Theorem pmapglb2xN 40067

Description: The projective map of the GLB of a set of lattice elements. Index-set version of pmapglb2N 40066, 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 39657	. . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP)
2		pmapglb2.g	. . . . . . . 8 ⊢ 𝐺 = (glb‘𝐾)
3		eqid 2735	. . . . . . . 8 ⊢ (1.‘𝐾) = (1.‘𝐾)
4	2, 3	glb0N 39488	. . . . . . 7 ⊢ (𝐾 ∈ OP → (𝐺‘∅) = (1.‘𝐾))
5	4	fveq2d 6837	. . . . . 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 40062	. . . . . 6 ⊢ (𝐾 ∈ OP → (𝑀‘(1.‘𝐾)) = 𝐴)
9	5, 8	eqtrd 2770	. . . . 5 ⊢ (𝐾 ∈ OP → (𝑀‘(𝐺‘∅)) = 𝐴)
10	1, 9	syl 17	. . . 4 ⊢ (𝐾 ∈ HL → (𝑀‘(𝐺‘∅)) = 𝐴)
11		rexeq 3291	. . . . . . . . 9 ⊢ (𝐼 = ∅ → (∃𝑖 ∈ 𝐼 𝑦 = 𝑆 ↔ ∃𝑖 ∈ ∅ 𝑦 = 𝑆))
12	11	abbidv 2801	. . . . . . . 8 ⊢ (𝐼 = ∅ → {𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆} = {𝑦 ∣ ∃𝑖 ∈ ∅ 𝑦 = 𝑆})
13		rex0 4311	. . . . . . . . 9 ⊢ ¬ ∃𝑖 ∈ ∅ 𝑦 = 𝑆
14	13	abf 4357	. . . . . . . 8 ⊢ {𝑦 ∣ ∃𝑖 ∈ ∅ 𝑦 = 𝑆} = ∅
15	12, 14	eqtrdi 2786	. . . . . . 7 ⊢ (𝐼 = ∅ → {𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆} = ∅)
16	15	fveq2d 6837	. . . . . 6 ⊢ (𝐼 = ∅ → (𝐺‘{𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆}) = (𝐺‘∅))
17	16	fveq2d 6837	. . . . 5 ⊢ (𝐼 = ∅ → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆})) = (𝑀‘(𝐺‘∅)))
18		riin0 5036	. . . . 5 ⊢ (𝐼 = ∅ → (𝐴 ∩ ∩ 𝑖 ∈ 𝐼 (𝑀‘𝑆)) = 𝐴)
19	17, 18	eqeq12d 2751	. . . 4 ⊢ (𝐼 = ∅ → ((𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆})) = (𝐴 ∩ ∩ 𝑖 ∈ 𝐼 (𝑀‘𝑆)) ↔ (𝑀‘(𝐺‘∅)) = 𝐴))
20	10, 19	syl5ibrcom 247	. . 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 40064	. . . 4 ⊢ ((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 ∧ 𝐼 ≠ ∅) → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆})) = ∩ 𝑖 ∈ 𝐼 (𝑀‘𝑆))
24		nfv 1916	. . . . . . . . . 10 ⊢ Ⅎ𝑖 𝐾 ∈ HL
25		nfra1 3259	. . . . . . . . . 10 ⊢ Ⅎ𝑖∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵
26	24, 25	nfan 1901	. . . . . . . . 9 ⊢ Ⅎ𝑖(𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵)
27		simpr 484	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) → 𝑖 ∈ 𝐼)
28		simpll 767	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) → 𝐾 ∈ HL)
29		rspa 3224	. . . . . . . . . . . . 13 ⊢ ((∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 ∧ 𝑖 ∈ 𝐼) → 𝑆 ∈ 𝐵)
30	29	adantll 715	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) → 𝑆 ∈ 𝐵)
31	22, 6, 7	pmapssat 40054	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ∈ 𝐵) → (𝑀‘𝑆) ⊆ 𝐴)
32	28, 30, 31	syl2anc 585	. . . . . . . . . . 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 2222	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵) → (∃𝑖 𝑖 ∈ 𝐼 → ∃𝑖(𝑖 ∈ 𝐼 ∧ (𝑀‘𝑆) ⊆ 𝐴)))
36		n0 4304	. . . . . . . 8 ⊢ (𝐼 ≠ ∅ ↔ ∃𝑖 𝑖 ∈ 𝐼)
37		df-rex 3060	. . . . . . . 8 ⊢ (∃𝑖 ∈ 𝐼 (𝑀‘𝑆) ⊆ 𝐴 ↔ ∃𝑖(𝑖 ∈ 𝐼 ∧ (𝑀‘𝑆) ⊆ 𝐴))
38	35, 36, 37	3imtr4g 296	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵) → (𝐼 ≠ ∅ → ∃𝑖 ∈ 𝐼 (𝑀‘𝑆) ⊆ 𝐴))
39	38	3impia 1118	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 ∧ 𝐼 ≠ ∅) → ∃𝑖 ∈ 𝐼 (𝑀‘𝑆) ⊆ 𝐴)
40		iinss 5011	. . . . . 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 4174	. . . . 5 ⊢ (∩ 𝑖 ∈ 𝐼 (𝑀‘𝑆) ⊆ 𝐴 ↔ (𝐴 ∩ ∩ 𝑖 ∈ 𝐼 (𝑀‘𝑆)) = ∩ 𝑖 ∈ 𝐼 (𝑀‘𝑆))
43	41, 42	sylib 218	. . . 4 ⊢ ((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 ∧ 𝐼 ≠ ∅) → (𝐴 ∩ ∩ 𝑖 ∈ 𝐼 (𝑀‘𝑆)) = ∩ 𝑖 ∈ 𝐼 (𝑀‘𝑆))
44	23, 43	eqtr4d 2773	. . 3 ⊢ ((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 ∧ 𝐼 ≠ ∅) → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆})) = (𝐴 ∩ ∩ 𝑖 ∈ 𝐼 (𝑀‘𝑆)))
45	44	3expia 1122	. 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 3017	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 1087   = wceq 1542  ∃wex 1781   ∈ wcel 2114  {cab 2713   ≠ wne 2931  ∀wral 3050  ∃wrex 3059   ∩ cin 3899   ⊆ wss 3900  ∅c0 4284  ∩ ciin 4946  ‘cfv 6491  Basecbs 17138  glbcglb 18235  1.cp1 18347  OPcops 39467  Atomscatm 39558  HLchlt 39645  pmapcpmap 39792

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2183  ax-ext 2707  ax-rep 5223  ax-sep 5240  ax-nul 5250  ax-pow 5309  ax-pr 5376  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rmo 3349  df-reu 3350  df-rab 3399  df-v 3441  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4947  df-iin 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6447  df-fun 6493  df-fn 6494  df-f 6495  df-f1 6496  df-fo 6497  df-f1o 6498  df-fv 6499  df-riota 7315  df-ov 7361  df-oprab 7362  df-proset 18219  df-poset 18238  df-lub 18269  df-glb 18270  df-join 18271  df-meet 18272  df-p1 18349  df-lat 18357  df-clat 18424  df-oposet 39471  df-ol 39473  df-oml 39474  df-ats 39562  df-hlat 39646  df-pmap 39799

This theorem is referenced by:  polval2N  40201