Theorem pmapglb 40262

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 3064	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝑆 𝑦 = 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑆 ∧ 𝑦 = 𝑥))
2		equcom 2025	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦)
3	2	anbi1ci 632	. . . . . . . 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 2822	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑆 ↔ 𝑦 ∈ 𝑆))
6	5	equsexvw 2012	. . . . . . 7 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝑥 ∈ 𝑆) ↔ 𝑦 ∈ 𝑆)
7	1, 4, 6	3bitri 298	. . . . . 6 ⊢ (∃𝑥 ∈ 𝑆 𝑦 = 𝑥 ↔ 𝑦 ∈ 𝑆)
8	7	abbii 2806	. . . . 5 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝑆 𝑦 = 𝑥} = {𝑦 ∣ 𝑦 ∈ 𝑆}
9		abid2 2876	. . . . 5 ⊢ {𝑦 ∣ 𝑦 ∈ 𝑆} = 𝑆
10	8, 9	eqtr2i 2763	. . . 4 ⊢ 𝑆 = {𝑦 ∣ ∃𝑥 ∈ 𝑆 𝑦 = 𝑥}
11	10	fveq2i 6830	. . 3 ⊢ (𝐺‘𝑆) = (𝐺‘{𝑦 ∣ ∃𝑥 ∈ 𝑆 𝑦 = 𝑥})
12	11	fveq2i 6830	. 2 ⊢ (𝑀‘(𝐺‘𝑆)) = (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑥 ∈ 𝑆 𝑦 = 𝑥}))
13		dfss3 3904	. . 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 40261	. . 3 ⊢ ((𝐾 ∈ HL ∧ ∀𝑥 ∈ 𝑆 𝑥 ∈ 𝐵 ∧ 𝑆 ≠ ∅) → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑥 ∈ 𝑆 𝑦 = 𝑥})) = ∩ 𝑥 ∈ 𝑆 (𝑀‘𝑥))
18	13, 17	syl3an2b 1412	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐵 ∧ 𝑆 ≠ ∅) → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑥 ∈ 𝑆 𝑦 = 𝑥})) = ∩ 𝑥 ∈ 𝑆 (𝑀‘𝑥))
19	12, 18	eqtrid 2786	1 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐵 ∧ 𝑆 ≠ ∅) → (𝑀‘(𝐺‘𝑆)) = ∩ 𝑥 ∈ 𝑆 (𝑀‘𝑥))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1092   = wceq 1547  ∃wex 1786   ∈ wcel 2119  {cab 2717   ≠ wne 2934  ∀wral 3053  ∃wrex 3063   ⊆ wss 3883  ∅c0 4261  ∩ ciin 4922  ‘cfv 6485  Basecbs 17170  glbcglb 18267  HLchlt 39842  pmapcpmap 39989

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-iin 4924  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-riota 7313  df-ov 7359  df-oprab 7360  df-poset 18270  df-lub 18301  df-glb 18302  df-join 18303  df-meet 18304  df-lat 18389  df-clat 18456  df-ats 39759  df-hlat 39843  df-pmap 39996

This theorem is referenced by:  pmapglb2N  40263  pmapmeet  40265