHilcphl 20331 ocvcocv 20367 ClSubSpccss 20368 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-tpos 7617 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-er 8009 df-map 8124 df-en 8223 df-dom 8224 df-sdom 8225 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-nn 11351 df-2 11414 df-3 11415 df-4 11416 df-5 11417 df-6 11418 df-7 11419 df-8 11420 df-ndx 16225 df-slot 16226 df-base 16228 df-sets 16229 df-plusg 16318 df-mulr 16319 df-sca 16321 df-vsca 16322 df-ip 16323 df-0g 16455 df-mre 16599 df-mrc 16600 df-mgm 17595 df-sgrp 17637 df-mnd 17648 df-mhm 17688 df-grp 17779 df-ghm 18009 df-mgp 18844 df-ur 18856 df-ring 18903 df-oppr 18977 df-rnghom 19071 df-staf 19201 df-srng 19202 df-lmod 19221 df-lmhm 19381 df-lvec 19462 df-sra 19533 df-rgmod 19534 df-phl 20333 df-ocv 20370 df-css 20371 |
This theorem is referenced by: (None) |
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