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Theorem 4atex2-0bOLDN 34062
Description: Same as 4atex2 34060 except that T is zero. (Contributed by NM, 27-May-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
4that.l |- .<_ = ( le ` K )
4that.j |- .\/ = ( join ` K )
4that.a |- A = ( Atoms ` K )
4that.h |- H = ( LHyp ` K )
Assertion
Ref Expression
4atex2-0bOLDN |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ T = ( 0. ` K ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> E. z e. A ( -. z .<_ W /\ ( S .\/ z ) = ( T .\/ z ) ) )
Distinct variable groups:   z, r, A   H, r   .\/ , r, z   K, r, z   .<_ , r, z   P, r, z   Q, r, z   S, r, z   W, r, z   T, r, z
Allowed substitution hint:   H( z)
Proof of Theorem 4atex2-0bOLDN
StepHypRef Expression
1 simp1 988 . . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ T = ( 0. ` K ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( K e. HL /\ W e. H ) )
2 simp21 1021 . . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ T = ( 0. ` K ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( P e. A /\ -. P .<_ W ) )
3 simp22 1022 . . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ T = ( 0. ` K ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( Q e. A /\ -. Q .<_ W ) )
4 simp32 1025 . . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ T = ( 0. ` K ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> T = ( 0. ` K ) )
5 simp31 1024 . . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ T = ( 0. ` K ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> P =/= Q )
6 simp23 1023 . . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ T = ( 0. ` K ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( S e. A /\ -. S .<_ W ) )
7 simp33 1026 . . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ T = ( 0. ` K ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) )
8 4that.l . . . 4 |- .<_ = ( le ` K )
9 4that.j . . . 4 |- .\/ = ( join ` K )
10 4that.a . . . 4 |- A = ( Atoms ` K )
11 4that.h . . . 4 |- H = ( LHyp ` K )
128, 9, 10, 114atex2-0aOLDN 34061 . . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ T = ( 0. ` K ) ) /\ ( P =/= Q /\ ( S e. A /\ -. S .<_ W ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> E. z e. A ( -. z .<_ W /\ ( T .\/ z ) = ( S .\/ z ) ) )
131, 2, 3, 4, 5, 6, 7, 12syl133anc 1242 . 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ T = ( 0. ` K ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> E. z e. A ( -. z .<_ W /\ ( T .\/ z ) = ( S .\/ z ) ) )
14 eqcom 2463 . . . 4 |- ( ( S .\/ z ) = ( T .\/ z ) <-> ( T .\/ z ) = ( S .\/ z ) )
1514anbi2i 694 . . 3 |- ( ( -. z .<_ W /\ ( S .\/ z ) = ( T .\/ z ) ) <-> ( -. z .<_ W /\ ( T .\/ z ) = ( S .\/ z ) ) )
1615rexbii 2862 . 2 |- ( E. z e. A ( -. z .<_ W /\ ( S .\/ z ) = ( T .\/ z ) ) <-> E. z e. A ( -. z .<_ W /\ ( T .\/ z ) = ( S .\/ z ) ) )
1713, 16sylibr 212 1 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ T = ( 0. ` K ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> E. z e. A ( -. z .<_ W /\ ( S .\/ z ) = ( T .\/ z ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   -> wi 4   /\ wa 369   /\ w3a 965   = wceq 1370   e. wcel 1758   =/= wne 2648   E.wrex 2800   class class class wbr 4401   ` cfv 5527  (class class class)co 6201   lecple 14365   joincjn 15234   0.cp0 15327   Atomscatm 33247   HLchlt 33334   LHypclh 33967
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-poset 15236  df-lub 15264  df-glb 15265  df-join 15266  df-meet 15267  df-p0 15329  df-lat 15336  df-oposet 33160  df-ol 33162  df-oml 33163  df-ats 33251  df-atl 33282  df-cvlat 33306  df-hlat 33335
This theorem is referenced by: (None)
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