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Mirrors > Home > MPE Home > Th. List > mrelatglb0 | Structured version Visualization version GIF version |
Description: The empty intersection in a Moore space is realized by the base set. (Contributed by Stefan O'Rear, 31-Jan-2015.) |
Ref | Expression |
---|---|
mreclat.i | ⊢ 𝐼 = (toInc‘𝐶) |
mrelatglb.g | ⊢ 𝐺 = (glb‘𝐼) |
Ref | Expression |
---|---|
mrelatglb0 | ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝐺‘∅) = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2651 | . 2 ⊢ (le‘𝐼) = (le‘𝐼) | |
2 | mreclat.i | . . 3 ⊢ 𝐼 = (toInc‘𝐶) | |
3 | 2 | ipobas 17202 | . 2 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐶 = (Base‘𝐼)) |
4 | mrelatglb.g | . . 3 ⊢ 𝐺 = (glb‘𝐼) | |
5 | 4 | a1i 11 | . 2 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐺 = (glb‘𝐼)) |
6 | 2 | ipopos 17207 | . . 3 ⊢ 𝐼 ∈ Poset |
7 | 6 | a1i 11 | . 2 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐼 ∈ Poset) |
8 | 0ss 4005 | . . 3 ⊢ ∅ ⊆ 𝐶 | |
9 | 8 | a1i 11 | . 2 ⊢ (𝐶 ∈ (Moore‘𝑋) → ∅ ⊆ 𝐶) |
10 | mre1cl 16301 | . 2 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝑋 ∈ 𝐶) | |
11 | ral0 4109 | . . . 4 ⊢ ∀𝑥 ∈ ∅ 𝑋(le‘𝐼)𝑥 | |
12 | 11 | rspec 2960 | . . 3 ⊢ (𝑥 ∈ ∅ → 𝑋(le‘𝐼)𝑥) |
13 | 12 | adantl 481 | . 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ∈ ∅) → 𝑋(le‘𝐼)𝑥) |
14 | mress 16300 | . . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑦 ∈ 𝐶) → 𝑦 ⊆ 𝑋) | |
15 | 10 | adantr 480 | . . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑦 ∈ 𝐶) → 𝑋 ∈ 𝐶) |
16 | 2, 1 | ipole 17205 | . . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑦 ∈ 𝐶 ∧ 𝑋 ∈ 𝐶) → (𝑦(le‘𝐼)𝑋 ↔ 𝑦 ⊆ 𝑋)) |
17 | 15, 16 | mpd3an3 1465 | . . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑦 ∈ 𝐶) → (𝑦(le‘𝐼)𝑋 ↔ 𝑦 ⊆ 𝑋)) |
18 | 14, 17 | mpbird 247 | . . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑦 ∈ 𝐶) → 𝑦(le‘𝐼)𝑋) |
19 | 18 | 3adant3 1101 | . 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑦 ∈ 𝐶 ∧ ∀𝑥 ∈ ∅ 𝑦(le‘𝐼)𝑥) → 𝑦(le‘𝐼)𝑋) |
20 | 1, 3, 5, 7, 9, 10, 13, 19 | posglbd 17197 | 1 ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝐺‘∅) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 = wceq 1523 ∈ wcel 2030 ∀wral 2941 ⊆ wss 3607 ∅c0 3948 class class class wbr 4685 ‘cfv 5926 lecple 15995 Moorecmre 16289 Posetcpo 16987 glbcglb 16990 toInccipo 17198 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-oadd 7609 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-5 11120 df-6 11121 df-7 11122 df-8 11123 df-9 11124 df-n0 11331 df-z 11416 df-dec 11532 df-uz 11726 df-fz 12365 df-struct 15906 df-ndx 15907 df-slot 15908 df-base 15910 df-sets 15911 df-tset 16007 df-ple 16008 df-ocomp 16010 df-mre 16293 df-preset 16975 df-poset 16993 df-lub 17021 df-glb 17022 df-odu 17176 df-ipo 17199 |
This theorem is referenced by: mreclatBAD 17234 |
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