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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 2610 | . 2 ⊢ (le‘𝐼) = (le‘𝐼) | |
2 | mreclat.i | . . 3 ⊢ 𝐼 = (toInc‘𝐶) | |
3 | 2 | ipobas 16978 | . 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 16983 | . . 3 ⊢ 𝐼 ∈ Poset |
7 | 6 | a1i 11 | . 2 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐼 ∈ Poset) |
8 | 0ss 3924 | . . 3 ⊢ ∅ ⊆ 𝐶 | |
9 | 8 | a1i 11 | . 2 ⊢ (𝐶 ∈ (Moore‘𝑋) → ∅ ⊆ 𝐶) |
10 | mre1cl 16077 | . 2 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝑋 ∈ 𝐶) | |
11 | ral0 4028 | . . . 4 ⊢ ∀𝑥 ∈ ∅ 𝑋(le‘𝐼)𝑥 | |
12 | 11 | rspec 2915 | . . 3 ⊢ (𝑥 ∈ ∅ → 𝑋(le‘𝐼)𝑥) |
13 | 12 | adantl 481 | . 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ∈ ∅) → 𝑋(le‘𝐼)𝑥) |
14 | mress 16076 | . . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑦 ∈ 𝐶) → 𝑦 ⊆ 𝑋) | |
15 | 10 | adantr 480 | . . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑦 ∈ 𝐶) → 𝑋 ∈ 𝐶) |
16 | 2, 1 | ipole 16981 | . . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑦 ∈ 𝐶 ∧ 𝑋 ∈ 𝐶) → (𝑦(le‘𝐼)𝑋 ↔ 𝑦 ⊆ 𝑋)) |
17 | 15, 16 | mpd3an3 1417 | . . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑦 ∈ 𝐶) → (𝑦(le‘𝐼)𝑋 ↔ 𝑦 ⊆ 𝑋)) |
18 | 14, 17 | mpbird 246 | . . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑦 ∈ 𝐶) → 𝑦(le‘𝐼)𝑋) |
19 | 18 | 3adant3 1074 | . 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 16973 | 1 ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝐺‘∅) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ⊆ wss 3540 ∅c0 3874 class class class wbr 4583 ‘cfv 5804 lecple 15775 Moorecmre 16065 Posetcpo 16763 glbcglb 16766 toInccipo 16974 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-fz 12198 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-tset 15787 df-ple 15788 df-ocomp 15790 df-mre 16069 df-preset 16751 df-poset 16769 df-lub 16797 df-glb 16798 df-odu 16952 df-ipo 16975 |
This theorem is referenced by: mreclatBAD 17010 |
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