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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 2771 | . 2 ⊢ (le‘𝐼) = (le‘𝐼) | |
2 | mreclat.i | . . 3 ⊢ 𝐼 = (toInc‘𝐶) | |
3 | 2 | ipobas 17635 | . 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 17640 | . . 3 ⊢ 𝐼 ∈ Poset |
7 | 6 | a1i 11 | . 2 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐼 ∈ Poset) |
8 | 0ss 4230 | . . 3 ⊢ ∅ ⊆ 𝐶 | |
9 | 8 | a1i 11 | . 2 ⊢ (𝐶 ∈ (Moore‘𝑋) → ∅ ⊆ 𝐶) |
10 | mre1cl 16735 | . 2 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝑋 ∈ 𝐶) | |
11 | ral0 4333 | . . . 4 ⊢ ∀𝑥 ∈ ∅ 𝑋(le‘𝐼)𝑥 | |
12 | 11 | rspec 3150 | . . 3 ⊢ (𝑥 ∈ ∅ → 𝑋(le‘𝐼)𝑥) |
13 | 12 | adantl 474 | . 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ∈ ∅) → 𝑋(le‘𝐼)𝑥) |
14 | mress 16734 | . . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑦 ∈ 𝐶) → 𝑦 ⊆ 𝑋) | |
15 | 10 | adantr 473 | . . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑦 ∈ 𝐶) → 𝑋 ∈ 𝐶) |
16 | 2, 1 | ipole 17638 | . . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑦 ∈ 𝐶 ∧ 𝑋 ∈ 𝐶) → (𝑦(le‘𝐼)𝑋 ↔ 𝑦 ⊆ 𝑋)) |
17 | 15, 16 | mpd3an3 1442 | . . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑦 ∈ 𝐶) → (𝑦(le‘𝐼)𝑋 ↔ 𝑦 ⊆ 𝑋)) |
18 | 14, 17 | mpbird 249 | . . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑦 ∈ 𝐶) → 𝑦(le‘𝐼)𝑋) |
19 | 18 | 3adant3 1113 | . 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 17630 | 1 ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝐺‘∅) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 = wceq 1508 ∈ wcel 2051 ∀wral 3081 ⊆ wss 3822 ∅c0 4172 class class class wbr 4925 ‘cfv 6185 lecple 16426 Moorecmre 16723 Posetcpo 17420 glbcglb 17423 toInccipo 17631 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-cnex 10389 ax-resscn 10390 ax-1cn 10391 ax-icn 10392 ax-addcl 10393 ax-addrcl 10394 ax-mulcl 10395 ax-mulrcl 10396 ax-mulcom 10397 ax-addass 10398 ax-mulass 10399 ax-distr 10400 ax-i2m1 10401 ax-1ne0 10402 ax-1rid 10403 ax-rnegex 10404 ax-rrecex 10405 ax-cnre 10406 ax-pre-lttri 10407 ax-pre-lttrn 10408 ax-pre-ltadd 10409 ax-pre-mulgt0 10410 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-nel 3067 df-ral 3086 df-rex 3087 df-reu 3088 df-rmo 3089 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-pss 3838 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-int 4746 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-om 7395 df-1st 7499 df-2nd 7500 df-wrecs 7748 df-recs 7810 df-rdg 7848 df-1o 7903 df-oadd 7907 df-er 8087 df-en 8305 df-dom 8306 df-sdom 8307 df-fin 8308 df-pnf 10474 df-mnf 10475 df-xr 10476 df-ltxr 10477 df-le 10478 df-sub 10670 df-neg 10671 df-nn 11438 df-2 11501 df-3 11502 df-4 11503 df-5 11504 df-6 11505 df-7 11506 df-8 11507 df-9 11508 df-n0 11706 df-z 11792 df-dec 11910 df-uz 12057 df-fz 12707 df-struct 16339 df-ndx 16340 df-slot 16341 df-base 16343 df-sets 16344 df-tset 16438 df-ple 16439 df-ocomp 16440 df-mre 16727 df-proset 17408 df-poset 17426 df-lub 17454 df-glb 17455 df-odu 17609 df-ipo 17632 |
This theorem is referenced by: mreclatBAD 17667 |
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