Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  isat3 Structured version   Visualization version   GIF version

Theorem isat3 37815
Description: The predicate "is an atom". (elat2 31324 analog.) (Contributed by NM, 27-Apr-2014.)
Hypotheses
Ref Expression
isat3.b 𝐡 = (Baseβ€˜πΎ)
isat3.l ≀ = (leβ€˜πΎ)
isat3.z 0 = (0.β€˜πΎ)
isat3.a 𝐴 = (Atomsβ€˜πΎ)
Assertion
Ref Expression
isat3 (𝐾 ∈ AtLat β†’ (𝑃 ∈ 𝐴 ↔ (𝑃 ∈ 𝐡 ∧ 𝑃 β‰  0 ∧ βˆ€π‘₯ ∈ 𝐡 (π‘₯ ≀ 𝑃 β†’ (π‘₯ = 𝑃 ∨ π‘₯ = 0 )))))
Distinct variable groups:   π‘₯,𝐡   π‘₯,𝐾   π‘₯,𝑃   π‘₯, 0
Allowed substitution hints:   𝐴(π‘₯)   ≀ (π‘₯)

Proof of Theorem isat3
StepHypRef Expression
1 isat3.b . . . 4 𝐡 = (Baseβ€˜πΎ)
2 isat3.z . . . 4 0 = (0.β€˜πΎ)
3 eqid 2733 . . . 4 ( β‹– β€˜πΎ) = ( β‹– β€˜πΎ)
4 isat3.a . . . 4 𝐴 = (Atomsβ€˜πΎ)
51, 2, 3, 4isat 37794 . . 3 (𝐾 ∈ AtLat β†’ (𝑃 ∈ 𝐴 ↔ (𝑃 ∈ 𝐡 ∧ 0 ( β‹– β€˜πΎ)𝑃)))
6 simpl 484 . . . . . 6 ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐡) β†’ 𝐾 ∈ AtLat)
71, 2atl0cl 37811 . . . . . . 7 (𝐾 ∈ AtLat β†’ 0 ∈ 𝐡)
87adantr 482 . . . . . 6 ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐡) β†’ 0 ∈ 𝐡)
9 simpr 486 . . . . . 6 ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐡) β†’ 𝑃 ∈ 𝐡)
10 isat3.l . . . . . . 7 ≀ = (leβ€˜πΎ)
11 eqid 2733 . . . . . . 7 (ltβ€˜πΎ) = (ltβ€˜πΎ)
121, 10, 11, 3cvrval2 37782 . . . . . 6 ((𝐾 ∈ AtLat ∧ 0 ∈ 𝐡 ∧ 𝑃 ∈ 𝐡) β†’ ( 0 ( β‹– β€˜πΎ)𝑃 ↔ ( 0 (ltβ€˜πΎ)𝑃 ∧ βˆ€π‘₯ ∈ 𝐡 (( 0 (ltβ€˜πΎ)π‘₯ ∧ π‘₯ ≀ 𝑃) β†’ π‘₯ = 𝑃))))
136, 8, 9, 12syl3anc 1372 . . . . 5 ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐡) β†’ ( 0 ( β‹– β€˜πΎ)𝑃 ↔ ( 0 (ltβ€˜πΎ)𝑃 ∧ βˆ€π‘₯ ∈ 𝐡 (( 0 (ltβ€˜πΎ)π‘₯ ∧ π‘₯ ≀ 𝑃) β†’ π‘₯ = 𝑃))))
141, 11, 2atlltn0 37814 . . . . . 6 ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐡) β†’ ( 0 (ltβ€˜πΎ)𝑃 ↔ 𝑃 β‰  0 ))
151, 11, 2atlltn0 37814 . . . . . . . . . . 11 ((𝐾 ∈ AtLat ∧ π‘₯ ∈ 𝐡) β†’ ( 0 (ltβ€˜πΎ)π‘₯ ↔ π‘₯ β‰  0 ))
1615adantlr 714 . . . . . . . . . 10 (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐡) ∧ π‘₯ ∈ 𝐡) β†’ ( 0 (ltβ€˜πΎ)π‘₯ ↔ π‘₯ β‰  0 ))
1716imbi1d 342 . . . . . . . . 9 (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐡) ∧ π‘₯ ∈ 𝐡) β†’ (( 0 (ltβ€˜πΎ)π‘₯ β†’ π‘₯ = 𝑃) ↔ (π‘₯ β‰  0 β†’ π‘₯ = 𝑃)))
1817imbi2d 341 . . . . . . . 8 (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐡) ∧ π‘₯ ∈ 𝐡) β†’ ((π‘₯ ≀ 𝑃 β†’ ( 0 (ltβ€˜πΎ)π‘₯ β†’ π‘₯ = 𝑃)) ↔ (π‘₯ ≀ 𝑃 β†’ (π‘₯ β‰  0 β†’ π‘₯ = 𝑃))))
19 impexp 452 . . . . . . . . 9 ((( 0 (ltβ€˜πΎ)π‘₯ ∧ π‘₯ ≀ 𝑃) β†’ π‘₯ = 𝑃) ↔ ( 0 (ltβ€˜πΎ)π‘₯ β†’ (π‘₯ ≀ 𝑃 β†’ π‘₯ = 𝑃)))
20 bi2.04 389 . . . . . . . . 9 (( 0 (ltβ€˜πΎ)π‘₯ β†’ (π‘₯ ≀ 𝑃 β†’ π‘₯ = 𝑃)) ↔ (π‘₯ ≀ 𝑃 β†’ ( 0 (ltβ€˜πΎ)π‘₯ β†’ π‘₯ = 𝑃)))
2119, 20bitri 275 . . . . . . . 8 ((( 0 (ltβ€˜πΎ)π‘₯ ∧ π‘₯ ≀ 𝑃) β†’ π‘₯ = 𝑃) ↔ (π‘₯ ≀ 𝑃 β†’ ( 0 (ltβ€˜πΎ)π‘₯ β†’ π‘₯ = 𝑃)))
22 orcom 869 . . . . . . . . . 10 ((π‘₯ = 𝑃 ∨ π‘₯ = 0 ) ↔ (π‘₯ = 0 ∨ π‘₯ = 𝑃))
23 neor 3033 . . . . . . . . . 10 ((π‘₯ = 0 ∨ π‘₯ = 𝑃) ↔ (π‘₯ β‰  0 β†’ π‘₯ = 𝑃))
2422, 23bitri 275 . . . . . . . . 9 ((π‘₯ = 𝑃 ∨ π‘₯ = 0 ) ↔ (π‘₯ β‰  0 β†’ π‘₯ = 𝑃))
2524imbi2i 336 . . . . . . . 8 ((π‘₯ ≀ 𝑃 β†’ (π‘₯ = 𝑃 ∨ π‘₯ = 0 )) ↔ (π‘₯ ≀ 𝑃 β†’ (π‘₯ β‰  0 β†’ π‘₯ = 𝑃)))
2618, 21, 253bitr4g 314 . . . . . . 7 (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐡) ∧ π‘₯ ∈ 𝐡) β†’ ((( 0 (ltβ€˜πΎ)π‘₯ ∧ π‘₯ ≀ 𝑃) β†’ π‘₯ = 𝑃) ↔ (π‘₯ ≀ 𝑃 β†’ (π‘₯ = 𝑃 ∨ π‘₯ = 0 ))))
2726ralbidva 3169 . . . . . 6 ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐡) β†’ (βˆ€π‘₯ ∈ 𝐡 (( 0 (ltβ€˜πΎ)π‘₯ ∧ π‘₯ ≀ 𝑃) β†’ π‘₯ = 𝑃) ↔ βˆ€π‘₯ ∈ 𝐡 (π‘₯ ≀ 𝑃 β†’ (π‘₯ = 𝑃 ∨ π‘₯ = 0 ))))
2814, 27anbi12d 632 . . . . 5 ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐡) β†’ (( 0 (ltβ€˜πΎ)𝑃 ∧ βˆ€π‘₯ ∈ 𝐡 (( 0 (ltβ€˜πΎ)π‘₯ ∧ π‘₯ ≀ 𝑃) β†’ π‘₯ = 𝑃)) ↔ (𝑃 β‰  0 ∧ βˆ€π‘₯ ∈ 𝐡 (π‘₯ ≀ 𝑃 β†’ (π‘₯ = 𝑃 ∨ π‘₯ = 0 )))))
2913, 28bitr2d 280 . . . 4 ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐡) β†’ ((𝑃 β‰  0 ∧ βˆ€π‘₯ ∈ 𝐡 (π‘₯ ≀ 𝑃 β†’ (π‘₯ = 𝑃 ∨ π‘₯ = 0 ))) ↔ 0 ( β‹– β€˜πΎ)𝑃))
3029pm5.32da 580 . . 3 (𝐾 ∈ AtLat β†’ ((𝑃 ∈ 𝐡 ∧ (𝑃 β‰  0 ∧ βˆ€π‘₯ ∈ 𝐡 (π‘₯ ≀ 𝑃 β†’ (π‘₯ = 𝑃 ∨ π‘₯ = 0 )))) ↔ (𝑃 ∈ 𝐡 ∧ 0 ( β‹– β€˜πΎ)𝑃)))
315, 30bitr4d 282 . 2 (𝐾 ∈ AtLat β†’ (𝑃 ∈ 𝐴 ↔ (𝑃 ∈ 𝐡 ∧ (𝑃 β‰  0 ∧ βˆ€π‘₯ ∈ 𝐡 (π‘₯ ≀ 𝑃 β†’ (π‘₯ = 𝑃 ∨ π‘₯ = 0 ))))))
32 3anass 1096 . 2 ((𝑃 ∈ 𝐡 ∧ 𝑃 β‰  0 ∧ βˆ€π‘₯ ∈ 𝐡 (π‘₯ ≀ 𝑃 β†’ (π‘₯ = 𝑃 ∨ π‘₯ = 0 ))) ↔ (𝑃 ∈ 𝐡 ∧ (𝑃 β‰  0 ∧ βˆ€π‘₯ ∈ 𝐡 (π‘₯ ≀ 𝑃 β†’ (π‘₯ = 𝑃 ∨ π‘₯ = 0 )))))
3331, 32bitr4di 289 1 (𝐾 ∈ AtLat β†’ (𝑃 ∈ 𝐴 ↔ (𝑃 ∈ 𝐡 ∧ 𝑃 β‰  0 ∧ βˆ€π‘₯ ∈ 𝐡 (π‘₯ ≀ 𝑃 β†’ (π‘₯ = 𝑃 ∨ π‘₯ = 0 )))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∨ wo 846   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2940  βˆ€wral 3061   class class class wbr 5106  β€˜cfv 6497  Basecbs 17088  lecple 17145  ltcplt 18202  0.cp0 18317   β‹– ccvr 37770  Atomscatm 37771  AtLatcal 37772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-plt 18224  df-glb 18241  df-p0 18319  df-covers 37774  df-ats 37775  df-atl 37806
This theorem is referenced by:  atn0  37816  dihlspsnat  39842
  Copyright terms: Public domain W3C validator