Theorem glbconxN 39381

Description: De Morgan's law for GLB and LUB. Index-set version of glbconN 39379, 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 ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵) → (𝐺‘{𝑥 ∣ ∃𝑖 ∈ 𝐼 𝑥 = 𝑆}) = ( ⊥ ‘(𝑈‘{𝑥 ∣ ∃𝑖 ∈ 𝐼 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.

Step	Hyp	Ref	Expression
1		vex 3483	. . . . . 6 ⊢ 𝑦 ∈ V
2		eqeq1 2740	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑆 ↔ 𝑦 = 𝑆))
3	2	rexbidv 3178	. . . . . 6 ⊢ (𝑥 = 𝑦 → (∃𝑖 ∈ 𝐼 𝑥 = 𝑆 ↔ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆))
4	1, 3	elab 3678	. . . . 5 ⊢ (𝑦 ∈ {𝑥 ∣ ∃𝑖 ∈ 𝐼 𝑥 = 𝑆} ↔ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆)
5		nfra1 3283	. . . . . 6 ⊢ Ⅎ𝑖∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵
6		nfv 1913	. . . . . 6 ⊢ Ⅎ𝑖 𝑦 ∈ 𝐵
7		rsp 3246	. . . . . . 7 ⊢ (∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 → (𝑖 ∈ 𝐼 → 𝑆 ∈ 𝐵))
8		eleq1a 2835	. . . . . . 7 ⊢ (𝑆 ∈ 𝐵 → (𝑦 = 𝑆 → 𝑦 ∈ 𝐵))
9	7, 8	syl6 35	. . . . . 6 ⊢ (∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 → (𝑖 ∈ 𝐼 → (𝑦 = 𝑆 → 𝑦 ∈ 𝐵)))
10	5, 6, 9	rexlimd 3265	. . . . 5 ⊢ (∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 → (∃𝑖 ∈ 𝐼 𝑦 = 𝑆 → 𝑦 ∈ 𝐵))
11	4, 10	biimtrid 242	. . . 4 ⊢ (∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 → (𝑦 ∈ {𝑥 ∣ ∃𝑖 ∈ 𝐼 𝑥 = 𝑆} → 𝑦 ∈ 𝐵))
12	11	ssrdv 3988	. . 3 ⊢ (∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 → {𝑥 ∣ ∃𝑖 ∈ 𝐼 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)
17	13, 14, 15, 16	glbconN 39379	. . 3 ⊢ ((𝐾 ∈ HL ∧ {𝑥 ∣ ∃𝑖 ∈ 𝐼 𝑥 = 𝑆} ⊆ 𝐵) → (𝐺‘{𝑥 ∣ ∃𝑖 ∈ 𝐼 𝑥 = 𝑆}) = ( ⊥ ‘(𝑈‘{𝑦 ∈ 𝐵 ∣ ( ⊥ ‘𝑦) ∈ {𝑥 ∣ ∃𝑖 ∈ 𝐼 𝑥 = 𝑆}})))
18	12, 17	sylan2 593	. 2 ⊢ ((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵) → (𝐺‘{𝑥 ∣ ∃𝑖 ∈ 𝐼 𝑥 = 𝑆}) = ( ⊥ ‘(𝑈‘{𝑦 ∈ 𝐵 ∣ ( ⊥ ‘𝑦) ∈ {𝑥 ∣ ∃𝑖 ∈ 𝐼 𝑥 = 𝑆}})))
19		fvex 6918	. . . . . . . 8 ⊢ ( ⊥ ‘𝑦) ∈ V
20		eqeq1 2740	. . . . . . . . 9 ⊢ (𝑥 = ( ⊥ ‘𝑦) → (𝑥 = 𝑆 ↔ ( ⊥ ‘𝑦) = 𝑆))
21	20	rexbidv 3178	. . . . . . . 8 ⊢ (𝑥 = ( ⊥ ‘𝑦) → (∃𝑖 ∈ 𝐼 𝑥 = 𝑆 ↔ ∃𝑖 ∈ 𝐼 ( ⊥ ‘𝑦) = 𝑆))
22	19, 21	elab 3678	. . . . . . 7 ⊢ (( ⊥ ‘𝑦) ∈ {𝑥 ∣ ∃𝑖 ∈ 𝐼 𝑥 = 𝑆} ↔ ∃𝑖 ∈ 𝐼 ( ⊥ ‘𝑦) = 𝑆)
23	22	rabbii 3441	. . . . . 6 ⊢ {𝑦 ∈ 𝐵 ∣ ( ⊥ ‘𝑦) ∈ {𝑥 ∣ ∃𝑖 ∈ 𝐼 𝑥 = 𝑆}} = {𝑦 ∈ 𝐵 ∣ ∃𝑖 ∈ 𝐼 ( ⊥ ‘𝑦) = 𝑆}
24		df-rab 3436	. . . . . 6 ⊢ {𝑦 ∈ 𝐵 ∣ ∃𝑖 ∈ 𝐼 ( ⊥ ‘𝑦) = 𝑆} = {𝑦 ∣ (𝑦 ∈ 𝐵 ∧ ∃𝑖 ∈ 𝐼 ( ⊥ ‘𝑦) = 𝑆)}
25	23, 24	eqtri 2764	. . . . 5 ⊢ {𝑦 ∈ 𝐵 ∣ ( ⊥ ‘𝑦) ∈ {𝑥 ∣ ∃𝑖 ∈ 𝐼 𝑥 = 𝑆}} = {𝑦 ∣ (𝑦 ∈ 𝐵 ∧ ∃𝑖 ∈ 𝐼 ( ⊥ ‘𝑦) = 𝑆)}
26		nfv 1913	. . . . . . . . . 10 ⊢ Ⅎ𝑖 𝐾 ∈ HL
27	26, 5	nfan 1898	. . . . . . . . 9 ⊢ Ⅎ𝑖(𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵)
28		rspa 3247	. . . . . . . . . . 11 ⊢ ((∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 ∧ 𝑖 ∈ 𝐼) → 𝑆 ∈ 𝐵)
29		hlop 39364	. . . . . . . . . . . . . . 15 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP)
30	13, 16	opoccl 39196	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ OP ∧ 𝑆 ∈ 𝐵) → ( ⊥ ‘𝑆) ∈ 𝐵)
31	29, 30	sylan 580	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ∈ 𝐵) → ( ⊥ ‘𝑆) ∈ 𝐵)
32		eleq1a 2835	. . . . . . . . . . . . . 14 ⊢ (( ⊥ ‘𝑆) ∈ 𝐵 → (𝑦 = ( ⊥ ‘𝑆) → 𝑦 ∈ 𝐵))
33	31, 32	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ∈ 𝐵) → (𝑦 = ( ⊥ ‘𝑆) → 𝑦 ∈ 𝐵))
34	33	pm4.71rd 562	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ∈ 𝐵) → (𝑦 = ( ⊥ ‘𝑆) ↔ (𝑦 ∈ 𝐵 ∧ 𝑦 = ( ⊥ ‘𝑆))))
35	13, 16	opcon2b 39199	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ OP ∧ 𝑆 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑆 = ( ⊥ ‘𝑦) ↔ 𝑦 = ( ⊥ ‘𝑆)))
36	29, 35	syl3an1 1163	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑆 = ( ⊥ ‘𝑦) ↔ 𝑦 = ( ⊥ ‘𝑆)))
37	36	3expa 1118	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑆 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → (𝑆 = ( ⊥ ‘𝑦) ↔ 𝑦 = ( ⊥ ‘𝑆)))
38		eqcom 2743	. . . . . . . . . . . . . 14 ⊢ (𝑆 = ( ⊥ ‘𝑦) ↔ ( ⊥ ‘𝑦) = 𝑆)
39	37, 38	bitr3di 286	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑆 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → (𝑦 = ( ⊥ ‘𝑆) ↔ ( ⊥ ‘𝑦) = 𝑆))
40	39	pm5.32da 579	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ∈ 𝐵) → ((𝑦 ∈ 𝐵 ∧ 𝑦 = ( ⊥ ‘𝑆)) ↔ (𝑦 ∈ 𝐵 ∧ ( ⊥ ‘𝑦) = 𝑆)))
41	34, 40	bitrd 279	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ∈ 𝐵) → (𝑦 = ( ⊥ ‘𝑆) ↔ (𝑦 ∈ 𝐵 ∧ ( ⊥ ‘𝑦) = 𝑆)))
42	28, 41	sylan2 593	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ (∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 ∧ 𝑖 ∈ 𝐼)) → (𝑦 = ( ⊥ ‘𝑆) ↔ (𝑦 ∈ 𝐵 ∧ ( ⊥ ‘𝑦) = 𝑆)))
43	42	anassrs 467	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) → (𝑦 = ( ⊥ ‘𝑆) ↔ (𝑦 ∈ 𝐵 ∧ ( ⊥ ‘𝑦) = 𝑆)))
44	27, 43	rexbida 3271	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵) → (∃𝑖 ∈ 𝐼 𝑦 = ( ⊥ ‘𝑆) ↔ ∃𝑖 ∈ 𝐼 (𝑦 ∈ 𝐵 ∧ ( ⊥ ‘𝑦) = 𝑆)))
45		r19.42v 3190	. . . . . . . 8 ⊢ (∃𝑖 ∈ 𝐼 (𝑦 ∈ 𝐵 ∧ ( ⊥ ‘𝑦) = 𝑆) ↔ (𝑦 ∈ 𝐵 ∧ ∃𝑖 ∈ 𝐼 ( ⊥ ‘𝑦) = 𝑆))
46	44, 45	bitr2di 288	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵) → ((𝑦 ∈ 𝐵 ∧ ∃𝑖 ∈ 𝐼 ( ⊥ ‘𝑦) = 𝑆) ↔ ∃𝑖 ∈ 𝐼 𝑦 = ( ⊥ ‘𝑆)))
47	46	abbidv 2807	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵) → {𝑦 ∣ (𝑦 ∈ 𝐵 ∧ ∃𝑖 ∈ 𝐼 ( ⊥ ‘𝑦) = 𝑆)} = {𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = ( ⊥ ‘𝑆)})
48		eqeq1 2740	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝑦 = ( ⊥ ‘𝑆) ↔ 𝑥 = ( ⊥ ‘𝑆)))
49	48	rexbidv 3178	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (∃𝑖 ∈ 𝐼 𝑦 = ( ⊥ ‘𝑆) ↔ ∃𝑖 ∈ 𝐼 𝑥 = ( ⊥ ‘𝑆)))
50	49	cbvabv 2811	. . . . . 6 ⊢ {𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = ( ⊥ ‘𝑆)} = {𝑥 ∣ ∃𝑖 ∈ 𝐼 𝑥 = ( ⊥ ‘𝑆)}
51	47, 50	eqtrdi 2792	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵) → {𝑦 ∣ (𝑦 ∈ 𝐵 ∧ ∃𝑖 ∈ 𝐼 ( ⊥ ‘𝑦) = 𝑆)} = {𝑥 ∣ ∃𝑖 ∈ 𝐼 𝑥 = ( ⊥ ‘𝑆)})
52	25, 51	eqtrid 2788	. . . 4 ⊢ ((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵) → {𝑦 ∈ 𝐵 ∣ ( ⊥ ‘𝑦) ∈ {𝑥 ∣ ∃𝑖 ∈ 𝐼 𝑥 = 𝑆}} = {𝑥 ∣ ∃𝑖 ∈ 𝐼 𝑥 = ( ⊥ ‘𝑆)})
53	52	fveq2d 6909	. . 3 ⊢ ((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵) → (𝑈‘{𝑦 ∈ 𝐵 ∣ ( ⊥ ‘𝑦) ∈ {𝑥 ∣ ∃𝑖 ∈ 𝐼 𝑥 = 𝑆}}) = (𝑈‘{𝑥 ∣ ∃𝑖 ∈ 𝐼 𝑥 = ( ⊥ ‘𝑆)}))
54	53	fveq2d 6909	. 2 ⊢ ((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵) → ( ⊥ ‘(𝑈‘{𝑦 ∈ 𝐵 ∣ ( ⊥ ‘𝑦) ∈ {𝑥 ∣ ∃𝑖 ∈ 𝐼 𝑥 = 𝑆}})) = ( ⊥ ‘(𝑈‘{𝑥 ∣ ∃𝑖 ∈ 𝐼 𝑥 = ( ⊥ ‘𝑆)})))
55	18, 54	eqtrd 2776	1 ⊢ ((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵) → (𝐺‘{𝑥 ∣ ∃𝑖 ∈ 𝐼 𝑥 = 𝑆}) = ( ⊥ ‘(𝑈‘{𝑥 ∣ ∃𝑖 ∈ 𝐼 𝑥 = ( ⊥ ‘𝑆)})))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1539   ∈ wcel 2107  {cab 2713  ∀wral 3060  ∃wrex 3069  {crab 3435   ⊆ wss 3950  ‘cfv 6560  Basecbs 17248  occoc 17306  lubclub 18356  glbcglb 18357  OPcops 39174  HLchlt 39352

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-lub 18392  df-glb 18393  df-clat 18545  df-oposet 39178  df-ol 39180  df-oml 39181  df-hlat 39353

This theorem is referenced by:  polval2N  39909