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Theorem glbconxN 37841
Description: De Morgan's law for GLB and LUB. Index-set version of glbconN 37839, 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 3449 . . . . . 6 𝑦 ∈ V
2 eqeq1 2740 . . . . . . 7 (𝑥 = 𝑦 → (𝑥 = 𝑆𝑦 = 𝑆))
32rexbidv 3175 . . . . . 6 (𝑥 = 𝑦 → (∃𝑖𝐼 𝑥 = 𝑆 ↔ ∃𝑖𝐼 𝑦 = 𝑆))
41, 3elab 3630 . . . . 5 (𝑦 ∈ {𝑥 ∣ ∃𝑖𝐼 𝑥 = 𝑆} ↔ ∃𝑖𝐼 𝑦 = 𝑆)
5 nfra1 3267 . . . . . 6 𝑖𝑖𝐼 𝑆𝐵
6 nfv 1917 . . . . . 6 𝑖 𝑦𝐵
7 rsp 3230 . . . . . . 7 (∀𝑖𝐼 𝑆𝐵 → (𝑖𝐼𝑆𝐵))
8 eleq1a 2833 . . . . . . 7 (𝑆𝐵 → (𝑦 = 𝑆𝑦𝐵))
97, 8syl6 35 . . . . . 6 (∀𝑖𝐼 𝑆𝐵 → (𝑖𝐼 → (𝑦 = 𝑆𝑦𝐵)))
105, 6, 9rexlimd 3249 . . . . 5 (∀𝑖𝐼 𝑆𝐵 → (∃𝑖𝐼 𝑦 = 𝑆𝑦𝐵))
114, 10biimtrid 241 . . . 4 (∀𝑖𝐼 𝑆𝐵 → (𝑦 ∈ {𝑥 ∣ ∃𝑖𝐼 𝑥 = 𝑆} → 𝑦𝐵))
1211ssrdv 3950 . . 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 37839 . . 3 ((𝐾 ∈ HL ∧ {𝑥 ∣ ∃𝑖𝐼 𝑥 = 𝑆} ⊆ 𝐵) → (𝐺‘{𝑥 ∣ ∃𝑖𝐼 𝑥 = 𝑆}) = ( ‘(𝑈‘{𝑦𝐵 ∣ ( 𝑦) ∈ {𝑥 ∣ ∃𝑖𝐼 𝑥 = 𝑆}})))
1812, 17sylan2 593 . 2 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) → (𝐺‘{𝑥 ∣ ∃𝑖𝐼 𝑥 = 𝑆}) = ( ‘(𝑈‘{𝑦𝐵 ∣ ( 𝑦) ∈ {𝑥 ∣ ∃𝑖𝐼 𝑥 = 𝑆}})))
19 fvex 6855 . . . . . . . 8 ( 𝑦) ∈ V
20 eqeq1 2740 . . . . . . . . 9 (𝑥 = ( 𝑦) → (𝑥 = 𝑆 ↔ ( 𝑦) = 𝑆))
2120rexbidv 3175 . . . . . . . 8 (𝑥 = ( 𝑦) → (∃𝑖𝐼 𝑥 = 𝑆 ↔ ∃𝑖𝐼 ( 𝑦) = 𝑆))
2219, 21elab 3630 . . . . . . 7 (( 𝑦) ∈ {𝑥 ∣ ∃𝑖𝐼 𝑥 = 𝑆} ↔ ∃𝑖𝐼 ( 𝑦) = 𝑆)
2322rabbii 3413 . . . . . 6 {𝑦𝐵 ∣ ( 𝑦) ∈ {𝑥 ∣ ∃𝑖𝐼 𝑥 = 𝑆}} = {𝑦𝐵 ∣ ∃𝑖𝐼 ( 𝑦) = 𝑆}
24 df-rab 3408 . . . . . 6 {𝑦𝐵 ∣ ∃𝑖𝐼 ( 𝑦) = 𝑆} = {𝑦 ∣ (𝑦𝐵 ∧ ∃𝑖𝐼 ( 𝑦) = 𝑆)}
2523, 24eqtri 2764 . . . . 5 {𝑦𝐵 ∣ ( 𝑦) ∈ {𝑥 ∣ ∃𝑖𝐼 𝑥 = 𝑆}} = {𝑦 ∣ (𝑦𝐵 ∧ ∃𝑖𝐼 ( 𝑦) = 𝑆)}
26 nfv 1917 . . . . . . . . . 10 𝑖 𝐾 ∈ HL
2726, 5nfan 1902 . . . . . . . . 9 𝑖(𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵)
28 rspa 3231 . . . . . . . . . . 11 ((∀𝑖𝐼 𝑆𝐵𝑖𝐼) → 𝑆𝐵)
29 hlop 37824 . . . . . . . . . . . . . . 15 (𝐾 ∈ HL → 𝐾 ∈ OP)
3013, 16opoccl 37656 . . . . . . . . . . . . . . 15 ((𝐾 ∈ OP ∧ 𝑆𝐵) → ( 𝑆) ∈ 𝐵)
3129, 30sylan 580 . . . . . . . . . . . . . 14 ((𝐾 ∈ HL ∧ 𝑆𝐵) → ( 𝑆) ∈ 𝐵)
32 eleq1a 2833 . . . . . . . . . . . . . 14 (( 𝑆) ∈ 𝐵 → (𝑦 = ( 𝑆) → 𝑦𝐵))
3331, 32syl 17 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ 𝑆𝐵) → (𝑦 = ( 𝑆) → 𝑦𝐵))
3433pm4.71rd 563 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑆𝐵) → (𝑦 = ( 𝑆) ↔ (𝑦𝐵𝑦 = ( 𝑆))))
3513, 16opcon2b 37659 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ OP ∧ 𝑆𝐵𝑦𝐵) → (𝑆 = ( 𝑦) ↔ 𝑦 = ( 𝑆)))
3629, 35syl3an1 1163 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ 𝑆𝐵𝑦𝐵) → (𝑆 = ( 𝑦) ↔ 𝑦 = ( 𝑆)))
37363expa 1118 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑆𝐵) ∧ 𝑦𝐵) → (𝑆 = ( 𝑦) ↔ 𝑦 = ( 𝑆)))
38 eqcom 2743 . . . . . . . . . . . . . 14 (𝑆 = ( 𝑦) ↔ ( 𝑦) = 𝑆)
3937, 38bitr3di 285 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ 𝑆𝐵) ∧ 𝑦𝐵) → (𝑦 = ( 𝑆) ↔ ( 𝑦) = 𝑆))
4039pm5.32da 579 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑆𝐵) → ((𝑦𝐵𝑦 = ( 𝑆)) ↔ (𝑦𝐵 ∧ ( 𝑦) = 𝑆)))
4134, 40bitrd 278 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝑆𝐵) → (𝑦 = ( 𝑆) ↔ (𝑦𝐵 ∧ ( 𝑦) = 𝑆)))
4228, 41sylan2 593 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ (∀𝑖𝐼 𝑆𝐵𝑖𝐼)) → (𝑦 = ( 𝑆) ↔ (𝑦𝐵 ∧ ( 𝑦) = 𝑆)))
4342anassrs 468 . . . . . . . . 9 (((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) ∧ 𝑖𝐼) → (𝑦 = ( 𝑆) ↔ (𝑦𝐵 ∧ ( 𝑦) = 𝑆)))
4427, 43rexbida 3255 . . . . . . . 8 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) → (∃𝑖𝐼 𝑦 = ( 𝑆) ↔ ∃𝑖𝐼 (𝑦𝐵 ∧ ( 𝑦) = 𝑆)))
45 r19.42v 3187 . . . . . . . 8 (∃𝑖𝐼 (𝑦𝐵 ∧ ( 𝑦) = 𝑆) ↔ (𝑦𝐵 ∧ ∃𝑖𝐼 ( 𝑦) = 𝑆))
4644, 45bitr2di 287 . . . . . . 7 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) → ((𝑦𝐵 ∧ ∃𝑖𝐼 ( 𝑦) = 𝑆) ↔ ∃𝑖𝐼 𝑦 = ( 𝑆)))
4746abbidv 2805 . . . . . 6 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) → {𝑦 ∣ (𝑦𝐵 ∧ ∃𝑖𝐼 ( 𝑦) = 𝑆)} = {𝑦 ∣ ∃𝑖𝐼 𝑦 = ( 𝑆)})
48 eqeq1 2740 . . . . . . . 8 (𝑦 = 𝑥 → (𝑦 = ( 𝑆) ↔ 𝑥 = ( 𝑆)))
4948rexbidv 3175 . . . . . . 7 (𝑦 = 𝑥 → (∃𝑖𝐼 𝑦 = ( 𝑆) ↔ ∃𝑖𝐼 𝑥 = ( 𝑆)))
5049cbvabv 2809 . . . . . 6 {𝑦 ∣ ∃𝑖𝐼 𝑦 = ( 𝑆)} = {𝑥 ∣ ∃𝑖𝐼 𝑥 = ( 𝑆)}
5147, 50eqtrdi 2792 . . . . 5 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) → {𝑦 ∣ (𝑦𝐵 ∧ ∃𝑖𝐼 ( 𝑦) = 𝑆)} = {𝑥 ∣ ∃𝑖𝐼 𝑥 = ( 𝑆)})
5225, 51eqtrid 2788 . . . 4 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) → {𝑦𝐵 ∣ ( 𝑦) ∈ {𝑥 ∣ ∃𝑖𝐼 𝑥 = 𝑆}} = {𝑥 ∣ ∃𝑖𝐼 𝑥 = ( 𝑆)})
5352fveq2d 6846 . . 3 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) → (𝑈‘{𝑦𝐵 ∣ ( 𝑦) ∈ {𝑥 ∣ ∃𝑖𝐼 𝑥 = 𝑆}}) = (𝑈‘{𝑥 ∣ ∃𝑖𝐼 𝑥 = ( 𝑆)}))
5453fveq2d 6846 . 2 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) → ( ‘(𝑈‘{𝑦𝐵 ∣ ( 𝑦) ∈ {𝑥 ∣ ∃𝑖𝐼 𝑥 = 𝑆}})) = ( ‘(𝑈‘{𝑥 ∣ ∃𝑖𝐼 𝑥 = ( 𝑆)})))
5518, 54eqtrd 2776 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 396   = wceq 1541  wcel 2106  {cab 2713  wral 3064  wrex 3073  {crab 3407  wss 3910  cfv 6496  Basecbs 17083  occoc 17141  lubclub 18198  glbcglb 18199  OPcops 37634  HLchlt 37812
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
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  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-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-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  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-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-lub 18235  df-glb 18236  df-clat 18388  df-oposet 37638  df-ol 37640  df-oml 37641  df-hlat 37813
This theorem is referenced by:  polval2N  38369
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