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Theorem glbconxN 35452
Description: De Morgan's law for GLB and LUB. Index-set version of glbconN 35451, 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 3417 . . . . . 6 𝑦 ∈ V
2 eqeq1 2829 . . . . . . 7 (𝑥 = 𝑦 → (𝑥 = 𝑆𝑦 = 𝑆))
32rexbidv 3262 . . . . . 6 (𝑥 = 𝑦 → (∃𝑖𝐼 𝑥 = 𝑆 ↔ ∃𝑖𝐼 𝑦 = 𝑆))
41, 3elab 3571 . . . . 5 (𝑦 ∈ {𝑥 ∣ ∃𝑖𝐼 𝑥 = 𝑆} ↔ ∃𝑖𝐼 𝑦 = 𝑆)
5 nfra1 3150 . . . . . 6 𝑖𝑖𝐼 𝑆𝐵
6 nfv 2013 . . . . . 6 𝑖 𝑦𝐵
7 rsp 3138 . . . . . . 7 (∀𝑖𝐼 𝑆𝐵 → (𝑖𝐼𝑆𝐵))
8 eleq1a 2901 . . . . . . 7 (𝑆𝐵 → (𝑦 = 𝑆𝑦𝐵))
97, 8syl6 35 . . . . . 6 (∀𝑖𝐼 𝑆𝐵 → (𝑖𝐼 → (𝑦 = 𝑆𝑦𝐵)))
105, 6, 9rexlimd 3235 . . . . 5 (∀𝑖𝐼 𝑆𝐵 → (∃𝑖𝐼 𝑦 = 𝑆𝑦𝐵))
114, 10syl5bi 234 . . . 4 (∀𝑖𝐼 𝑆𝐵 → (𝑦 ∈ {𝑥 ∣ ∃𝑖𝐼 𝑥 = 𝑆} → 𝑦𝐵))
1211ssrdv 3833 . . 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 35451 . . 3 ((𝐾 ∈ HL ∧ {𝑥 ∣ ∃𝑖𝐼 𝑥 = 𝑆} ⊆ 𝐵) → (𝐺‘{𝑥 ∣ ∃𝑖𝐼 𝑥 = 𝑆}) = ( ‘(𝑈‘{𝑦𝐵 ∣ ( 𝑦) ∈ {𝑥 ∣ ∃𝑖𝐼 𝑥 = 𝑆}})))
1812, 17sylan2 586 . 2 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) → (𝐺‘{𝑥 ∣ ∃𝑖𝐼 𝑥 = 𝑆}) = ( ‘(𝑈‘{𝑦𝐵 ∣ ( 𝑦) ∈ {𝑥 ∣ ∃𝑖𝐼 𝑥 = 𝑆}})))
19 fvex 6450 . . . . . . . 8 ( 𝑦) ∈ V
20 eqeq1 2829 . . . . . . . . 9 (𝑥 = ( 𝑦) → (𝑥 = 𝑆 ↔ ( 𝑦) = 𝑆))
2120rexbidv 3262 . . . . . . . 8 (𝑥 = ( 𝑦) → (∃𝑖𝐼 𝑥 = 𝑆 ↔ ∃𝑖𝐼 ( 𝑦) = 𝑆))
2219, 21elab 3571 . . . . . . 7 (( 𝑦) ∈ {𝑥 ∣ ∃𝑖𝐼 𝑥 = 𝑆} ↔ ∃𝑖𝐼 ( 𝑦) = 𝑆)
2322rabbii 3398 . . . . . 6 {𝑦𝐵 ∣ ( 𝑦) ∈ {𝑥 ∣ ∃𝑖𝐼 𝑥 = 𝑆}} = {𝑦𝐵 ∣ ∃𝑖𝐼 ( 𝑦) = 𝑆}
24 df-rab 3126 . . . . . 6 {𝑦𝐵 ∣ ∃𝑖𝐼 ( 𝑦) = 𝑆} = {𝑦 ∣ (𝑦𝐵 ∧ ∃𝑖𝐼 ( 𝑦) = 𝑆)}
2523, 24eqtri 2849 . . . . 5 {𝑦𝐵 ∣ ( 𝑦) ∈ {𝑥 ∣ ∃𝑖𝐼 𝑥 = 𝑆}} = {𝑦 ∣ (𝑦𝐵 ∧ ∃𝑖𝐼 ( 𝑦) = 𝑆)}
26 nfv 2013 . . . . . . . . . 10 𝑖 𝐾 ∈ HL
2726, 5nfan 2002 . . . . . . . . 9 𝑖(𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵)
28 rspa 3139 . . . . . . . . . . 11 ((∀𝑖𝐼 𝑆𝐵𝑖𝐼) → 𝑆𝐵)
29 hlop 35436 . . . . . . . . . . . . . . 15 (𝐾 ∈ HL → 𝐾 ∈ OP)
3013, 16opoccl 35268 . . . . . . . . . . . . . . 15 ((𝐾 ∈ OP ∧ 𝑆𝐵) → ( 𝑆) ∈ 𝐵)
3129, 30sylan 575 . . . . . . . . . . . . . 14 ((𝐾 ∈ HL ∧ 𝑆𝐵) → ( 𝑆) ∈ 𝐵)
32 eleq1a 2901 . . . . . . . . . . . . . 14 (( 𝑆) ∈ 𝐵 → (𝑦 = ( 𝑆) → 𝑦𝐵))
3331, 32syl 17 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ 𝑆𝐵) → (𝑦 = ( 𝑆) → 𝑦𝐵))
3433pm4.71rd 558 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑆𝐵) → (𝑦 = ( 𝑆) ↔ (𝑦𝐵𝑦 = ( 𝑆))))
35 eqcom 2832 . . . . . . . . . . . . . 14 (𝑆 = ( 𝑦) ↔ ( 𝑦) = 𝑆)
3613, 16opcon2b 35271 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ OP ∧ 𝑆𝐵𝑦𝐵) → (𝑆 = ( 𝑦) ↔ 𝑦 = ( 𝑆)))
3729, 36syl3an1 1206 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ 𝑆𝐵𝑦𝐵) → (𝑆 = ( 𝑦) ↔ 𝑦 = ( 𝑆)))
38373expa 1151 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑆𝐵) ∧ 𝑦𝐵) → (𝑆 = ( 𝑦) ↔ 𝑦 = ( 𝑆)))
3935, 38syl5rbbr 278 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ 𝑆𝐵) ∧ 𝑦𝐵) → (𝑦 = ( 𝑆) ↔ ( 𝑦) = 𝑆))
4039pm5.32da 574 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑆𝐵) → ((𝑦𝐵𝑦 = ( 𝑆)) ↔ (𝑦𝐵 ∧ ( 𝑦) = 𝑆)))
4134, 40bitrd 271 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝑆𝐵) → (𝑦 = ( 𝑆) ↔ (𝑦𝐵 ∧ ( 𝑦) = 𝑆)))
4228, 41sylan2 586 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ (∀𝑖𝐼 𝑆𝐵𝑖𝐼)) → (𝑦 = ( 𝑆) ↔ (𝑦𝐵 ∧ ( 𝑦) = 𝑆)))
4342anassrs 461 . . . . . . . . 9 (((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) ∧ 𝑖𝐼) → (𝑦 = ( 𝑆) ↔ (𝑦𝐵 ∧ ( 𝑦) = 𝑆)))
4427, 43rexbida 3257 . . . . . . . 8 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) → (∃𝑖𝐼 𝑦 = ( 𝑆) ↔ ∃𝑖𝐼 (𝑦𝐵 ∧ ( 𝑦) = 𝑆)))
45 r19.42v 3302 . . . . . . . 8 (∃𝑖𝐼 (𝑦𝐵 ∧ ( 𝑦) = 𝑆) ↔ (𝑦𝐵 ∧ ∃𝑖𝐼 ( 𝑦) = 𝑆))
4644, 45syl6rbb 280 . . . . . . 7 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) → ((𝑦𝐵 ∧ ∃𝑖𝐼 ( 𝑦) = 𝑆) ↔ ∃𝑖𝐼 𝑦 = ( 𝑆)))
4746abbidv 2946 . . . . . 6 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) → {𝑦 ∣ (𝑦𝐵 ∧ ∃𝑖𝐼 ( 𝑦) = 𝑆)} = {𝑦 ∣ ∃𝑖𝐼 𝑦 = ( 𝑆)})
48 eqeq1 2829 . . . . . . . 8 (𝑦 = 𝑥 → (𝑦 = ( 𝑆) ↔ 𝑥 = ( 𝑆)))
4948rexbidv 3262 . . . . . . 7 (𝑦 = 𝑥 → (∃𝑖𝐼 𝑦 = ( 𝑆) ↔ ∃𝑖𝐼 𝑥 = ( 𝑆)))
5049cbvabv 2952 . . . . . 6 {𝑦 ∣ ∃𝑖𝐼 𝑦 = ( 𝑆)} = {𝑥 ∣ ∃𝑖𝐼 𝑥 = ( 𝑆)}
5147, 50syl6eq 2877 . . . . 5 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) → {𝑦 ∣ (𝑦𝐵 ∧ ∃𝑖𝐼 ( 𝑦) = 𝑆)} = {𝑥 ∣ ∃𝑖𝐼 𝑥 = ( 𝑆)})
5225, 51syl5eq 2873 . . . 4 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) → {𝑦𝐵 ∣ ( 𝑦) ∈ {𝑥 ∣ ∃𝑖𝐼 𝑥 = 𝑆}} = {𝑥 ∣ ∃𝑖𝐼 𝑥 = ( 𝑆)})
5352fveq2d 6441 . . 3 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) → (𝑈‘{𝑦𝐵 ∣ ( 𝑦) ∈ {𝑥 ∣ ∃𝑖𝐼 𝑥 = 𝑆}}) = (𝑈‘{𝑥 ∣ ∃𝑖𝐼 𝑥 = ( 𝑆)}))
5453fveq2d 6441 . 2 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) → ( ‘(𝑈‘{𝑦𝐵 ∣ ( 𝑦) ∈ {𝑥 ∣ ∃𝑖𝐼 𝑥 = 𝑆}})) = ( ‘(𝑈‘{𝑥 ∣ ∃𝑖𝐼 𝑥 = ( 𝑆)})))
5518, 54eqtrd 2861 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 198  wa 386   = wceq 1656  wcel 2164  {cab 2811  wral 3117  wrex 3118  {crab 3121  wss 3798  cfv 6127  Basecbs 16229  occoc 16320  lubclub 17302  glbcglb 17303  OPcops 35246  HLchlt 35424
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214  ax-riotaBAD 35027
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-riota 6871  df-ov 6913  df-undef 7669  df-lub 17334  df-glb 17335  df-clat 17468  df-oposet 35250  df-ol 35252  df-oml 35253  df-hlat 35425
This theorem is referenced by:  polval2N  35980
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