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Theorem glbconxN 38565
Description: De Morgan's law for GLB and LUB. Index-set version of glbconN 38563, where we read 𝑆 as 𝑆(𝑖). (Contributed by NM, 17-Jan-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
glbcon.b 𝐵 = (Base‘𝐾)
glbcon.u 𝑈 = (lub‘𝐾)
glbcon.g 𝐺 = (glb‘𝐾)
glbcon.o = (oc‘𝐾)
Assertion
Ref Expression
glbconxN ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) → (𝐺‘{𝑥 ∣ ∃𝑖𝐼 𝑥 = 𝑆}) = ( ‘(𝑈‘{𝑥 ∣ ∃𝑖𝐼 𝑥 = ( 𝑆)})))
Distinct variable groups:   𝑥,𝐵   𝑥,   𝑥,𝑆   𝐵,𝑖   𝑥,𝐼   𝑖,𝐾   ,𝑖,𝑥
Allowed substitution hints:   𝑆(𝑖)   𝑈(𝑥,𝑖)   𝐺(𝑥,𝑖)   𝐼(𝑖)   𝐾(𝑥)

Proof of Theorem glbconxN
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 vex 3477 . . . . . 6 𝑦 ∈ V
2 eqeq1 2735 . . . . . . 7 (𝑥 = 𝑦 → (𝑥 = 𝑆𝑦 = 𝑆))
32rexbidv 3177 . . . . . 6 (𝑥 = 𝑦 → (∃𝑖𝐼 𝑥 = 𝑆 ↔ ∃𝑖𝐼 𝑦 = 𝑆))
41, 3elab 3668 . . . . 5 (𝑦 ∈ {𝑥 ∣ ∃𝑖𝐼 𝑥 = 𝑆} ↔ ∃𝑖𝐼 𝑦 = 𝑆)
5 nfra1 3280 . . . . . 6 𝑖𝑖𝐼 𝑆𝐵
6 nfv 1916 . . . . . 6 𝑖 𝑦𝐵
7 rsp 3243 . . . . . . 7 (∀𝑖𝐼 𝑆𝐵 → (𝑖𝐼𝑆𝐵))
8 eleq1a 2827 . . . . . . 7 (𝑆𝐵 → (𝑦 = 𝑆𝑦𝐵))
97, 8syl6 35 . . . . . 6 (∀𝑖𝐼 𝑆𝐵 → (𝑖𝐼 → (𝑦 = 𝑆𝑦𝐵)))
105, 6, 9rexlimd 3262 . . . . 5 (∀𝑖𝐼 𝑆𝐵 → (∃𝑖𝐼 𝑦 = 𝑆𝑦𝐵))
114, 10biimtrid 241 . . . 4 (∀𝑖𝐼 𝑆𝐵 → (𝑦 ∈ {𝑥 ∣ ∃𝑖𝐼 𝑥 = 𝑆} → 𝑦𝐵))
1211ssrdv 3988 . . 3 (∀𝑖𝐼 𝑆𝐵 → {𝑥 ∣ ∃𝑖𝐼 𝑥 = 𝑆} ⊆ 𝐵)
13 glbcon.b . . . 4 𝐵 = (Base‘𝐾)
14 glbcon.u . . . 4 𝑈 = (lub‘𝐾)
15 glbcon.g . . . 4 𝐺 = (glb‘𝐾)
16 glbcon.o . . . 4 = (oc‘𝐾)
1713, 14, 15, 16glbconN 38563 . . 3 ((𝐾 ∈ HL ∧ {𝑥 ∣ ∃𝑖𝐼 𝑥 = 𝑆} ⊆ 𝐵) → (𝐺‘{𝑥 ∣ ∃𝑖𝐼 𝑥 = 𝑆}) = ( ‘(𝑈‘{𝑦𝐵 ∣ ( 𝑦) ∈ {𝑥 ∣ ∃𝑖𝐼 𝑥 = 𝑆}})))
1812, 17sylan2 592 . 2 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) → (𝐺‘{𝑥 ∣ ∃𝑖𝐼 𝑥 = 𝑆}) = ( ‘(𝑈‘{𝑦𝐵 ∣ ( 𝑦) ∈ {𝑥 ∣ ∃𝑖𝐼 𝑥 = 𝑆}})))
19 fvex 6904 . . . . . . . 8 ( 𝑦) ∈ V
20 eqeq1 2735 . . . . . . . . 9 (𝑥 = ( 𝑦) → (𝑥 = 𝑆 ↔ ( 𝑦) = 𝑆))
2120rexbidv 3177 . . . . . . . 8 (𝑥 = ( 𝑦) → (∃𝑖𝐼 𝑥 = 𝑆 ↔ ∃𝑖𝐼 ( 𝑦) = 𝑆))
2219, 21elab 3668 . . . . . . 7 (( 𝑦) ∈ {𝑥 ∣ ∃𝑖𝐼 𝑥 = 𝑆} ↔ ∃𝑖𝐼 ( 𝑦) = 𝑆)
2322rabbii 3437 . . . . . 6 {𝑦𝐵 ∣ ( 𝑦) ∈ {𝑥 ∣ ∃𝑖𝐼 𝑥 = 𝑆}} = {𝑦𝐵 ∣ ∃𝑖𝐼 ( 𝑦) = 𝑆}
24 df-rab 3432 . . . . . 6 {𝑦𝐵 ∣ ∃𝑖𝐼 ( 𝑦) = 𝑆} = {𝑦 ∣ (𝑦𝐵 ∧ ∃𝑖𝐼 ( 𝑦) = 𝑆)}
2523, 24eqtri 2759 . . . . 5 {𝑦𝐵 ∣ ( 𝑦) ∈ {𝑥 ∣ ∃𝑖𝐼 𝑥 = 𝑆}} = {𝑦 ∣ (𝑦𝐵 ∧ ∃𝑖𝐼 ( 𝑦) = 𝑆)}
26 nfv 1916 . . . . . . . . . 10 𝑖 𝐾 ∈ HL
2726, 5nfan 1901 . . . . . . . . 9 𝑖(𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵)
28 rspa 3244 . . . . . . . . . . 11 ((∀𝑖𝐼 𝑆𝐵𝑖𝐼) → 𝑆𝐵)
29 hlop 38548 . . . . . . . . . . . . . . 15 (𝐾 ∈ HL → 𝐾 ∈ OP)
3013, 16opoccl 38380 . . . . . . . . . . . . . . 15 ((𝐾 ∈ OP ∧ 𝑆𝐵) → ( 𝑆) ∈ 𝐵)
3129, 30sylan 579 . . . . . . . . . . . . . 14 ((𝐾 ∈ HL ∧ 𝑆𝐵) → ( 𝑆) ∈ 𝐵)
32 eleq1a 2827 . . . . . . . . . . . . . 14 (( 𝑆) ∈ 𝐵 → (𝑦 = ( 𝑆) → 𝑦𝐵))
3331, 32syl 17 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ 𝑆𝐵) → (𝑦 = ( 𝑆) → 𝑦𝐵))
3433pm4.71rd 562 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑆𝐵) → (𝑦 = ( 𝑆) ↔ (𝑦𝐵𝑦 = ( 𝑆))))
3513, 16opcon2b 38383 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ OP ∧ 𝑆𝐵𝑦𝐵) → (𝑆 = ( 𝑦) ↔ 𝑦 = ( 𝑆)))
3629, 35syl3an1 1162 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ 𝑆𝐵𝑦𝐵) → (𝑆 = ( 𝑦) ↔ 𝑦 = ( 𝑆)))
37363expa 1117 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑆𝐵) ∧ 𝑦𝐵) → (𝑆 = ( 𝑦) ↔ 𝑦 = ( 𝑆)))
38 eqcom 2738 . . . . . . . . . . . . . 14 (𝑆 = ( 𝑦) ↔ ( 𝑦) = 𝑆)
3937, 38bitr3di 286 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ 𝑆𝐵) ∧ 𝑦𝐵) → (𝑦 = ( 𝑆) ↔ ( 𝑦) = 𝑆))
4039pm5.32da 578 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑆𝐵) → ((𝑦𝐵𝑦 = ( 𝑆)) ↔ (𝑦𝐵 ∧ ( 𝑦) = 𝑆)))
4134, 40bitrd 279 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝑆𝐵) → (𝑦 = ( 𝑆) ↔ (𝑦𝐵 ∧ ( 𝑦) = 𝑆)))
4228, 41sylan2 592 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ (∀𝑖𝐼 𝑆𝐵𝑖𝐼)) → (𝑦 = ( 𝑆) ↔ (𝑦𝐵 ∧ ( 𝑦) = 𝑆)))
4342anassrs 467 . . . . . . . . 9 (((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) ∧ 𝑖𝐼) → (𝑦 = ( 𝑆) ↔ (𝑦𝐵 ∧ ( 𝑦) = 𝑆)))
4427, 43rexbida 3268 . . . . . . . 8 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) → (∃𝑖𝐼 𝑦 = ( 𝑆) ↔ ∃𝑖𝐼 (𝑦𝐵 ∧ ( 𝑦) = 𝑆)))
45 r19.42v 3189 . . . . . . . 8 (∃𝑖𝐼 (𝑦𝐵 ∧ ( 𝑦) = 𝑆) ↔ (𝑦𝐵 ∧ ∃𝑖𝐼 ( 𝑦) = 𝑆))
4644, 45bitr2di 288 . . . . . . 7 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) → ((𝑦𝐵 ∧ ∃𝑖𝐼 ( 𝑦) = 𝑆) ↔ ∃𝑖𝐼 𝑦 = ( 𝑆)))
4746abbidv 2800 . . . . . 6 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) → {𝑦 ∣ (𝑦𝐵 ∧ ∃𝑖𝐼 ( 𝑦) = 𝑆)} = {𝑦 ∣ ∃𝑖𝐼 𝑦 = ( 𝑆)})
48 eqeq1 2735 . . . . . . . 8 (𝑦 = 𝑥 → (𝑦 = ( 𝑆) ↔ 𝑥 = ( 𝑆)))
4948rexbidv 3177 . . . . . . 7 (𝑦 = 𝑥 → (∃𝑖𝐼 𝑦 = ( 𝑆) ↔ ∃𝑖𝐼 𝑥 = ( 𝑆)))
5049cbvabv 2804 . . . . . 6 {𝑦 ∣ ∃𝑖𝐼 𝑦 = ( 𝑆)} = {𝑥 ∣ ∃𝑖𝐼 𝑥 = ( 𝑆)}
5147, 50eqtrdi 2787 . . . . 5 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) → {𝑦 ∣ (𝑦𝐵 ∧ ∃𝑖𝐼 ( 𝑦) = 𝑆)} = {𝑥 ∣ ∃𝑖𝐼 𝑥 = ( 𝑆)})
5225, 51eqtrid 2783 . . . 4 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) → {𝑦𝐵 ∣ ( 𝑦) ∈ {𝑥 ∣ ∃𝑖𝐼 𝑥 = 𝑆}} = {𝑥 ∣ ∃𝑖𝐼 𝑥 = ( 𝑆)})
5352fveq2d 6895 . . 3 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) → (𝑈‘{𝑦𝐵 ∣ ( 𝑦) ∈ {𝑥 ∣ ∃𝑖𝐼 𝑥 = 𝑆}}) = (𝑈‘{𝑥 ∣ ∃𝑖𝐼 𝑥 = ( 𝑆)}))
5453fveq2d 6895 . 2 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) → ( ‘(𝑈‘{𝑦𝐵 ∣ ( 𝑦) ∈ {𝑥 ∣ ∃𝑖𝐼 𝑥 = 𝑆}})) = ( ‘(𝑈‘{𝑥 ∣ ∃𝑖𝐼 𝑥 = ( 𝑆)})))
5518, 54eqtrd 2771 1 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) → (𝐺‘{𝑥 ∣ ∃𝑖𝐼 𝑥 = 𝑆}) = ( ‘(𝑈‘{𝑥 ∣ ∃𝑖𝐼 𝑥 = ( 𝑆)})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1540  wcel 2105  {cab 2708  wral 3060  wrex 3069  {crab 3431  wss 3948  cfv 6543  Basecbs 17151  occoc 17212  lubclub 18269  glbcglb 18270  OPcops 38358  HLchlt 38536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  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 7368  df-ov 7415  df-lub 18306  df-glb 18307  df-clat 18459  df-oposet 38362  df-ol 38364  df-oml 38365  df-hlat 38537
This theorem is referenced by:  polval2N  39093
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