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Theorem dihglblem2aN 39806
Description: Lemma for isomorphism H of a GLB. (Contributed by NM, 19-Mar-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
dihglblem.b 𝐡 = (Baseβ€˜πΎ)
dihglblem.l ≀ = (leβ€˜πΎ)
dihglblem.m ∧ = (meetβ€˜πΎ)
dihglblem.g 𝐺 = (glbβ€˜πΎ)
dihglblem.h 𝐻 = (LHypβ€˜πΎ)
dihglblem.t 𝑇 = {𝑒 ∈ 𝐡 ∣ βˆƒπ‘£ ∈ 𝑆 𝑒 = (𝑣 ∧ π‘Š)}
Assertion
Ref Expression
dihglblem2aN (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑆 βŠ† 𝐡 ∧ 𝑆 β‰  βˆ…)) β†’ 𝑇 β‰  βˆ…)
Distinct variable groups:   𝑣,𝑒, ∧   𝑒,𝐡   𝑒,𝑆,𝑣   𝑒,π‘Š,𝑣
Allowed substitution hints:   𝐡(𝑣)   𝑇(𝑣,𝑒)   𝐺(𝑣,𝑒)   𝐻(𝑣,𝑒)   𝐾(𝑣,𝑒)   ≀ (𝑣,𝑒)

Proof of Theorem dihglblem2aN
Dummy variables 𝑀 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dihglblem.t . . 3 𝑇 = {𝑒 ∈ 𝐡 ∣ βˆƒπ‘£ ∈ 𝑆 𝑒 = (𝑣 ∧ π‘Š)}
21a1i 11 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑆 βŠ† 𝐡 ∧ 𝑆 β‰  βˆ…)) β†’ 𝑇 = {𝑒 ∈ 𝐡 ∣ βˆƒπ‘£ ∈ 𝑆 𝑒 = (𝑣 ∧ π‘Š)})
3 simprr 772 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑆 βŠ† 𝐡 ∧ 𝑆 β‰  βˆ…)) β†’ 𝑆 β‰  βˆ…)
4 n0 4310 . . . 4 (𝑆 β‰  βˆ… ↔ βˆƒπ‘§ 𝑧 ∈ 𝑆)
53, 4sylib 217 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑆 βŠ† 𝐡 ∧ 𝑆 β‰  βˆ…)) β†’ βˆƒπ‘§ 𝑧 ∈ 𝑆)
6 hllat 37875 . . . . . . 7 (𝐾 ∈ HL β†’ 𝐾 ∈ Lat)
76ad3antrrr 729 . . . . . 6 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑆 βŠ† 𝐡 ∧ 𝑆 β‰  βˆ…)) ∧ 𝑧 ∈ 𝑆) β†’ 𝐾 ∈ Lat)
8 simplrl 776 . . . . . . 7 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑆 βŠ† 𝐡 ∧ 𝑆 β‰  βˆ…)) ∧ 𝑧 ∈ 𝑆) β†’ 𝑆 βŠ† 𝐡)
9 simpr 486 . . . . . . 7 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑆 βŠ† 𝐡 ∧ 𝑆 β‰  βˆ…)) ∧ 𝑧 ∈ 𝑆) β†’ 𝑧 ∈ 𝑆)
108, 9sseldd 3949 . . . . . 6 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑆 βŠ† 𝐡 ∧ 𝑆 β‰  βˆ…)) ∧ 𝑧 ∈ 𝑆) β†’ 𝑧 ∈ 𝐡)
11 dihglblem.b . . . . . . . 8 𝐡 = (Baseβ€˜πΎ)
12 dihglblem.h . . . . . . . 8 𝐻 = (LHypβ€˜πΎ)
1311, 12lhpbase 38511 . . . . . . 7 (π‘Š ∈ 𝐻 β†’ π‘Š ∈ 𝐡)
1413ad3antlr 730 . . . . . 6 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑆 βŠ† 𝐡 ∧ 𝑆 β‰  βˆ…)) ∧ 𝑧 ∈ 𝑆) β†’ π‘Š ∈ 𝐡)
15 dihglblem.m . . . . . . 7 ∧ = (meetβ€˜πΎ)
1611, 15latmcl 18337 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑧 ∈ 𝐡 ∧ π‘Š ∈ 𝐡) β†’ (𝑧 ∧ π‘Š) ∈ 𝐡)
177, 10, 14, 16syl3anc 1372 . . . . 5 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑆 βŠ† 𝐡 ∧ 𝑆 β‰  βˆ…)) ∧ 𝑧 ∈ 𝑆) β†’ (𝑧 ∧ π‘Š) ∈ 𝐡)
18 eqidd 2734 . . . . . 6 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑆 βŠ† 𝐡 ∧ 𝑆 β‰  βˆ…)) ∧ 𝑧 ∈ 𝑆) β†’ (𝑧 ∧ π‘Š) = (𝑧 ∧ π‘Š))
19 oveq1 7368 . . . . . . 7 (𝑣 = 𝑧 β†’ (𝑣 ∧ π‘Š) = (𝑧 ∧ π‘Š))
2019rspceeqv 3599 . . . . . 6 ((𝑧 ∈ 𝑆 ∧ (𝑧 ∧ π‘Š) = (𝑧 ∧ π‘Š)) β†’ βˆƒπ‘£ ∈ 𝑆 (𝑧 ∧ π‘Š) = (𝑣 ∧ π‘Š))
219, 18, 20syl2anc 585 . . . . 5 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑆 βŠ† 𝐡 ∧ 𝑆 β‰  βˆ…)) ∧ 𝑧 ∈ 𝑆) β†’ βˆƒπ‘£ ∈ 𝑆 (𝑧 ∧ π‘Š) = (𝑣 ∧ π‘Š))
22 ovex 7394 . . . . . 6 (𝑧 ∧ π‘Š) ∈ V
23 eleq1 2822 . . . . . . 7 (𝑀 = (𝑧 ∧ π‘Š) β†’ (𝑀 ∈ {𝑒 ∈ 𝐡 ∣ βˆƒπ‘£ ∈ 𝑆 𝑒 = (𝑣 ∧ π‘Š)} ↔ (𝑧 ∧ π‘Š) ∈ {𝑒 ∈ 𝐡 ∣ βˆƒπ‘£ ∈ 𝑆 𝑒 = (𝑣 ∧ π‘Š)}))
24 eqeq1 2737 . . . . . . . . 9 (𝑒 = (𝑧 ∧ π‘Š) β†’ (𝑒 = (𝑣 ∧ π‘Š) ↔ (𝑧 ∧ π‘Š) = (𝑣 ∧ π‘Š)))
2524rexbidv 3172 . . . . . . . 8 (𝑒 = (𝑧 ∧ π‘Š) β†’ (βˆƒπ‘£ ∈ 𝑆 𝑒 = (𝑣 ∧ π‘Š) ↔ βˆƒπ‘£ ∈ 𝑆 (𝑧 ∧ π‘Š) = (𝑣 ∧ π‘Š)))
2625elrab 3649 . . . . . . 7 ((𝑧 ∧ π‘Š) ∈ {𝑒 ∈ 𝐡 ∣ βˆƒπ‘£ ∈ 𝑆 𝑒 = (𝑣 ∧ π‘Š)} ↔ ((𝑧 ∧ π‘Š) ∈ 𝐡 ∧ βˆƒπ‘£ ∈ 𝑆 (𝑧 ∧ π‘Š) = (𝑣 ∧ π‘Š)))
2723, 26bitrdi 287 . . . . . 6 (𝑀 = (𝑧 ∧ π‘Š) β†’ (𝑀 ∈ {𝑒 ∈ 𝐡 ∣ βˆƒπ‘£ ∈ 𝑆 𝑒 = (𝑣 ∧ π‘Š)} ↔ ((𝑧 ∧ π‘Š) ∈ 𝐡 ∧ βˆƒπ‘£ ∈ 𝑆 (𝑧 ∧ π‘Š) = (𝑣 ∧ π‘Š))))
2822, 27spcev 3567 . . . . 5 (((𝑧 ∧ π‘Š) ∈ 𝐡 ∧ βˆƒπ‘£ ∈ 𝑆 (𝑧 ∧ π‘Š) = (𝑣 ∧ π‘Š)) β†’ βˆƒπ‘€ 𝑀 ∈ {𝑒 ∈ 𝐡 ∣ βˆƒπ‘£ ∈ 𝑆 𝑒 = (𝑣 ∧ π‘Š)})
2917, 21, 28syl2anc 585 . . . 4 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑆 βŠ† 𝐡 ∧ 𝑆 β‰  βˆ…)) ∧ 𝑧 ∈ 𝑆) β†’ βˆƒπ‘€ 𝑀 ∈ {𝑒 ∈ 𝐡 ∣ βˆƒπ‘£ ∈ 𝑆 𝑒 = (𝑣 ∧ π‘Š)})
30 n0 4310 . . . 4 ({𝑒 ∈ 𝐡 ∣ βˆƒπ‘£ ∈ 𝑆 𝑒 = (𝑣 ∧ π‘Š)} β‰  βˆ… ↔ βˆƒπ‘€ 𝑀 ∈ {𝑒 ∈ 𝐡 ∣ βˆƒπ‘£ ∈ 𝑆 𝑒 = (𝑣 ∧ π‘Š)})
3129, 30sylibr 233 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑆 βŠ† 𝐡 ∧ 𝑆 β‰  βˆ…)) ∧ 𝑧 ∈ 𝑆) β†’ {𝑒 ∈ 𝐡 ∣ βˆƒπ‘£ ∈ 𝑆 𝑒 = (𝑣 ∧ π‘Š)} β‰  βˆ…)
325, 31exlimddv 1939 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑆 βŠ† 𝐡 ∧ 𝑆 β‰  βˆ…)) β†’ {𝑒 ∈ 𝐡 ∣ βˆƒπ‘£ ∈ 𝑆 𝑒 = (𝑣 ∧ π‘Š)} β‰  βˆ…)
332, 32eqnetrd 3008 1 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑆 βŠ† 𝐡 ∧ 𝑆 β‰  βˆ…)) β†’ 𝑇 β‰  βˆ…)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107   β‰  wne 2940  βˆƒwrex 3070  {crab 3406   βŠ† wss 3914  βˆ…c0 4286  β€˜cfv 6500  (class class class)co 7361  Basecbs 17091  lecple 17148  glbcglb 18207  meetcmee 18209  Latclat 18328  HLchlt 37862  LHypclh 38497
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 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7317  df-ov 7364  df-oprab 7365  df-lub 18243  df-glb 18244  df-join 18245  df-meet 18246  df-lat 18329  df-atl 37810  df-cvlat 37834  df-hlat 37863  df-lhyp 38501
This theorem is referenced by:  dihglblem3N  39808
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