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Theorem glbconxN 36506
Description: De Morgan's law for GLB and LUB. Index-set version of glbconN 36505, 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 3496 . . . . . 6 𝑦 ∈ V
2 eqeq1 2823 . . . . . . 7 (𝑥 = 𝑦 → (𝑥 = 𝑆𝑦 = 𝑆))
32rexbidv 3295 . . . . . 6 (𝑥 = 𝑦 → (∃𝑖𝐼 𝑥 = 𝑆 ↔ ∃𝑖𝐼 𝑦 = 𝑆))
41, 3elab 3665 . . . . 5 (𝑦 ∈ {𝑥 ∣ ∃𝑖𝐼 𝑥 = 𝑆} ↔ ∃𝑖𝐼 𝑦 = 𝑆)
5 nfra1 3217 . . . . . 6 𝑖𝑖𝐼 𝑆𝐵
6 nfv 1908 . . . . . 6 𝑖 𝑦𝐵
7 rsp 3203 . . . . . . 7 (∀𝑖𝐼 𝑆𝐵 → (𝑖𝐼𝑆𝐵))
8 eleq1a 2906 . . . . . . 7 (𝑆𝐵 → (𝑦 = 𝑆𝑦𝐵))
97, 8syl6 35 . . . . . 6 (∀𝑖𝐼 𝑆𝐵 → (𝑖𝐼 → (𝑦 = 𝑆𝑦𝐵)))
105, 6, 9rexlimd 3315 . . . . 5 (∀𝑖𝐼 𝑆𝐵 → (∃𝑖𝐼 𝑦 = 𝑆𝑦𝐵))
114, 10syl5bi 244 . . . 4 (∀𝑖𝐼 𝑆𝐵 → (𝑦 ∈ {𝑥 ∣ ∃𝑖𝐼 𝑥 = 𝑆} → 𝑦𝐵))
1211ssrdv 3971 . . 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 36505 . . 3 ((𝐾 ∈ HL ∧ {𝑥 ∣ ∃𝑖𝐼 𝑥 = 𝑆} ⊆ 𝐵) → (𝐺‘{𝑥 ∣ ∃𝑖𝐼 𝑥 = 𝑆}) = ( ‘(𝑈‘{𝑦𝐵 ∣ ( 𝑦) ∈ {𝑥 ∣ ∃𝑖𝐼 𝑥 = 𝑆}})))
1812, 17sylan2 594 . 2 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) → (𝐺‘{𝑥 ∣ ∃𝑖𝐼 𝑥 = 𝑆}) = ( ‘(𝑈‘{𝑦𝐵 ∣ ( 𝑦) ∈ {𝑥 ∣ ∃𝑖𝐼 𝑥 = 𝑆}})))
19 fvex 6676 . . . . . . . 8 ( 𝑦) ∈ V
20 eqeq1 2823 . . . . . . . . 9 (𝑥 = ( 𝑦) → (𝑥 = 𝑆 ↔ ( 𝑦) = 𝑆))
2120rexbidv 3295 . . . . . . . 8 (𝑥 = ( 𝑦) → (∃𝑖𝐼 𝑥 = 𝑆 ↔ ∃𝑖𝐼 ( 𝑦) = 𝑆))
2219, 21elab 3665 . . . . . . 7 (( 𝑦) ∈ {𝑥 ∣ ∃𝑖𝐼 𝑥 = 𝑆} ↔ ∃𝑖𝐼 ( 𝑦) = 𝑆)
2322rabbii 3472 . . . . . 6 {𝑦𝐵 ∣ ( 𝑦) ∈ {𝑥 ∣ ∃𝑖𝐼 𝑥 = 𝑆}} = {𝑦𝐵 ∣ ∃𝑖𝐼 ( 𝑦) = 𝑆}
24 df-rab 3145 . . . . . 6 {𝑦𝐵 ∣ ∃𝑖𝐼 ( 𝑦) = 𝑆} = {𝑦 ∣ (𝑦𝐵 ∧ ∃𝑖𝐼 ( 𝑦) = 𝑆)}
2523, 24eqtri 2842 . . . . 5 {𝑦𝐵 ∣ ( 𝑦) ∈ {𝑥 ∣ ∃𝑖𝐼 𝑥 = 𝑆}} = {𝑦 ∣ (𝑦𝐵 ∧ ∃𝑖𝐼 ( 𝑦) = 𝑆)}
26 nfv 1908 . . . . . . . . . 10 𝑖 𝐾 ∈ HL
2726, 5nfan 1893 . . . . . . . . 9 𝑖(𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵)
28 rspa 3204 . . . . . . . . . . 11 ((∀𝑖𝐼 𝑆𝐵𝑖𝐼) → 𝑆𝐵)
29 hlop 36490 . . . . . . . . . . . . . . 15 (𝐾 ∈ HL → 𝐾 ∈ OP)
3013, 16opoccl 36322 . . . . . . . . . . . . . . 15 ((𝐾 ∈ OP ∧ 𝑆𝐵) → ( 𝑆) ∈ 𝐵)
3129, 30sylan 582 . . . . . . . . . . . . . 14 ((𝐾 ∈ HL ∧ 𝑆𝐵) → ( 𝑆) ∈ 𝐵)
32 eleq1a 2906 . . . . . . . . . . . . . 14 (( 𝑆) ∈ 𝐵 → (𝑦 = ( 𝑆) → 𝑦𝐵))
3331, 32syl 17 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ 𝑆𝐵) → (𝑦 = ( 𝑆) → 𝑦𝐵))
3433pm4.71rd 565 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑆𝐵) → (𝑦 = ( 𝑆) ↔ (𝑦𝐵𝑦 = ( 𝑆))))
35 eqcom 2826 . . . . . . . . . . . . . 14 (𝑆 = ( 𝑦) ↔ ( 𝑦) = 𝑆)
3613, 16opcon2b 36325 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ OP ∧ 𝑆𝐵𝑦𝐵) → (𝑆 = ( 𝑦) ↔ 𝑦 = ( 𝑆)))
3729, 36syl3an1 1157 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ 𝑆𝐵𝑦𝐵) → (𝑆 = ( 𝑦) ↔ 𝑦 = ( 𝑆)))
38373expa 1112 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑆𝐵) ∧ 𝑦𝐵) → (𝑆 = ( 𝑦) ↔ 𝑦 = ( 𝑆)))
3935, 38syl5rbbr 288 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ 𝑆𝐵) ∧ 𝑦𝐵) → (𝑦 = ( 𝑆) ↔ ( 𝑦) = 𝑆))
4039pm5.32da 581 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑆𝐵) → ((𝑦𝐵𝑦 = ( 𝑆)) ↔ (𝑦𝐵 ∧ ( 𝑦) = 𝑆)))
4134, 40bitrd 281 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝑆𝐵) → (𝑦 = ( 𝑆) ↔ (𝑦𝐵 ∧ ( 𝑦) = 𝑆)))
4228, 41sylan2 594 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ (∀𝑖𝐼 𝑆𝐵𝑖𝐼)) → (𝑦 = ( 𝑆) ↔ (𝑦𝐵 ∧ ( 𝑦) = 𝑆)))
4342anassrs 470 . . . . . . . . 9 (((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) ∧ 𝑖𝐼) → (𝑦 = ( 𝑆) ↔ (𝑦𝐵 ∧ ( 𝑦) = 𝑆)))
4427, 43rexbida 3316 . . . . . . . 8 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) → (∃𝑖𝐼 𝑦 = ( 𝑆) ↔ ∃𝑖𝐼 (𝑦𝐵 ∧ ( 𝑦) = 𝑆)))
45 r19.42v 3348 . . . . . . . 8 (∃𝑖𝐼 (𝑦𝐵 ∧ ( 𝑦) = 𝑆) ↔ (𝑦𝐵 ∧ ∃𝑖𝐼 ( 𝑦) = 𝑆))
4644, 45syl6rbb 290 . . . . . . 7 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) → ((𝑦𝐵 ∧ ∃𝑖𝐼 ( 𝑦) = 𝑆) ↔ ∃𝑖𝐼 𝑦 = ( 𝑆)))
4746abbidv 2883 . . . . . 6 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) → {𝑦 ∣ (𝑦𝐵 ∧ ∃𝑖𝐼 ( 𝑦) = 𝑆)} = {𝑦 ∣ ∃𝑖𝐼 𝑦 = ( 𝑆)})
48 eqeq1 2823 . . . . . . . 8 (𝑦 = 𝑥 → (𝑦 = ( 𝑆) ↔ 𝑥 = ( 𝑆)))
4948rexbidv 3295 . . . . . . 7 (𝑦 = 𝑥 → (∃𝑖𝐼 𝑦 = ( 𝑆) ↔ ∃𝑖𝐼 𝑥 = ( 𝑆)))
5049cbvabv 2887 . . . . . 6 {𝑦 ∣ ∃𝑖𝐼 𝑦 = ( 𝑆)} = {𝑥 ∣ ∃𝑖𝐼 𝑥 = ( 𝑆)}
5147, 50syl6eq 2870 . . . . 5 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) → {𝑦 ∣ (𝑦𝐵 ∧ ∃𝑖𝐼 ( 𝑦) = 𝑆)} = {𝑥 ∣ ∃𝑖𝐼 𝑥 = ( 𝑆)})
5225, 51syl5eq 2866 . . . 4 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) → {𝑦𝐵 ∣ ( 𝑦) ∈ {𝑥 ∣ ∃𝑖𝐼 𝑥 = 𝑆}} = {𝑥 ∣ ∃𝑖𝐼 𝑥 = ( 𝑆)})
5352fveq2d 6667 . . 3 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) → (𝑈‘{𝑦𝐵 ∣ ( 𝑦) ∈ {𝑥 ∣ ∃𝑖𝐼 𝑥 = 𝑆}}) = (𝑈‘{𝑥 ∣ ∃𝑖𝐼 𝑥 = ( 𝑆)}))
5453fveq2d 6667 . 2 ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) → ( ‘(𝑈‘{𝑦𝐵 ∣ ( 𝑦) ∈ {𝑥 ∣ ∃𝑖𝐼 𝑥 = 𝑆}})) = ( ‘(𝑈‘{𝑥 ∣ ∃𝑖𝐼 𝑥 = ( 𝑆)})))
5518, 54eqtrd 2854 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 208  wa 398   = wceq 1530  wcel 2107  {cab 2797  wral 3136  wrex 3137  {crab 3140  wss 3934  cfv 6348  Basecbs 16475  occoc 16565  lubclub 17544  glbcglb 17545  OPcops 36300  HLchlt 36478
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-riotaBAD 36081
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7106  df-ov 7151  df-undef 7931  df-lub 17576  df-glb 17577  df-clat 17710  df-oposet 36304  df-ol 36306  df-oml 36307  df-hlat 36479
This theorem is referenced by:  polval2N  37034
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