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Theorem glbeldm2d 48090
Description: Member of the domain of the greatest lower bound function of a poset. (Contributed by Zhi Wang, 29-Sep-2024.)
Hypotheses
Ref Expression
lubeldm2d.b (πœ‘ β†’ 𝐡 = (Baseβ€˜πΎ))
lubeldm2d.l (πœ‘ β†’ ≀ = (leβ€˜πΎ))
glbeldm2d.g (πœ‘ β†’ 𝐺 = (glbβ€˜πΎ))
glbeldm2d.p ((πœ‘ ∧ π‘₯ ∈ 𝐡) β†’ (πœ“ ↔ (βˆ€π‘¦ ∈ 𝑆 π‘₯ ≀ 𝑦 ∧ βˆ€π‘§ ∈ 𝐡 (βˆ€π‘¦ ∈ 𝑆 𝑧 ≀ 𝑦 β†’ 𝑧 ≀ π‘₯))))
glbeldm2d.k (πœ‘ β†’ 𝐾 ∈ Poset)
Assertion
Ref Expression
glbeldm2d (πœ‘ β†’ (𝑆 ∈ dom 𝐺 ↔ (𝑆 βŠ† 𝐡 ∧ βˆƒπ‘₯ ∈ 𝐡 πœ“)))
Distinct variable groups:   π‘₯,𝐾,𝑦,𝑧   π‘₯,𝑆,𝑦,𝑧   πœ‘,π‘₯,𝑦,𝑧
Allowed substitution hints:   πœ“(π‘₯,𝑦,𝑧)   𝐡(π‘₯,𝑦,𝑧)   𝐺(π‘₯,𝑦,𝑧)   ≀ (π‘₯,𝑦,𝑧)

Proof of Theorem glbeldm2d
StepHypRef Expression
1 eqid 2725 . . 3 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
2 eqid 2725 . . 3 (leβ€˜πΎ) = (leβ€˜πΎ)
3 eqid 2725 . . 3 (glbβ€˜πΎ) = (glbβ€˜πΎ)
4 biid 260 . . 3 ((βˆ€π‘¦ ∈ 𝑆 π‘₯(leβ€˜πΎ)𝑦 ∧ βˆ€π‘§ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ 𝑆 𝑧(leβ€˜πΎ)𝑦 β†’ 𝑧(leβ€˜πΎ)π‘₯)) ↔ (βˆ€π‘¦ ∈ 𝑆 π‘₯(leβ€˜πΎ)𝑦 ∧ βˆ€π‘§ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ 𝑆 𝑧(leβ€˜πΎ)𝑦 β†’ 𝑧(leβ€˜πΎ)π‘₯)))
5 glbeldm2d.k . . 3 (πœ‘ β†’ 𝐾 ∈ Poset)
61, 2, 3, 4, 5glbeldm2 48088 . 2 (πœ‘ β†’ (𝑆 ∈ dom (glbβ€˜πΎ) ↔ (𝑆 βŠ† (Baseβ€˜πΎ) ∧ βˆƒπ‘₯ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ 𝑆 π‘₯(leβ€˜πΎ)𝑦 ∧ βˆ€π‘§ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ 𝑆 𝑧(leβ€˜πΎ)𝑦 β†’ 𝑧(leβ€˜πΎ)π‘₯)))))
7 glbeldm2d.g . . . 4 (πœ‘ β†’ 𝐺 = (glbβ€˜πΎ))
87dmeqd 5902 . . 3 (πœ‘ β†’ dom 𝐺 = dom (glbβ€˜πΎ))
98eleq2d 2811 . 2 (πœ‘ β†’ (𝑆 ∈ dom 𝐺 ↔ 𝑆 ∈ dom (glbβ€˜πΎ)))
10 lubeldm2d.b . . . 4 (πœ‘ β†’ 𝐡 = (Baseβ€˜πΎ))
1110sseq2d 4005 . . 3 (πœ‘ β†’ (𝑆 βŠ† 𝐡 ↔ 𝑆 βŠ† (Baseβ€˜πΎ)))
12 glbeldm2d.p . . . . . . 7 ((πœ‘ ∧ π‘₯ ∈ 𝐡) β†’ (πœ“ ↔ (βˆ€π‘¦ ∈ 𝑆 π‘₯ ≀ 𝑦 ∧ βˆ€π‘§ ∈ 𝐡 (βˆ€π‘¦ ∈ 𝑆 𝑧 ≀ 𝑦 β†’ 𝑧 ≀ π‘₯))))
13 lubeldm2d.l . . . . . . . . . . 11 (πœ‘ β†’ ≀ = (leβ€˜πΎ))
1413breqd 5154 . . . . . . . . . 10 (πœ‘ β†’ (π‘₯ ≀ 𝑦 ↔ π‘₯(leβ€˜πΎ)𝑦))
1514ralbidv 3168 . . . . . . . . 9 (πœ‘ β†’ (βˆ€π‘¦ ∈ 𝑆 π‘₯ ≀ 𝑦 ↔ βˆ€π‘¦ ∈ 𝑆 π‘₯(leβ€˜πΎ)𝑦))
1613breqd 5154 . . . . . . . . . . . 12 (πœ‘ β†’ (𝑧 ≀ 𝑦 ↔ 𝑧(leβ€˜πΎ)𝑦))
1716ralbidv 3168 . . . . . . . . . . 11 (πœ‘ β†’ (βˆ€π‘¦ ∈ 𝑆 𝑧 ≀ 𝑦 ↔ βˆ€π‘¦ ∈ 𝑆 𝑧(leβ€˜πΎ)𝑦))
1813breqd 5154 . . . . . . . . . . 11 (πœ‘ β†’ (𝑧 ≀ π‘₯ ↔ 𝑧(leβ€˜πΎ)π‘₯))
1917, 18imbi12d 343 . . . . . . . . . 10 (πœ‘ β†’ ((βˆ€π‘¦ ∈ 𝑆 𝑧 ≀ 𝑦 β†’ 𝑧 ≀ π‘₯) ↔ (βˆ€π‘¦ ∈ 𝑆 𝑧(leβ€˜πΎ)𝑦 β†’ 𝑧(leβ€˜πΎ)π‘₯)))
2010, 19raleqbidv 3330 . . . . . . . . 9 (πœ‘ β†’ (βˆ€π‘§ ∈ 𝐡 (βˆ€π‘¦ ∈ 𝑆 𝑧 ≀ 𝑦 β†’ 𝑧 ≀ π‘₯) ↔ βˆ€π‘§ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ 𝑆 𝑧(leβ€˜πΎ)𝑦 β†’ 𝑧(leβ€˜πΎ)π‘₯)))
2115, 20anbi12d 630 . . . . . . . 8 (πœ‘ β†’ ((βˆ€π‘¦ ∈ 𝑆 π‘₯ ≀ 𝑦 ∧ βˆ€π‘§ ∈ 𝐡 (βˆ€π‘¦ ∈ 𝑆 𝑧 ≀ 𝑦 β†’ 𝑧 ≀ π‘₯)) ↔ (βˆ€π‘¦ ∈ 𝑆 π‘₯(leβ€˜πΎ)𝑦 ∧ βˆ€π‘§ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ 𝑆 𝑧(leβ€˜πΎ)𝑦 β†’ 𝑧(leβ€˜πΎ)π‘₯))))
2221adantr 479 . . . . . . 7 ((πœ‘ ∧ π‘₯ ∈ 𝐡) β†’ ((βˆ€π‘¦ ∈ 𝑆 π‘₯ ≀ 𝑦 ∧ βˆ€π‘§ ∈ 𝐡 (βˆ€π‘¦ ∈ 𝑆 𝑧 ≀ 𝑦 β†’ 𝑧 ≀ π‘₯)) ↔ (βˆ€π‘¦ ∈ 𝑆 π‘₯(leβ€˜πΎ)𝑦 ∧ βˆ€π‘§ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ 𝑆 𝑧(leβ€˜πΎ)𝑦 β†’ 𝑧(leβ€˜πΎ)π‘₯))))
2312, 22bitrd 278 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ 𝐡) β†’ (πœ“ ↔ (βˆ€π‘¦ ∈ 𝑆 π‘₯(leβ€˜πΎ)𝑦 ∧ βˆ€π‘§ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ 𝑆 𝑧(leβ€˜πΎ)𝑦 β†’ 𝑧(leβ€˜πΎ)π‘₯))))
2423pm5.32da 577 . . . . 5 (πœ‘ β†’ ((π‘₯ ∈ 𝐡 ∧ πœ“) ↔ (π‘₯ ∈ 𝐡 ∧ (βˆ€π‘¦ ∈ 𝑆 π‘₯(leβ€˜πΎ)𝑦 ∧ βˆ€π‘§ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ 𝑆 𝑧(leβ€˜πΎ)𝑦 β†’ 𝑧(leβ€˜πΎ)π‘₯)))))
2510eleq2d 2811 . . . . . 6 (πœ‘ β†’ (π‘₯ ∈ 𝐡 ↔ π‘₯ ∈ (Baseβ€˜πΎ)))
2625anbi1d 629 . . . . 5 (πœ‘ β†’ ((π‘₯ ∈ 𝐡 ∧ (βˆ€π‘¦ ∈ 𝑆 π‘₯(leβ€˜πΎ)𝑦 ∧ βˆ€π‘§ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ 𝑆 𝑧(leβ€˜πΎ)𝑦 β†’ 𝑧(leβ€˜πΎ)π‘₯))) ↔ (π‘₯ ∈ (Baseβ€˜πΎ) ∧ (βˆ€π‘¦ ∈ 𝑆 π‘₯(leβ€˜πΎ)𝑦 ∧ βˆ€π‘§ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ 𝑆 𝑧(leβ€˜πΎ)𝑦 β†’ 𝑧(leβ€˜πΎ)π‘₯)))))
2724, 26bitrd 278 . . . 4 (πœ‘ β†’ ((π‘₯ ∈ 𝐡 ∧ πœ“) ↔ (π‘₯ ∈ (Baseβ€˜πΎ) ∧ (βˆ€π‘¦ ∈ 𝑆 π‘₯(leβ€˜πΎ)𝑦 ∧ βˆ€π‘§ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ 𝑆 𝑧(leβ€˜πΎ)𝑦 β†’ 𝑧(leβ€˜πΎ)π‘₯)))))
2827rexbidv2 3165 . . 3 (πœ‘ β†’ (βˆƒπ‘₯ ∈ 𝐡 πœ“ ↔ βˆƒπ‘₯ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ 𝑆 π‘₯(leβ€˜πΎ)𝑦 ∧ βˆ€π‘§ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ 𝑆 𝑧(leβ€˜πΎ)𝑦 β†’ 𝑧(leβ€˜πΎ)π‘₯))))
2911, 28anbi12d 630 . 2 (πœ‘ β†’ ((𝑆 βŠ† 𝐡 ∧ βˆƒπ‘₯ ∈ 𝐡 πœ“) ↔ (𝑆 βŠ† (Baseβ€˜πΎ) ∧ βˆƒπ‘₯ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ 𝑆 π‘₯(leβ€˜πΎ)𝑦 ∧ βˆ€π‘§ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ 𝑆 𝑧(leβ€˜πΎ)𝑦 β†’ 𝑧(leβ€˜πΎ)π‘₯)))))
306, 9, 293bitr4d 310 1 (πœ‘ β†’ (𝑆 ∈ dom 𝐺 ↔ (𝑆 βŠ† 𝐡 ∧ βˆƒπ‘₯ ∈ 𝐡 πœ“)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098  βˆ€wral 3051  βˆƒwrex 3060   βŠ† wss 3939   class class class wbr 5143  dom cdm 5672  β€˜cfv 6543  Basecbs 17179  lecple 17239  Posetcpo 18298  glbcglb 18301
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7372  df-proset 18286  df-poset 18304  df-glb 18338
This theorem is referenced by:  ipoglbdm  48113
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