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Mirrors > Home > MPE Home > Th. List > mrelatglb | Structured version Visualization version GIF version |
Description: Greatest lower bounds in a Moore space are realized by intersections. (Contributed by Stefan O'Rear, 31-Jan-2015.) |
Ref | Expression |
---|---|
mreclat.i | ⊢ 𝐼 = (toInc‘𝐶) |
mrelatglb.g | ⊢ 𝐺 = (glb‘𝐼) |
Ref | Expression |
---|---|
mrelatglb | ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶 ∧ 𝑈 ≠ ∅) → (𝐺‘𝑈) = ∩ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2821 | . 2 ⊢ (le‘𝐼) = (le‘𝐼) | |
2 | mreclat.i | . . . 4 ⊢ 𝐼 = (toInc‘𝐶) | |
3 | 2 | ipobas 17759 | . . 3 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐶 = (Base‘𝐼)) |
4 | 3 | 3ad2ant1 1129 | . 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶 ∧ 𝑈 ≠ ∅) → 𝐶 = (Base‘𝐼)) |
5 | mrelatglb.g | . . 3 ⊢ 𝐺 = (glb‘𝐼) | |
6 | 5 | a1i 11 | . 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶 ∧ 𝑈 ≠ ∅) → 𝐺 = (glb‘𝐼)) |
7 | 2 | ipopos 17764 | . . 3 ⊢ 𝐼 ∈ Poset |
8 | 7 | a1i 11 | . 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶 ∧ 𝑈 ≠ ∅) → 𝐼 ∈ Poset) |
9 | simp2 1133 | . 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶 ∧ 𝑈 ≠ ∅) → 𝑈 ⊆ 𝐶) | |
10 | mreintcl 16860 | . 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶 ∧ 𝑈 ≠ ∅) → ∩ 𝑈 ∈ 𝐶) | |
11 | intss1 4883 | . . . 4 ⊢ (𝑥 ∈ 𝑈 → ∩ 𝑈 ⊆ 𝑥) | |
12 | 11 | adantl 484 | . . 3 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶 ∧ 𝑈 ≠ ∅) ∧ 𝑥 ∈ 𝑈) → ∩ 𝑈 ⊆ 𝑥) |
13 | simpl1 1187 | . . . 4 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶 ∧ 𝑈 ≠ ∅) ∧ 𝑥 ∈ 𝑈) → 𝐶 ∈ (Moore‘𝑋)) | |
14 | 10 | adantr 483 | . . . 4 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶 ∧ 𝑈 ≠ ∅) ∧ 𝑥 ∈ 𝑈) → ∩ 𝑈 ∈ 𝐶) |
15 | 9 | sselda 3966 | . . . 4 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶 ∧ 𝑈 ≠ ∅) ∧ 𝑥 ∈ 𝑈) → 𝑥 ∈ 𝐶) |
16 | 2, 1 | ipole 17762 | . . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ ∩ 𝑈 ∈ 𝐶 ∧ 𝑥 ∈ 𝐶) → (∩ 𝑈(le‘𝐼)𝑥 ↔ ∩ 𝑈 ⊆ 𝑥)) |
17 | 13, 14, 15, 16 | syl3anc 1367 | . . 3 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶 ∧ 𝑈 ≠ ∅) ∧ 𝑥 ∈ 𝑈) → (∩ 𝑈(le‘𝐼)𝑥 ↔ ∩ 𝑈 ⊆ 𝑥)) |
18 | 12, 17 | mpbird 259 | . 2 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶 ∧ 𝑈 ≠ ∅) ∧ 𝑥 ∈ 𝑈) → ∩ 𝑈(le‘𝐼)𝑥) |
19 | simpll1 1208 | . . . . . . . 8 ⊢ ((((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶 ∧ 𝑈 ≠ ∅) ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ∈ 𝑈) → 𝐶 ∈ (Moore‘𝑋)) | |
20 | simplr 767 | . . . . . . . 8 ⊢ ((((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶 ∧ 𝑈 ≠ ∅) ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ∈ 𝑈) → 𝑦 ∈ 𝐶) | |
21 | simpl2 1188 | . . . . . . . . 9 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶 ∧ 𝑈 ≠ ∅) ∧ 𝑦 ∈ 𝐶) → 𝑈 ⊆ 𝐶) | |
22 | 21 | sselda 3966 | . . . . . . . 8 ⊢ ((((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶 ∧ 𝑈 ≠ ∅) ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ∈ 𝑈) → 𝑥 ∈ 𝐶) |
23 | 2, 1 | ipole 17762 | . . . . . . . 8 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑦 ∈ 𝐶 ∧ 𝑥 ∈ 𝐶) → (𝑦(le‘𝐼)𝑥 ↔ 𝑦 ⊆ 𝑥)) |
24 | 19, 20, 22, 23 | syl3anc 1367 | . . . . . . 7 ⊢ ((((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶 ∧ 𝑈 ≠ ∅) ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ∈ 𝑈) → (𝑦(le‘𝐼)𝑥 ↔ 𝑦 ⊆ 𝑥)) |
25 | 24 | biimpd 231 | . . . . . 6 ⊢ ((((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶 ∧ 𝑈 ≠ ∅) ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ∈ 𝑈) → (𝑦(le‘𝐼)𝑥 → 𝑦 ⊆ 𝑥)) |
26 | 25 | ralimdva 3177 | . . . . 5 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶 ∧ 𝑈 ≠ ∅) ∧ 𝑦 ∈ 𝐶) → (∀𝑥 ∈ 𝑈 𝑦(le‘𝐼)𝑥 → ∀𝑥 ∈ 𝑈 𝑦 ⊆ 𝑥)) |
27 | 26 | 3impia 1113 | . . . 4 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶 ∧ 𝑈 ≠ ∅) ∧ 𝑦 ∈ 𝐶 ∧ ∀𝑥 ∈ 𝑈 𝑦(le‘𝐼)𝑥) → ∀𝑥 ∈ 𝑈 𝑦 ⊆ 𝑥) |
28 | ssint 4884 | . . . 4 ⊢ (𝑦 ⊆ ∩ 𝑈 ↔ ∀𝑥 ∈ 𝑈 𝑦 ⊆ 𝑥) | |
29 | 27, 28 | sylibr 236 | . . 3 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶 ∧ 𝑈 ≠ ∅) ∧ 𝑦 ∈ 𝐶 ∧ ∀𝑥 ∈ 𝑈 𝑦(le‘𝐼)𝑥) → 𝑦 ⊆ ∩ 𝑈) |
30 | simp11 1199 | . . . 4 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶 ∧ 𝑈 ≠ ∅) ∧ 𝑦 ∈ 𝐶 ∧ ∀𝑥 ∈ 𝑈 𝑦(le‘𝐼)𝑥) → 𝐶 ∈ (Moore‘𝑋)) | |
31 | simp2 1133 | . . . 4 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶 ∧ 𝑈 ≠ ∅) ∧ 𝑦 ∈ 𝐶 ∧ ∀𝑥 ∈ 𝑈 𝑦(le‘𝐼)𝑥) → 𝑦 ∈ 𝐶) | |
32 | 10 | 3ad2ant1 1129 | . . . 4 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶 ∧ 𝑈 ≠ ∅) ∧ 𝑦 ∈ 𝐶 ∧ ∀𝑥 ∈ 𝑈 𝑦(le‘𝐼)𝑥) → ∩ 𝑈 ∈ 𝐶) |
33 | 2, 1 | ipole 17762 | . . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑦 ∈ 𝐶 ∧ ∩ 𝑈 ∈ 𝐶) → (𝑦(le‘𝐼)∩ 𝑈 ↔ 𝑦 ⊆ ∩ 𝑈)) |
34 | 30, 31, 32, 33 | syl3anc 1367 | . . 3 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶 ∧ 𝑈 ≠ ∅) ∧ 𝑦 ∈ 𝐶 ∧ ∀𝑥 ∈ 𝑈 𝑦(le‘𝐼)𝑥) → (𝑦(le‘𝐼)∩ 𝑈 ↔ 𝑦 ⊆ ∩ 𝑈)) |
35 | 29, 34 | mpbird 259 | . 2 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶 ∧ 𝑈 ≠ ∅) ∧ 𝑦 ∈ 𝐶 ∧ ∀𝑥 ∈ 𝑈 𝑦(le‘𝐼)𝑥) → 𝑦(le‘𝐼)∩ 𝑈) |
36 | 1, 4, 6, 8, 9, 10, 18, 35 | posglbd 17754 | 1 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶 ∧ 𝑈 ≠ ∅) → (𝐺‘𝑈) = ∩ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∀wral 3138 ⊆ wss 3935 ∅c0 4290 ∩ cint 4868 class class class wbr 5058 ‘cfv 6349 Basecbs 16477 lecple 16566 Moorecmre 16847 Posetcpo 17544 glbcglb 17547 toInccipo 17755 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-fz 12887 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-tset 16578 df-ple 16579 df-ocomp 16580 df-mre 16851 df-proset 17532 df-poset 17550 df-lub 17578 df-glb 17579 df-odu 17733 df-ipo 17756 |
This theorem is referenced by: mreclatBAD 17791 |
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