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Theorem cdlemg7N 37648
Description: TODO: FIX COMMENT. (Contributed by NM, 28-Apr-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
cdlemg7.b 𝐵 = (Base‘𝐾)
cdlemg7.l = (le‘𝐾)
cdlemg7.a 𝐴 = (Atoms‘𝐾)
cdlemg7.h 𝐻 = (LHyp‘𝐾)
cdlemg7.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
Assertion
Ref Expression
cdlemg7N (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑋𝐵) ∧ (𝐹𝑇𝐺𝑇 ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) → (𝐹‘(𝐺𝑋)) = 𝑋)

Proof of Theorem cdlemg7N
StepHypRef Expression
1 simpl1 1185 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑋𝐵) ∧ (𝐹𝑇𝐺𝑇 ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) ∧ 𝑋 𝑊) → (𝐾 ∈ HL ∧ 𝑊𝐻))
2 simpl31 1248 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑋𝐵) ∧ (𝐹𝑇𝐺𝑇 ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) ∧ 𝑋 𝑊) → 𝐹𝑇)
3 simpl32 1249 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑋𝐵) ∧ (𝐹𝑇𝐺𝑇 ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) ∧ 𝑋 𝑊) → 𝐺𝑇)
4 simpl2r 1221 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑋𝐵) ∧ (𝐹𝑇𝐺𝑇 ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) ∧ 𝑋 𝑊) → 𝑋𝐵)
5 cdlemg7.b . . . . . 6 𝐵 = (Base‘𝐾)
6 cdlemg7.h . . . . . 6 𝐻 = (LHyp‘𝐾)
7 cdlemg7.t . . . . . 6 𝑇 = ((LTrn‘𝐾)‘𝑊)
85, 6, 7ltrncl 37147 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐺𝑇𝑋𝐵) → (𝐺𝑋) ∈ 𝐵)
91, 3, 4, 8syl3anc 1365 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑋𝐵) ∧ (𝐹𝑇𝐺𝑇 ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) ∧ 𝑋 𝑊) → (𝐺𝑋) ∈ 𝐵)
10 simpr 485 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑋𝐵) ∧ (𝐹𝑇𝐺𝑇 ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) ∧ 𝑋 𝑊) → 𝑋 𝑊)
11 cdlemg7.l . . . . . . 7 = (le‘𝐾)
125, 11, 6, 7ltrnval1 37156 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐺𝑇 ∧ (𝑋𝐵𝑋 𝑊)) → (𝐺𝑋) = 𝑋)
131, 3, 4, 10, 12syl112anc 1368 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑋𝐵) ∧ (𝐹𝑇𝐺𝑇 ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) ∧ 𝑋 𝑊) → (𝐺𝑋) = 𝑋)
1413, 10eqbrtrd 5085 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑋𝐵) ∧ (𝐹𝑇𝐺𝑇 ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) ∧ 𝑋 𝑊) → (𝐺𝑋) 𝑊)
155, 11, 6, 7ltrnval1 37156 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝐺𝑋) ∈ 𝐵 ∧ (𝐺𝑋) 𝑊)) → (𝐹‘(𝐺𝑋)) = (𝐺𝑋))
161, 2, 9, 14, 15syl112anc 1368 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑋𝐵) ∧ (𝐹𝑇𝐺𝑇 ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) ∧ 𝑋 𝑊) → (𝐹‘(𝐺𝑋)) = (𝐺𝑋))
1716, 13eqtrd 2861 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑋𝐵) ∧ (𝐹𝑇𝐺𝑇 ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) ∧ 𝑋 𝑊) → (𝐹‘(𝐺𝑋)) = 𝑋)
18 simpl1 1185 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑋𝐵) ∧ (𝐹𝑇𝐺𝑇 ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) ∧ ¬ 𝑋 𝑊) → (𝐾 ∈ HL ∧ 𝑊𝐻))
19 simpl2l 1220 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑋𝐵) ∧ (𝐹𝑇𝐺𝑇 ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) ∧ ¬ 𝑋 𝑊) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
20 simpl2r 1221 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑋𝐵) ∧ (𝐹𝑇𝐺𝑇 ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) ∧ ¬ 𝑋 𝑊) → 𝑋𝐵)
21 simpr 485 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑋𝐵) ∧ (𝐹𝑇𝐺𝑇 ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) ∧ ¬ 𝑋 𝑊) → ¬ 𝑋 𝑊)
2220, 21jca 512 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑋𝐵) ∧ (𝐹𝑇𝐺𝑇 ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) ∧ ¬ 𝑋 𝑊) → (𝑋𝐵 ∧ ¬ 𝑋 𝑊))
23 simpl31 1248 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑋𝐵) ∧ (𝐹𝑇𝐺𝑇 ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) ∧ ¬ 𝑋 𝑊) → 𝐹𝑇)
24 simpl32 1249 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑋𝐵) ∧ (𝐹𝑇𝐺𝑇 ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) ∧ ¬ 𝑋 𝑊) → 𝐺𝑇)
25 simpl33 1250 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑋𝐵) ∧ (𝐹𝑇𝐺𝑇 ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) ∧ ¬ 𝑋 𝑊) → (𝐹‘(𝐺𝑃)) = 𝑃)
26 cdlemg7.a . . . 4 𝐴 = (Atoms‘𝐾)
275, 11, 26, 6, 7cdlemg7aN 37647 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) ∧ (𝐹𝑇𝐺𝑇 ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) → (𝐹‘(𝐺𝑋)) = 𝑋)
2818, 19, 22, 23, 24, 25, 27syl123anc 1381 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑋𝐵) ∧ (𝐹𝑇𝐺𝑇 ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) ∧ ¬ 𝑋 𝑊) → (𝐹‘(𝐺𝑋)) = 𝑋)
2917, 28pm2.61dan 809 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑋𝐵) ∧ (𝐹𝑇𝐺𝑇 ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) → (𝐹‘(𝐺𝑋)) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  w3a 1081   = wceq 1530  wcel 2107   class class class wbr 5063  cfv 6354  Basecbs 16478  lecple 16567  Atomscatm 36285  HLchlt 36372  LHypclh 37006  LTrncltrn 37123
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-13 2385  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455  ax-riotaBAD 35975
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-iun 4919  df-iin 4920  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-1st 7685  df-2nd 7686  df-undef 7935  df-map 8403  df-proset 17533  df-poset 17551  df-plt 17563  df-lub 17579  df-glb 17580  df-join 17581  df-meet 17582  df-p0 17644  df-p1 17645  df-lat 17651  df-clat 17713  df-oposet 36198  df-ol 36200  df-oml 36201  df-covers 36288  df-ats 36289  df-atl 36320  df-cvlat 36344  df-hlat 36373  df-llines 36520  df-lplanes 36521  df-lvols 36522  df-lines 36523  df-psubsp 36525  df-pmap 36526  df-padd 36818  df-lhyp 37010  df-laut 37011  df-ldil 37126  df-ltrn 37127  df-trl 37181
This theorem is referenced by: (None)
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