Theorem pmapval 39876

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 39875	. . 3 ⊢ (𝐾 ∈ 𝐶 → 𝑀 = (𝑥 ∈ 𝐵 ↦ {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑥}))
6	5	fveq1d 6830	. 2 ⊢ (𝐾 ∈ 𝐶 → (𝑀‘𝑋) = ((𝑥 ∈ 𝐵 ↦ {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑥})‘𝑋))
7		breq2 5097	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑎 ≤ 𝑥 ↔ 𝑎 ≤ 𝑋))
8	7	rabbidv 3403	. . 3 ⊢ (𝑥 = 𝑋 → {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑥} = {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑋})
9		eqid 2733	. . 3 ⊢ (𝑥 ∈ 𝐵 ↦ {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑥}) = (𝑥 ∈ 𝐵 ↦ {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑥})
10	3	fvexi 6842	. . . 4 ⊢ 𝐴 ∈ V
11	10	rabex 5279	. . 3 ⊢ {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑋} ∈ V
12	8, 9, 11	fvmpt 6935	. 2 ⊢ (𝑋 ∈ 𝐵 → ((𝑥 ∈ 𝐵 ↦ {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑥})‘𝑋) = {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑋})
13	6, 12	sylan9eq 2788	1 ⊢ ((𝐾 ∈ 𝐶 ∧ 𝑋 ∈ 𝐵) → (𝑀‘𝑋) = {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑋})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  {crab 3396   class class class wbr 5093   ↦ cmpt 5174  ‘cfv 6486  Basecbs 17122  lecple 17170  Atomscatm 39382  pmapcpmap 39616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-pmap 39623

This theorem is referenced by:  elpmap  39877  pmapssat  39878  pmaple  39880  pmapat  39882  pmap0  39884  pmap1N  39886  pmapsub  39887  pmapglbx  39888  isline2  39893  linepmap  39894  polpmapN  40031  2polssN  40034  pmaplubN  40043