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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 2739 | . 2 ⊢ (le‘𝐼) = (le‘𝐼) | |
2 | mreclat.i | . . 3 ⊢ 𝐼 = (toInc‘𝐶) | |
3 | 2 | ipobas 18258 | . 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 18263 | . . 3 ⊢ 𝐼 ∈ Poset |
7 | 6 | a1i 11 | . 2 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐼 ∈ Poset) |
8 | 0ss 4331 | . . 3 ⊢ ∅ ⊆ 𝐶 | |
9 | 8 | a1i 11 | . 2 ⊢ (𝐶 ∈ (Moore‘𝑋) → ∅ ⊆ 𝐶) |
10 | mre1cl 17312 | . 2 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝑋 ∈ 𝐶) | |
11 | ral0 4444 | . . . 4 ⊢ ∀𝑥 ∈ ∅ 𝑋(le‘𝐼)𝑥 | |
12 | 11 | rspec 3134 | . . 3 ⊢ (𝑥 ∈ ∅ → 𝑋(le‘𝐼)𝑥) |
13 | 12 | adantl 482 | . 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ∈ ∅) → 𝑋(le‘𝐼)𝑥) |
14 | mress 17311 | . . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑦 ∈ 𝐶) → 𝑦 ⊆ 𝑋) | |
15 | 10 | adantr 481 | . . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑦 ∈ 𝐶) → 𝑋 ∈ 𝐶) |
16 | 2, 1 | ipole 18261 | . . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑦 ∈ 𝐶 ∧ 𝑋 ∈ 𝐶) → (𝑦(le‘𝐼)𝑋 ↔ 𝑦 ⊆ 𝑋)) |
17 | 15, 16 | mpd3an3 1461 | . . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑦 ∈ 𝐶) → (𝑦(le‘𝐼)𝑋 ↔ 𝑦 ⊆ 𝑋)) |
18 | 14, 17 | mpbird 256 | . . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑦 ∈ 𝐶) → 𝑦(le‘𝐼)𝑋) |
19 | 18 | 3adant3 1131 | . 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑦 ∈ 𝐶 ∧ ∀𝑥 ∈ ∅ 𝑦(le‘𝐼)𝑥) → 𝑦(le‘𝐼)𝑋) |
20 | 1, 3, 5, 7, 9, 10, 13, 19 | posglbdg 18142 | 1 ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝐺‘∅) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1539 ∈ wcel 2107 ∀wral 3065 ⊆ wss 3888 ∅c0 4257 class class class wbr 5075 ‘cfv 6437 lecple 16978 Moorecmre 17300 Posetcpo 18034 glbcglb 18037 toInccipo 18254 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2710 ax-rep 5210 ax-sep 5224 ax-nul 5231 ax-pow 5289 ax-pr 5353 ax-un 7597 ax-cnex 10936 ax-resscn 10937 ax-1cn 10938 ax-icn 10939 ax-addcl 10940 ax-addrcl 10941 ax-mulcl 10942 ax-mulrcl 10943 ax-mulcom 10944 ax-addass 10945 ax-mulass 10946 ax-distr 10947 ax-i2m1 10948 ax-1ne0 10949 ax-1rid 10950 ax-rnegex 10951 ax-rrecex 10952 ax-cnre 10953 ax-pre-lttri 10954 ax-pre-lttrn 10955 ax-pre-ltadd 10956 ax-pre-mulgt0 10957 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-rmo 3072 df-reu 3073 df-rab 3074 df-v 3435 df-sbc 3718 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-pss 3907 df-nul 4258 df-if 4461 df-pw 4536 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4841 df-iun 4927 df-br 5076 df-opab 5138 df-mpt 5159 df-tr 5193 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6206 df-ord 6273 df-on 6274 df-lim 6275 df-suc 6276 df-iota 6395 df-fun 6439 df-fn 6440 df-f 6441 df-f1 6442 df-fo 6443 df-f1o 6444 df-fv 6445 df-riota 7241 df-ov 7287 df-oprab 7288 df-mpo 7289 df-om 7722 df-1st 7840 df-2nd 7841 df-frecs 8106 df-wrecs 8137 df-recs 8211 df-rdg 8250 df-1o 8306 df-er 8507 df-en 8743 df-dom 8744 df-sdom 8745 df-fin 8746 df-pnf 11020 df-mnf 11021 df-xr 11022 df-ltxr 11023 df-le 11024 df-sub 11216 df-neg 11217 df-nn 11983 df-2 12045 df-3 12046 df-4 12047 df-5 12048 df-6 12049 df-7 12050 df-8 12051 df-9 12052 df-n0 12243 df-z 12329 df-dec 12447 df-uz 12592 df-fz 13249 df-struct 16857 df-sets 16874 df-slot 16892 df-ndx 16904 df-base 16922 df-tset 16990 df-ple 16991 df-ocomp 16992 df-mre 17304 df-odu 18014 df-proset 18022 df-poset 18040 df-lub 18073 df-glb 18074 df-ipo 18255 |
This theorem is referenced by: mreclatBAD 18290 |
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