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Theorem hlatexch4 35369
Description: Exchange 2 atoms. (Contributed by NM, 13-May-2013.)
Hypotheses
Ref Expression
hlatexch4.j = (join‘𝐾)
hlatexch4.a 𝐴 = (Atoms‘𝐾)
Assertion
Ref Expression
hlatexch4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → (𝑃 𝑅) = (𝑄 𝑆))

Proof of Theorem hlatexch4
StepHypRef Expression
1 simp11 1260 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → 𝐾 ∈ HL)
2 simp2l 1256 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → 𝑅𝐴)
3 simp2r 1257 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → 𝑆𝐴)
4 eqid 2765 . . . . . . . 8 (le‘𝐾) = (le‘𝐾)
5 hlatexch4.j . . . . . . . 8 = (join‘𝐾)
6 hlatexch4.a . . . . . . . 8 𝐴 = (Atoms‘𝐾)
74, 5, 6hlatlej2 35264 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑅𝐴𝑆𝐴) → 𝑆(le‘𝐾)(𝑅 𝑆))
81, 2, 3, 7syl3anc 1490 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → 𝑆(le‘𝐾)(𝑅 𝑆))
9 simp33 1268 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → (𝑃 𝑄) = (𝑅 𝑆))
108, 9breqtrrd 4837 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → 𝑆(le‘𝐾)(𝑃 𝑄))
11 simp12 1261 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → 𝑃𝐴)
12 simp13 1262 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → 𝑄𝐴)
13 simp32 1267 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → 𝑄𝑆)
1413necomd 2992 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → 𝑆𝑄)
154, 5, 6hlatexch2 35284 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑆𝐴𝑃𝐴𝑄𝐴) ∧ 𝑆𝑄) → (𝑆(le‘𝐾)(𝑃 𝑄) → 𝑃(le‘𝐾)(𝑆 𝑄)))
161, 3, 11, 12, 14, 15syl131anc 1502 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → (𝑆(le‘𝐾)(𝑃 𝑄) → 𝑃(le‘𝐾)(𝑆 𝑄)))
1710, 16mpd 15 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → 𝑃(le‘𝐾)(𝑆 𝑄))
185, 6hlatjcom 35256 . . . . 5 ((𝐾 ∈ HL ∧ 𝑆𝐴𝑄𝐴) → (𝑆 𝑄) = (𝑄 𝑆))
191, 3, 12, 18syl3anc 1490 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → (𝑆 𝑄) = (𝑄 𝑆))
2017, 19breqtrd 4835 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → 𝑃(le‘𝐾)(𝑄 𝑆))
214, 5, 6hlatlej2 35264 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → 𝑄(le‘𝐾)(𝑃 𝑄))
221, 11, 12, 21syl3anc 1490 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → 𝑄(le‘𝐾)(𝑃 𝑄))
2322, 9breqtrd 4835 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → 𝑄(le‘𝐾)(𝑅 𝑆))
244, 5, 6hlatexch2 35284 . . . . 5 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑅𝐴𝑆𝐴) ∧ 𝑄𝑆) → (𝑄(le‘𝐾)(𝑅 𝑆) → 𝑅(le‘𝐾)(𝑄 𝑆)))
251, 12, 2, 3, 13, 24syl131anc 1502 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → (𝑄(le‘𝐾)(𝑅 𝑆) → 𝑅(le‘𝐾)(𝑄 𝑆)))
2623, 25mpd 15 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → 𝑅(le‘𝐾)(𝑄 𝑆))
271hllatd 35252 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → 𝐾 ∈ Lat)
28 eqid 2765 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
2928, 6atbase 35177 . . . . 5 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
3011, 29syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → 𝑃 ∈ (Base‘𝐾))
3128, 6atbase 35177 . . . . 5 (𝑅𝐴𝑅 ∈ (Base‘𝐾))
322, 31syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → 𝑅 ∈ (Base‘𝐾))
3328, 5, 6hlatjcl 35255 . . . . 5 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑆𝐴) → (𝑄 𝑆) ∈ (Base‘𝐾))
341, 12, 3, 33syl3anc 1490 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → (𝑄 𝑆) ∈ (Base‘𝐾))
3528, 4, 5latjle12 17330 . . . 4 ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾) ∧ (𝑄 𝑆) ∈ (Base‘𝐾))) → ((𝑃(le‘𝐾)(𝑄 𝑆) ∧ 𝑅(le‘𝐾)(𝑄 𝑆)) ↔ (𝑃 𝑅)(le‘𝐾)(𝑄 𝑆)))
3627, 30, 32, 34, 35syl13anc 1491 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → ((𝑃(le‘𝐾)(𝑄 𝑆) ∧ 𝑅(le‘𝐾)(𝑄 𝑆)) ↔ (𝑃 𝑅)(le‘𝐾)(𝑄 𝑆)))
3720, 26, 36mpbi2and 703 . 2 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → (𝑃 𝑅)(le‘𝐾)(𝑄 𝑆))
38 simp31 1266 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → 𝑃𝑅)
394, 5, 6ps-1 35365 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑅𝐴𝑃𝑅) ∧ (𝑄𝐴𝑆𝐴)) → ((𝑃 𝑅)(le‘𝐾)(𝑄 𝑆) ↔ (𝑃 𝑅) = (𝑄 𝑆)))
401, 11, 2, 38, 12, 3, 39syl132anc 1507 . 2 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → ((𝑃 𝑅)(le‘𝐾)(𝑄 𝑆) ↔ (𝑃 𝑅) = (𝑄 𝑆)))
4137, 40mpbid 223 1 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → (𝑃 𝑅) = (𝑄 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  wne 2937   class class class wbr 4809  cfv 6068  (class class class)co 6842  Basecbs 16132  lecple 16223  joincjn 17212  Latclat 17313  Atomscatm 35151  HLchlt 35238
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-proset 17196  df-poset 17214  df-plt 17226  df-lub 17242  df-glb 17243  df-join 17244  df-meet 17245  df-p0 17307  df-lat 17314  df-covers 35154  df-ats 35155  df-atl 35186  df-cvlat 35210  df-hlat 35239
This theorem is referenced by:  cdlemg18a  36566
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