Theorem pmapglb2xN 37713

Description: The projective map of the GLB of a set of lattice elements. Index-set version of pmapglb2N 37712, where we read 𝑆 as 𝑆(𝑖). Extension of Theorem 15.5.2 of [MaedaMaeda] p. 62 that allows 𝐼 = ∅. (Contributed by NM, 21-Jan-2012.) (New usage is discouraged.)

Hypotheses

Ref	Expression
pmapglb2.b	⊢ 𝐵 = (Base‘𝐾)
pmapglb2.g	⊢ 𝐺 = (glb‘𝐾)
pmapglb2.a	⊢ 𝐴 = (Atoms‘𝐾)
pmapglb2.m	⊢ 𝑀 = (pmap‘𝐾)

Assertion

Ref	Expression
pmapglb2xN	⊢ ((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵) → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆})) = (𝐴 ∩ ∩ 𝑖 ∈ 𝐼 (𝑀‘𝑆)))

Distinct variable groups:   𝐴,𝑖   𝑦,𝑖,𝐵   𝑖,𝐼,𝑦   𝑖,𝐾,𝑦   𝑦,𝑆

Allowed substitution hints:   𝐴(𝑦)   𝑆(𝑖)   𝐺(𝑦,𝑖)   𝑀(𝑦,𝑖)

Proof of Theorem pmapglb2xN

Step	Hyp	Ref	Expression
1		hlop 37303	. . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP)
2		pmapglb2.g	. . . . . . . 8 ⊢ 𝐺 = (glb‘𝐾)
3		eqid 2738	. . . . . . . 8 ⊢ (1.‘𝐾) = (1.‘𝐾)
4	2, 3	glb0N 37134	. . . . . . 7 ⊢ (𝐾 ∈ OP → (𝐺‘∅) = (1.‘𝐾))
5	4	fveq2d 6760	. . . . . 6 ⊢ (𝐾 ∈ OP → (𝑀‘(𝐺‘∅)) = (𝑀‘(1.‘𝐾)))
6		pmapglb2.a	. . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾)
7		pmapglb2.m	. . . . . . 7 ⊢ 𝑀 = (pmap‘𝐾)
8	3, 6, 7	pmap1N 37708	. . . . . 6 ⊢ (𝐾 ∈ OP → (𝑀‘(1.‘𝐾)) = 𝐴)
9	5, 8	eqtrd 2778	. . . . 5 ⊢ (𝐾 ∈ OP → (𝑀‘(𝐺‘∅)) = 𝐴)
10	1, 9	syl 17	. . . 4 ⊢ (𝐾 ∈ HL → (𝑀‘(𝐺‘∅)) = 𝐴)
11		rexeq 3334	. . . . . . . . 9 ⊢ (𝐼 = ∅ → (∃𝑖 ∈ 𝐼 𝑦 = 𝑆 ↔ ∃𝑖 ∈ ∅ 𝑦 = 𝑆))
12	11	abbidv 2808	. . . . . . . 8 ⊢ (𝐼 = ∅ → {𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆} = {𝑦 ∣ ∃𝑖 ∈ ∅ 𝑦 = 𝑆})
13		rex0 4288	. . . . . . . . 9 ⊢ ¬ ∃𝑖 ∈ ∅ 𝑦 = 𝑆
14	13	abf 4333	. . . . . . . 8 ⊢ {𝑦 ∣ ∃𝑖 ∈ ∅ 𝑦 = 𝑆} = ∅
15	12, 14	eqtrdi 2795	. . . . . . 7 ⊢ (𝐼 = ∅ → {𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆} = ∅)
16	15	fveq2d 6760	. . . . . 6 ⊢ (𝐼 = ∅ → (𝐺‘{𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆}) = (𝐺‘∅))
17	16	fveq2d 6760	. . . . 5 ⊢ (𝐼 = ∅ → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆})) = (𝑀‘(𝐺‘∅)))
18		riin0 5007	. . . . 5 ⊢ (𝐼 = ∅ → (𝐴 ∩ ∩ 𝑖 ∈ 𝐼 (𝑀‘𝑆)) = 𝐴)
19	17, 18	eqeq12d 2754	. . . 4 ⊢ (𝐼 = ∅ → ((𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆})) = (𝐴 ∩ ∩ 𝑖 ∈ 𝐼 (𝑀‘𝑆)) ↔ (𝑀‘(𝐺‘∅)) = 𝐴))
20	10, 19	syl5ibrcom 246	. . 3 ⊢ (𝐾 ∈ HL → (𝐼 = ∅ → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆})) = (𝐴 ∩ ∩ 𝑖 ∈ 𝐼 (𝑀‘𝑆))))
21	20	adantr 480	. 2 ⊢ ((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵) → (𝐼 = ∅ → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆})) = (𝐴 ∩ ∩ 𝑖 ∈ 𝐼 (𝑀‘𝑆))))
22		pmapglb2.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐾)
23	22, 2, 7	pmapglbx 37710	. . . 4 ⊢ ((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 ∧ 𝐼 ≠ ∅) → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆})) = ∩ 𝑖 ∈ 𝐼 (𝑀‘𝑆))
24		nfv 1918	. . . . . . . . . 10 ⊢ Ⅎ𝑖 𝐾 ∈ HL
25		nfra1 3142	. . . . . . . . . 10 ⊢ Ⅎ𝑖∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵
26	24, 25	nfan 1903	. . . . . . . . 9 ⊢ Ⅎ𝑖(𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵)
27		simpr 484	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) → 𝑖 ∈ 𝐼)
28		simpll 763	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) → 𝐾 ∈ HL)
29		rspa 3130	. . . . . . . . . . . . 13 ⊢ ((∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 ∧ 𝑖 ∈ 𝐼) → 𝑆 ∈ 𝐵)
30	29	adantll 710	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) → 𝑆 ∈ 𝐵)
31	22, 6, 7	pmapssat 37700	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ∈ 𝐵) → (𝑀‘𝑆) ⊆ 𝐴)
32	28, 30, 31	syl2anc 583	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) → (𝑀‘𝑆) ⊆ 𝐴)
33	27, 32	jca 511	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) → (𝑖 ∈ 𝐼 ∧ (𝑀‘𝑆) ⊆ 𝐴))
34	33	ex 412	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵) → (𝑖 ∈ 𝐼 → (𝑖 ∈ 𝐼 ∧ (𝑀‘𝑆) ⊆ 𝐴)))
35	26, 34	eximd 2212	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵) → (∃𝑖 𝑖 ∈ 𝐼 → ∃𝑖(𝑖 ∈ 𝐼 ∧ (𝑀‘𝑆) ⊆ 𝐴)))
36		n0 4277	. . . . . . . 8 ⊢ (𝐼 ≠ ∅ ↔ ∃𝑖 𝑖 ∈ 𝐼)
37		df-rex 3069	. . . . . . . 8 ⊢ (∃𝑖 ∈ 𝐼 (𝑀‘𝑆) ⊆ 𝐴 ↔ ∃𝑖(𝑖 ∈ 𝐼 ∧ (𝑀‘𝑆) ⊆ 𝐴))
38	35, 36, 37	3imtr4g 295	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵) → (𝐼 ≠ ∅ → ∃𝑖 ∈ 𝐼 (𝑀‘𝑆) ⊆ 𝐴))
39	38	3impia 1115	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 ∧ 𝐼 ≠ ∅) → ∃𝑖 ∈ 𝐼 (𝑀‘𝑆) ⊆ 𝐴)
40		iinss 4982	. . . . . 6 ⊢ (∃𝑖 ∈ 𝐼 (𝑀‘𝑆) ⊆ 𝐴 → ∩ 𝑖 ∈ 𝐼 (𝑀‘𝑆) ⊆ 𝐴)
41	39, 40	syl 17	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 ∧ 𝐼 ≠ ∅) → ∩ 𝑖 ∈ 𝐼 (𝑀‘𝑆) ⊆ 𝐴)
42		sseqin2 4146	. . . . 5 ⊢ (∩ 𝑖 ∈ 𝐼 (𝑀‘𝑆) ⊆ 𝐴 ↔ (𝐴 ∩ ∩ 𝑖 ∈ 𝐼 (𝑀‘𝑆)) = ∩ 𝑖 ∈ 𝐼 (𝑀‘𝑆))
43	41, 42	sylib 217	. . . 4 ⊢ ((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 ∧ 𝐼 ≠ ∅) → (𝐴 ∩ ∩ 𝑖 ∈ 𝐼 (𝑀‘𝑆)) = ∩ 𝑖 ∈ 𝐼 (𝑀‘𝑆))
44	23, 43	eqtr4d 2781	. . 3 ⊢ ((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 ∧ 𝐼 ≠ ∅) → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆})) = (𝐴 ∩ ∩ 𝑖 ∈ 𝐼 (𝑀‘𝑆)))
45	44	3expia 1119	. 2 ⊢ ((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵) → (𝐼 ≠ ∅ → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆})) = (𝐴 ∩ ∩ 𝑖 ∈ 𝐼 (𝑀‘𝑆))))
46	21, 45	pm2.61dne 3030	1 ⊢ ((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵) → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆})) = (𝐴 ∩ ∩ 𝑖 ∈ 𝐼 (𝑀‘𝑆)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1539  ∃wex 1783   ∈ wcel 2108  {cab 2715   ≠ wne 2942  ∀wral 3063  ∃wrex 3064   ∩ cin 3882   ⊆ wss 3883  ∅c0 4253  ∩ ciin 4922  ‘cfv 6418  Basecbs 16840  glbcglb 17943  1.cp1 18057  OPcops 37113  Atomscatm 37204  HLchlt 37291  pmapcpmap 37438

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-iin 4924  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-proset 17928  df-poset 17946  df-lub 17979  df-glb 17980  df-join 17981  df-meet 17982  df-p1 18059  df-lat 18065  df-clat 18132  df-oposet 37117  df-ol 37119  df-oml 37120  df-ats 37208  df-hlat 37292  df-pmap 37445

This theorem is referenced by:  polval2N  37847