Theorem pmapval 40256

Description: Value of the projective map of a Hilbert lattice. Definition in Theorem 15.5 of [MaedaMaeda] p. 62. (Contributed by NM, 2-Oct-2011.)

Hypotheses

Ref	Expression
pmapfval.b	⊢ 𝐵 = (Base‘𝐾)
pmapfval.l	⊢ ≤ = (le‘𝐾)
pmapfval.a	⊢ 𝐴 = (Atoms‘𝐾)
pmapfval.m	⊢ 𝑀 = (pmap‘𝐾)

Assertion

Ref	Expression
pmapval	⊢ ((𝐾 ∈ 𝐶 ∧ 𝑋 ∈ 𝐵) → (𝑀‘𝑋) = {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑋})

Distinct variable groups:   𝐴,𝑎   𝐾,𝑎   𝑋,𝑎

Allowed substitution hints:   𝐵(𝑎)   𝐶(𝑎)   ≤ (𝑎)   𝑀(𝑎)

Proof of Theorem pmapval

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pmapfval.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
2		pmapfval.l	. . . 4 ⊢ ≤ = (le‘𝐾)
3		pmapfval.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
4		pmapfval.m	. . . 4 ⊢ 𝑀 = (pmap‘𝐾)
5	1, 2, 3, 4	pmapfval 40255	. . 3 ⊢ (𝐾 ∈ 𝐶 → 𝑀 = (𝑥 ∈ 𝐵 ↦ {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑥}))
6	5	fveq1d 6836	. 2 ⊢ (𝐾 ∈ 𝐶 → (𝑀‘𝑋) = ((𝑥 ∈ 𝐵 ↦ {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑥})‘𝑋))
7		breq2 5083	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑎 ≤ 𝑥 ↔ 𝑎 ≤ 𝑋))
8	7	rabbidv 3399	. . 3 ⊢ (𝑥 = 𝑋 → {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑥} = {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑋})
9		eqid 2740	. . 3 ⊢ (𝑥 ∈ 𝐵 ↦ {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑥}) = (𝑥 ∈ 𝐵 ↦ {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑥})
10	3	fvexi 6848	. . . 4 ⊢ 𝐴 ∈ V
11	10	rabex 5274	. . 3 ⊢ {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑋} ∈ V
12	8, 9, 11	fvmpt 6942	. 2 ⊢ (𝑋 ∈ 𝐵 → ((𝑥 ∈ 𝐵 ↦ {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑥})‘𝑋) = {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑋})
13	6, 12	sylan9eq 2795	1 ⊢ ((𝐾 ∈ 𝐶 ∧ 𝑋 ∈ 𝐵) → (𝑀‘𝑋) = {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑋})

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  {crab 3392   class class class wbr 5079   ↦ cmpt 5160  ‘cfv 6492  Basecbs 17177  lecple 17225  Atomscatm 39762  pmapcpmap 39996

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 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pr 5369

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 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-pmap 40003

This theorem is referenced by:  elpmap  40257  pmapssat  40258  pmaple  40260  pmapat  40262  pmap0  40264  pmap1N  40266  pmapsub  40267  pmapglbx  40268  isline2  40273  linepmap  40274  polpmapN  40411  2polssN  40414  pmaplubN  40423