Theorem pmapglb 37470

Description: The projective map of the GLB of a set of lattice elements 𝑆. Variant of Theorem 15.5.2 of [MaedaMaeda] p. 62. (Contributed by NM, 5-Dec-2011.)

Hypotheses

Ref	Expression
pmapglb.b	⊢ 𝐵 = (Base‘𝐾)
pmapglb.g	⊢ 𝐺 = (glb‘𝐾)
pmapglb.m	⊢ 𝑀 = (pmap‘𝐾)

Assertion

Ref	Expression
pmapglb	⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐵 ∧ 𝑆 ≠ ∅) → (𝑀‘(𝐺‘𝑆)) = ∩ 𝑥 ∈ 𝑆 (𝑀‘𝑥))

Distinct variable groups:   𝑥,𝐵   𝑥,𝐾   𝑥,𝑆

Allowed substitution hints:   𝐺(𝑥)   𝑀(𝑥)

Proof of Theorem pmapglb

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-rex 3057	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝑆 𝑦 = 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑆 ∧ 𝑦 = 𝑥))
2		equcom 2028	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦)
3	2	anbi1ci 629	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 = 𝑥) ↔ (𝑥 = 𝑦 ∧ 𝑥 ∈ 𝑆))
4	3	exbii 1855	. . . . . . 7 ⊢ (∃𝑥(𝑥 ∈ 𝑆 ∧ 𝑦 = 𝑥) ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝑥 ∈ 𝑆))
5		eleq1w 2813	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑆 ↔ 𝑦 ∈ 𝑆))
6	5	equsexvw 2014	. . . . . . 7 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝑥 ∈ 𝑆) ↔ 𝑦 ∈ 𝑆)
7	1, 4, 6	3bitri 300	. . . . . 6 ⊢ (∃𝑥 ∈ 𝑆 𝑦 = 𝑥 ↔ 𝑦 ∈ 𝑆)
8	7	abbii 2801	. . . . 5 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝑆 𝑦 = 𝑥} = {𝑦 ∣ 𝑦 ∈ 𝑆}
9		abid2 2872	. . . . 5 ⊢ {𝑦 ∣ 𝑦 ∈ 𝑆} = 𝑆
10	8, 9	eqtr2i 2760	. . . 4 ⊢ 𝑆 = {𝑦 ∣ ∃𝑥 ∈ 𝑆 𝑦 = 𝑥}
11	10	fveq2i 6698	. . 3 ⊢ (𝐺‘𝑆) = (𝐺‘{𝑦 ∣ ∃𝑥 ∈ 𝑆 𝑦 = 𝑥})
12	11	fveq2i 6698	. 2 ⊢ (𝑀‘(𝐺‘𝑆)) = (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑥 ∈ 𝑆 𝑦 = 𝑥}))
13		dfss3 3875	. . 3 ⊢ (𝑆 ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝑆 𝑥 ∈ 𝐵)
14		pmapglb.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
15		pmapglb.g	. . . 4 ⊢ 𝐺 = (glb‘𝐾)
16		pmapglb.m	. . . 4 ⊢ 𝑀 = (pmap‘𝐾)
17	14, 15, 16	pmapglbx 37469	. . 3 ⊢ ((𝐾 ∈ HL ∧ ∀𝑥 ∈ 𝑆 𝑥 ∈ 𝐵 ∧ 𝑆 ≠ ∅) → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑥 ∈ 𝑆 𝑦 = 𝑥})) = ∩ 𝑥 ∈ 𝑆 (𝑀‘𝑥))
18	13, 17	syl3an2b 1406	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐵 ∧ 𝑆 ≠ ∅) → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑥 ∈ 𝑆 𝑦 = 𝑥})) = ∩ 𝑥 ∈ 𝑆 (𝑀‘𝑥))
19	12, 18	syl5eq 2783	1 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐵 ∧ 𝑆 ≠ ∅) → (𝑀‘(𝐺‘𝑆)) = ∩ 𝑥 ∈ 𝑆 (𝑀‘𝑥))

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1089   = wceq 1543  ∃wex 1787   ∈ wcel 2112  {cab 2714   ≠ wne 2932  ∀wral 3051  ∃wrex 3052   ⊆ wss 3853  ∅c0 4223  ∩ ciin 4891  ‘cfv 6358  Basecbs 16666  glbcglb 17771  HLchlt 37050  pmapcpmap 37197

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 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pow 5243  ax-pr 5307  ax-un 7501

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3400  df-sbc 3684  df-csb 3799  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-pw 4501  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4806  df-iun 4892  df-iin 4893  df-br 5040  df-opab 5102  df-mpt 5121  df-id 5440  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-iota 6316  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-riota 7148  df-ov 7194  df-oprab 7195  df-poset 17774  df-lub 17806  df-glb 17807  df-join 17808  df-meet 17809  df-lat 17892  df-clat 17959  df-ats 36967  df-hlat 37051  df-pmap 37204

This theorem is referenced by:  pmapglb2N  37471  pmapmeet  37473