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Theorem glbeldm2d 47866
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 2726 . . 3 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
2 eqid 2726 . . 3 (leβ€˜πΎ) = (leβ€˜πΎ)
3 eqid 2726 . . 3 (glbβ€˜πΎ) = (glbβ€˜πΎ)
4 biid 261 . . 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 47864 . 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 5899 . . 3 (πœ‘ β†’ dom 𝐺 = dom (glbβ€˜πΎ))
98eleq2d 2813 . 2 (πœ‘ β†’ (𝑆 ∈ dom 𝐺 ↔ 𝑆 ∈ dom (glbβ€˜πΎ)))
10 lubeldm2d.b . . . 4 (πœ‘ β†’ 𝐡 = (Baseβ€˜πΎ))
1110sseq2d 4009 . . 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 5152 . . . . . . . . . 10 (πœ‘ β†’ (π‘₯ ≀ 𝑦 ↔ π‘₯(leβ€˜πΎ)𝑦))
1514ralbidv 3171 . . . . . . . . 9 (πœ‘ β†’ (βˆ€π‘¦ ∈ 𝑆 π‘₯ ≀ 𝑦 ↔ βˆ€π‘¦ ∈ 𝑆 π‘₯(leβ€˜πΎ)𝑦))
1613breqd 5152 . . . . . . . . . . . 12 (πœ‘ β†’ (𝑧 ≀ 𝑦 ↔ 𝑧(leβ€˜πΎ)𝑦))
1716ralbidv 3171 . . . . . . . . . . 11 (πœ‘ β†’ (βˆ€π‘¦ ∈ 𝑆 𝑧 ≀ 𝑦 ↔ βˆ€π‘¦ ∈ 𝑆 𝑧(leβ€˜πΎ)𝑦))
1813breqd 5152 . . . . . . . . . . 11 (πœ‘ β†’ (𝑧 ≀ π‘₯ ↔ 𝑧(leβ€˜πΎ)π‘₯))
1917, 18imbi12d 344 . . . . . . . . . 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 3336 . . . . . . . . 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 480 . . . . . . 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 279 . . . . . 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 578 . . . . 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 2813 . . . . . 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 279 . . . 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 3168 . . 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 311 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 395   = wceq 1533   ∈ wcel 2098  βˆ€wral 3055  βˆƒwrex 3064   βŠ† wss 3943   class class class wbr 5141  dom cdm 5669  β€˜cfv 6537  Basecbs 17153  lecple 17213  Posetcpo 18272  glbcglb 18275
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-proset 18260  df-poset 18278  df-glb 18312
This theorem is referenced by:  ipoglbdm  47889
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