Theorem pmapglb2xN 40471

Description: The projective map of the GLB of a set of lattice elements. Index-set version of pmapglb2N 40470, 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 40061	. . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP)
2		pmapglb2.g	. . . . . . . 8 ⊢ 𝐺 = (glb‘𝐾)
3		eqid 2769	. . . . . . . 8 ⊢ (1.‘𝐾) = (1.‘𝐾)
4	2, 3	glb0N 39892	. . . . . . 7 ⊢ (𝐾 ∈ OP → (𝐺‘∅) = (1.‘𝐾))
5	4	fveq2d 6886	. . . . . 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 40466	. . . . . 6 ⊢ (𝐾 ∈ OP → (𝑀‘(1.‘𝐾)) = 𝐴)
9	5, 8	eqtrd 2804	. . . . 5 ⊢ (𝐾 ∈ OP → (𝑀‘(𝐺‘∅)) = 𝐴)
10	1, 9	syl 18	. . . 4 ⊢ (𝐾 ∈ HL → (𝑀‘(𝐺‘∅)) = 𝐴)
11		rexeq 3325	. . . . . . . . 9 ⊢ (𝐼 = ∅ → (∃𝑖 ∈ 𝐼 𝑦 = 𝑆 ↔ ∃𝑖 ∈ ∅ 𝑦 = 𝑆))
12	11	abbidv 2835	. . . . . . . 8 ⊢ (𝐼 = ∅ → {𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆} = {𝑦 ∣ ∃𝑖 ∈ ∅ 𝑦 = 𝑆})
13		rex0 4321	. . . . . . . . 9 ⊢ ¬ ∃𝑖 ∈ ∅ 𝑦 = 𝑆
14	13	abf 4375	. . . . . . . 8 ⊢ {𝑦 ∣ ∃𝑖 ∈ ∅ 𝑦 = 𝑆} = ∅
15	12, 14	eqtrdi 2820	. . . . . . 7 ⊢ (𝐼 = ∅ → {𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆} = ∅)
16	15	fveq2d 6886	. . . . . 6 ⊢ (𝐼 = ∅ → (𝐺‘{𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆}) = (𝐺‘∅))
17	16	fveq2d 6886	. . . . 5 ⊢ (𝐼 = ∅ → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆})) = (𝑀‘(𝐺‘∅)))
18		riin0 5050	. . . . 5 ⊢ (𝐼 = ∅ → (𝐴 ∩ ∩ 𝑖 ∈ 𝐼 (𝑀‘𝑆)) = 𝐴)
19	17, 18	eqeq12d 2785	. . . 4 ⊢ (𝐼 = ∅ → ((𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆})) = (𝐴 ∩ ∩ 𝑖 ∈ 𝐼 (𝑀‘𝑆)) ↔ (𝑀‘(𝐺‘∅)) = 𝐴))
20	10, 19	syl5ibrcom 250	. . 3 ⊢ (𝐾 ∈ HL → (𝐼 = ∅ → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆})) = (𝐴 ∩ ∩ 𝑖 ∈ 𝐼 (𝑀‘𝑆))))
21	20	adantr 485	. 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 40468	. . . 4 ⊢ ((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 ∧ 𝐼 ≠ ∅) → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆})) = ∩ 𝑖 ∈ 𝐼 (𝑀‘𝑆))
24		nfv 1941	. . . . . . . . . 10 ⊢ Ⅎ𝑖 𝐾 ∈ HL
25		nfra1 3295	. . . . . . . . . 10 ⊢ Ⅎ𝑖∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵
26	24, 25	nfan 1926	. . . . . . . . 9 ⊢ Ⅎ𝑖(𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵)
27		simpr 489	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) → 𝑖 ∈ 𝐼)
28		simpll 778	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) → 𝐾 ∈ HL)
29		rspa 3260	. . . . . . . . . . . . 13 ⊢ ((∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 ∧ 𝑖 ∈ 𝐼) → 𝑆 ∈ 𝐵)
30	29	adantll 726	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) → 𝑆 ∈ 𝐵)
31	22, 6, 7	pmapssat 40458	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ∈ 𝐵) → (𝑀‘𝑆) ⊆ 𝐴)
32	28, 30, 31	syl2anc 595	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) → (𝑀‘𝑆) ⊆ 𝐴)
33	27, 32	jca 520	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) → (𝑖 ∈ 𝐼 ∧ (𝑀‘𝑆) ⊆ 𝐴))
34	33	ex 417	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵) → (𝑖 ∈ 𝐼 → (𝑖 ∈ 𝐼 ∧ (𝑀‘𝑆) ⊆ 𝐴)))
35	26, 34	eximd 2258	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵) → (∃𝑖 𝑖 ∈ 𝐼 → ∃𝑖(𝑖 ∈ 𝐼 ∧ (𝑀‘𝑆) ⊆ 𝐴)))
36		n0 4313	. . . . . . . 8 ⊢ (𝐼 ≠ ∅ ↔ ∃𝑖 𝑖 ∈ 𝐼)
37		df-rex 3096	. . . . . . . 8 ⊢ (∃𝑖 ∈ 𝐼 (𝑀‘𝑆) ⊆ 𝐴 ↔ ∃𝑖(𝑖 ∈ 𝐼 ∧ (𝑀‘𝑆) ⊆ 𝐴))
38	35, 36, 37	3imtr4g 299	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵) → (𝐼 ≠ ∅ → ∃𝑖 ∈ 𝐼 (𝑀‘𝑆) ⊆ 𝐴))
39	38	3impia 1133	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 ∧ 𝐼 ≠ ∅) → ∃𝑖 ∈ 𝐼 (𝑀‘𝑆) ⊆ 𝐴)
40		iinss 5023	. . . . . 6 ⊢ (∃𝑖 ∈ 𝐼 (𝑀‘𝑆) ⊆ 𝐴 → ∩ 𝑖 ∈ 𝐼 (𝑀‘𝑆) ⊆ 𝐴)
41	39, 40	syl 18	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 ∧ 𝐼 ≠ ∅) → ∩ 𝑖 ∈ 𝐼 (𝑀‘𝑆) ⊆ 𝐴)
42		sseqin2 4182	. . . . 5 ⊢ (∩ 𝑖 ∈ 𝐼 (𝑀‘𝑆) ⊆ 𝐴 ↔ (𝐴 ∩ ∩ 𝑖 ∈ 𝐼 (𝑀‘𝑆)) = ∩ 𝑖 ∈ 𝐼 (𝑀‘𝑆))
43	41, 42	sylib 221	. . . 4 ⊢ ((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 ∧ 𝐼 ≠ ∅) → (𝐴 ∩ ∩ 𝑖 ∈ 𝐼 (𝑀‘𝑆)) = ∩ 𝑖 ∈ 𝐼 (𝑀‘𝑆))
44	23, 43	eqtr4d 2807	. . 3 ⊢ ((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 ∧ 𝐼 ≠ ∅) → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆})) = (𝐴 ∩ ∩ 𝑖 ∈ 𝐼 (𝑀‘𝑆)))
45	44	3expia 1137	. 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 3050	1 ⊢ ((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵) → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆})) = (𝐴 ∩ ∩ 𝑖 ∈ 𝐼 (𝑀‘𝑆)))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1567  ∃wex 1806   ∈ wcel 2149  {cab 2747   ≠ wne 2964  ∀wral 3085  ∃wrex 3095   ∩ cin 3910   ⊆ wss 3911  ∅c0 4292  ∩ ciin 4959  ‘cfv 6537  Basecbs 17269  glbcglb 18366  1.cp1 18478  OPcops 39871  Atomscatm 39962  HLchlt 40049  pmapcpmap 40196

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3375  df-reu 3376  df-rab 3423  df-v 3463  df-sbc 3752  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4491  df-pw 4567  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-iin 4961  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-proset 18350  df-poset 18369  df-lub 18400  df-glb 18401  df-join 18402  df-meet 18403  df-p1 18480  df-lat 18488  df-clat 18555  df-oposet 39875  df-ol 39877  df-oml 39878  df-ats 39966  df-hlat 40050  df-pmap 40203

This theorem is referenced by:  polval2N  40605