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Mirrors > Home > MPE Home > Th. List > Mathboxes > toplatglb | Structured version Visualization version GIF version |
Description: Greatest lower bounds in a topology are realized by the interior of the intersection. (Contributed by Zhi Wang, 30-Sep-2024.) |
Ref | Expression |
---|---|
topclat.i | ⊢ 𝐼 = (toInc‘𝐽) |
toplatlub.j | ⊢ (𝜑 → 𝐽 ∈ Top) |
toplatlub.s | ⊢ (𝜑 → 𝑆 ⊆ 𝐽) |
toplatglb.g | ⊢ 𝐺 = (glb‘𝐼) |
toplatglb.e | ⊢ (𝜑 → 𝑆 ≠ ∅) |
Ref | Expression |
---|---|
toplatglb | ⊢ (𝜑 → (𝐺‘𝑆) = ((int‘𝐽)‘∩ 𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | topclat.i | . 2 ⊢ 𝐼 = (toInc‘𝐽) | |
2 | toplatlub.j | . 2 ⊢ (𝜑 → 𝐽 ∈ Top) | |
3 | toplatlub.s | . 2 ⊢ (𝜑 → 𝑆 ⊆ 𝐽) | |
4 | toplatglb.g | . . 3 ⊢ 𝐺 = (glb‘𝐼) | |
5 | 4 | a1i 11 | . 2 ⊢ (𝜑 → 𝐺 = (glb‘𝐼)) |
6 | toplatglb.e | . . . . . 6 ⊢ (𝜑 → 𝑆 ≠ ∅) | |
7 | intssuni 4931 | . . . . . 6 ⊢ (𝑆 ≠ ∅ → ∩ 𝑆 ⊆ ∪ 𝑆) | |
8 | 6, 7 | syl 17 | . . . . 5 ⊢ (𝜑 → ∩ 𝑆 ⊆ ∪ 𝑆) |
9 | 3 | unissd 4875 | . . . . 5 ⊢ (𝜑 → ∪ 𝑆 ⊆ ∪ 𝐽) |
10 | 8, 9 | sstrd 3954 | . . . 4 ⊢ (𝜑 → ∩ 𝑆 ⊆ ∪ 𝐽) |
11 | eqid 2736 | . . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
12 | 11 | ntrval 22387 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ ∩ 𝑆 ⊆ ∪ 𝐽) → ((int‘𝐽)‘∩ 𝑆) = ∪ (𝐽 ∩ 𝒫 ∩ 𝑆)) |
13 | 2, 10, 12 | syl2anc 584 | . . 3 ⊢ (𝜑 → ((int‘𝐽)‘∩ 𝑆) = ∪ (𝐽 ∩ 𝒫 ∩ 𝑆)) |
14 | 2 | uniexd 7679 | . . . . 5 ⊢ (𝜑 → ∪ 𝐽 ∈ V) |
15 | 14, 10 | ssexd 5281 | . . . 4 ⊢ (𝜑 → ∩ 𝑆 ∈ V) |
16 | inpw 46893 | . . . . 5 ⊢ (∩ 𝑆 ∈ V → (𝐽 ∩ 𝒫 ∩ 𝑆) = {𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ ∩ 𝑆}) | |
17 | 16 | unieqd 4879 | . . . 4 ⊢ (∩ 𝑆 ∈ V → ∪ (𝐽 ∩ 𝒫 ∩ 𝑆) = ∪ {𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ ∩ 𝑆}) |
18 | 15, 17 | syl 17 | . . 3 ⊢ (𝜑 → ∪ (𝐽 ∩ 𝒫 ∩ 𝑆) = ∪ {𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ ∩ 𝑆}) |
19 | 13, 18 | eqtrd 2776 | . 2 ⊢ (𝜑 → ((int‘𝐽)‘∩ 𝑆) = ∪ {𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ ∩ 𝑆}) |
20 | 11 | ntropn 22400 | . . 3 ⊢ ((𝐽 ∈ Top ∧ ∩ 𝑆 ⊆ ∪ 𝐽) → ((int‘𝐽)‘∩ 𝑆) ∈ 𝐽) |
21 | 2, 10, 20 | syl2anc 584 | . 2 ⊢ (𝜑 → ((int‘𝐽)‘∩ 𝑆) ∈ 𝐽) |
22 | 1, 2, 3, 5, 19, 21 | ipoglb 47006 | 1 ⊢ (𝜑 → (𝐺‘𝑆) = ((int‘𝐽)‘∩ 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 {crab 3407 Vcvv 3445 ∩ cin 3909 ⊆ wss 3910 ∅c0 4282 𝒫 cpw 4560 ∪ cuni 4865 ∩ cint 4907 ‘cfv 6496 glbcglb 18199 toInccipo 18416 Topctop 22242 intcnt 22368 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-er 8648 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-dec 12619 df-uz 12764 df-fz 13425 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-tset 17152 df-ple 17153 df-ocomp 17154 df-odu 18176 df-proset 18184 df-poset 18202 df-lub 18235 df-glb 18236 df-ipo 18417 df-top 22243 df-ntr 22371 |
This theorem is referenced by: (None) |
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