Theorem pmapval 40127

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 40126	. . 3 ⊢ (𝐾 ∈ 𝐶 → 𝑀 = (𝑥 ∈ 𝐵 ↦ {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑥}))
6	5	fveq1d 6844	. 2 ⊢ (𝐾 ∈ 𝐶 → (𝑀‘𝑋) = ((𝑥 ∈ 𝐵 ↦ {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑥})‘𝑋))
7		breq2 5104	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑎 ≤ 𝑥 ↔ 𝑎 ≤ 𝑋))
8	7	rabbidv 3408	. . 3 ⊢ (𝑥 = 𝑋 → {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑥} = {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑋})
9		eqid 2737	. . 3 ⊢ (𝑥 ∈ 𝐵 ↦ {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑥}) = (𝑥 ∈ 𝐵 ↦ {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑥})
10	3	fvexi 6856	. . . 4 ⊢ 𝐴 ∈ V
11	10	rabex 5286	. . 3 ⊢ {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑋} ∈ V
12	8, 9, 11	fvmpt 6949	. 2 ⊢ (𝑋 ∈ 𝐵 → ((𝑥 ∈ 𝐵 ↦ {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑥})‘𝑋) = {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑋})
13	6, 12	sylan9eq 2792	1 ⊢ ((𝐾 ∈ 𝐶 ∧ 𝑋 ∈ 𝐵) → (𝑀‘𝑋) = {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑋})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {crab 3401   class class class wbr 5100   ↦ cmpt 5181  ‘cfv 6500  Basecbs 17148  lecple 17196  Atomscatm 39633  pmapcpmap 39867

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 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pr 5379

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-pmap 39874

This theorem is referenced by:  elpmap  40128  pmapssat  40129  pmaple  40131  pmapat  40133  pmap0  40135  pmap1N  40137  pmapsub  40138  pmapglbx  40139  isline2  40144  linepmap  40145  polpmapN  40282  2polssN  40285  pmaplubN  40294