Theorem pmapglb2xN 40279

Description: The projective map of the GLB of a set of lattice elements. Index-set version of pmapglb2N 40278, 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 39869	. . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP)
2		pmapglb2.g	. . . . . . . 8 ⊢ 𝐺 = (glb‘𝐾)
3		eqid 2741	. . . . . . . 8 ⊢ (1.‘𝐾) = (1.‘𝐾)
4	2, 3	glb0N 39700	. . . . . . 7 ⊢ (𝐾 ∈ OP → (𝐺‘∅) = (1.‘𝐾))
5	4	fveq2d 6835	. . . . . 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 40274	. . . . . 6 ⊢ (𝐾 ∈ OP → (𝑀‘(1.‘𝐾)) = 𝐴)
9	5, 8	eqtrd 2776	. . . . 5 ⊢ (𝐾 ∈ OP → (𝑀‘(𝐺‘∅)) = 𝐴)
10	1, 9	syl 17	. . . 4 ⊢ (𝐾 ∈ HL → (𝑀‘(𝐺‘∅)) = 𝐴)
11		rexeq 3295	. . . . . . . . 9 ⊢ (𝐼 = ∅ → (∃𝑖 ∈ 𝐼 𝑦 = 𝑆 ↔ ∃𝑖 ∈ ∅ 𝑦 = 𝑆))
12	11	abbidv 2807	. . . . . . . 8 ⊢ (𝐼 = ∅ → {𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆} = {𝑦 ∣ ∃𝑖 ∈ ∅ 𝑦 = 𝑆})
13		rex0 4291	. . . . . . . . 9 ⊢ ¬ ∃𝑖 ∈ ∅ 𝑦 = 𝑆
14	13	abf 4337	. . . . . . . 8 ⊢ {𝑦 ∣ ∃𝑖 ∈ ∅ 𝑦 = 𝑆} = ∅
15	12, 14	eqtrdi 2792	. . . . . . 7 ⊢ (𝐼 = ∅ → {𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆} = ∅)
16	15	fveq2d 6835	. . . . . 6 ⊢ (𝐼 = ∅ → (𝐺‘{𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆}) = (𝐺‘∅))
17	16	fveq2d 6835	. . . . 5 ⊢ (𝐼 = ∅ → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆})) = (𝑀‘(𝐺‘∅)))
18		riin0 5014	. . . . 5 ⊢ (𝐼 = ∅ → (𝐴 ∩ ∩ 𝑖 ∈ 𝐼 (𝑀‘𝑆)) = 𝐴)
19	17, 18	eqeq12d 2757	. . . 4 ⊢ (𝐼 = ∅ → ((𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆})) = (𝐴 ∩ ∩ 𝑖 ∈ 𝐼 (𝑀‘𝑆)) ↔ (𝑀‘(𝐺‘∅)) = 𝐴))
20	10, 19	syl5ibrcom 249	. . 3 ⊢ (𝐾 ∈ HL → (𝐼 = ∅ → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆})) = (𝐴 ∩ ∩ 𝑖 ∈ 𝐼 (𝑀‘𝑆))))
21	20	adantr 482	. 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 40276	. . . 4 ⊢ ((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 ∧ 𝐼 ≠ ∅) → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆})) = ∩ 𝑖 ∈ 𝐼 (𝑀‘𝑆))
24		nfv 1922	. . . . . . . . . 10 ⊢ Ⅎ𝑖 𝐾 ∈ HL
25		nfra1 3265	. . . . . . . . . 10 ⊢ Ⅎ𝑖∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵
26	24, 25	nfan 1907	. . . . . . . . 9 ⊢ Ⅎ𝑖(𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵)
27		simpr 486	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) → 𝑖 ∈ 𝐼)
28		simpll 773	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) → 𝐾 ∈ HL)
29		rspa 3230	. . . . . . . . . . . . 13 ⊢ ((∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 ∧ 𝑖 ∈ 𝐼) → 𝑆 ∈ 𝐵)
30	29	adantll 721	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) → 𝑆 ∈ 𝐵)
31	22, 6, 7	pmapssat 40266	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ∈ 𝐵) → (𝑀‘𝑆) ⊆ 𝐴)
32	28, 30, 31	syl2anc 591	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) → (𝑀‘𝑆) ⊆ 𝐴)
33	27, 32	jca 517	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) → (𝑖 ∈ 𝐼 ∧ (𝑀‘𝑆) ⊆ 𝐴))
34	33	ex 414	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵) → (𝑖 ∈ 𝐼 → (𝑖 ∈ 𝐼 ∧ (𝑀‘𝑆) ⊆ 𝐴)))
35	26, 34	eximd 2230	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵) → (∃𝑖 𝑖 ∈ 𝐼 → ∃𝑖(𝑖 ∈ 𝐼 ∧ (𝑀‘𝑆) ⊆ 𝐴)))
36		n0 4284	. . . . . . . 8 ⊢ (𝐼 ≠ ∅ ↔ ∃𝑖 𝑖 ∈ 𝐼)
37		df-rex 3066	. . . . . . . 8 ⊢ (∃𝑖 ∈ 𝐼 (𝑀‘𝑆) ⊆ 𝐴 ↔ ∃𝑖(𝑖 ∈ 𝐼 ∧ (𝑀‘𝑆) ⊆ 𝐴))
38	35, 36, 37	3imtr4g 298	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵) → (𝐼 ≠ ∅ → ∃𝑖 ∈ 𝐼 (𝑀‘𝑆) ⊆ 𝐴))
39	38	3impia 1124	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 ∧ 𝐼 ≠ ∅) → ∃𝑖 ∈ 𝐼 (𝑀‘𝑆) ⊆ 𝐴)
40		iinss 4989	. . . . . 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 4155	. . . . 5 ⊢ (∩ 𝑖 ∈ 𝐼 (𝑀‘𝑆) ⊆ 𝐴 ↔ (𝐴 ∩ ∩ 𝑖 ∈ 𝐼 (𝑀‘𝑆)) = ∩ 𝑖 ∈ 𝐼 (𝑀‘𝑆))
43	41, 42	sylib 220	. . . 4 ⊢ ((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 ∧ 𝐼 ≠ ∅) → (𝐴 ∩ ∩ 𝑖 ∈ 𝐼 (𝑀‘𝑆)) = ∩ 𝑖 ∈ 𝐼 (𝑀‘𝑆))
44	23, 43	eqtr4d 2779	. . 3 ⊢ ((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 ∧ 𝐼 ≠ ∅) → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆})) = (𝐴 ∩ ∩ 𝑖 ∈ 𝐼 (𝑀‘𝑆)))
45	44	3expia 1128	. 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 3022	1 ⊢ ((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵) → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆})) = (𝐴 ∩ ∩ 𝑖 ∈ 𝐼 (𝑀‘𝑆)))

Syntax hints:   → wi 4   ∧ wa 397   ∧ w3a 1093   = wceq 1548  ∃wex 1787   ∈ wcel 2121  {cab 2719   ≠ wne 2936  ∀wral 3055  ∃wrex 3065   ∩ cin 3884   ⊆ wss 3885  ∅c0 4264  ∩ ciin 4925  ‘cfv 6489  Basecbs 17174  glbcglb 18271  1.cp1 18383  OPcops 39679  Atomscatm 39770  HLchlt 39857  pmapcpmap 40004

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-rmo 3346  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-iun 4926  df-iin 4927  df-br 5076  df-opab 5138  df-mpt 5157  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-riota 7317  df-ov 7363  df-oprab 7364  df-proset 18255  df-poset 18274  df-lub 18305  df-glb 18306  df-join 18307  df-meet 18308  df-p1 18385  df-lat 18393  df-clat 18460  df-oposet 39683  df-ol 39685  df-oml 39686  df-ats 39774  df-hlat 39858  df-pmap 40011

This theorem is referenced by:  polval2N  40413