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Theorem cdlemn11b 40535
Description: Part of proof of Lemma N of [Crawley] p. 121 line 37. (Contributed by NM, 27-Feb-2014.)
Hypotheses
Ref Expression
cdlemn11a.b 𝐡 = (Baseβ€˜πΎ)
cdlemn11a.l ≀ = (leβ€˜πΎ)
cdlemn11a.j ∨ = (joinβ€˜πΎ)
cdlemn11a.a 𝐴 = (Atomsβ€˜πΎ)
cdlemn11a.h 𝐻 = (LHypβ€˜πΎ)
cdlemn11a.p 𝑃 = ((ocβ€˜πΎ)β€˜π‘Š)
cdlemn11a.o 𝑂 = (β„Ž ∈ 𝑇 ↦ ( I β†Ύ 𝐡))
cdlemn11a.t 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
cdlemn11a.r 𝑅 = ((trLβ€˜πΎ)β€˜π‘Š)
cdlemn11a.e 𝐸 = ((TEndoβ€˜πΎ)β€˜π‘Š)
cdlemn11a.i 𝐼 = ((DIsoBβ€˜πΎ)β€˜π‘Š)
cdlemn11a.J 𝐽 = ((DIsoCβ€˜πΎ)β€˜π‘Š)
cdlemn11a.u π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)
cdlemn11a.d + = (+gβ€˜π‘ˆ)
cdlemn11a.s βŠ• = (LSSumβ€˜π‘ˆ)
cdlemn11a.f 𝐹 = (β„©β„Ž ∈ 𝑇 (β„Žβ€˜π‘ƒ) = 𝑄)
cdlemn11a.g 𝐺 = (β„©β„Ž ∈ 𝑇 (β„Žβ€˜π‘ƒ) = 𝑁)
Assertion
Ref Expression
cdlemn11b (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑁 ∈ 𝐴 ∧ Β¬ 𝑁 ≀ π‘Š) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š)) ∧ (π½β€˜π‘) βŠ† ((π½β€˜π‘„) βŠ• (πΌβ€˜π‘‹))) β†’ ⟨𝐺, ( I β†Ύ 𝑇)⟩ ∈ ((π½β€˜π‘„) βŠ• (πΌβ€˜π‘‹)))
Distinct variable groups:   ≀ ,β„Ž   𝐴,β„Ž   𝐡,β„Ž   β„Ž,𝐻   β„Ž,𝐾   β„Ž,𝑁   𝑃,β„Ž   𝑄,β„Ž   𝑇,β„Ž   β„Ž,π‘Š
Allowed substitution hints:   + (β„Ž)   βŠ• (β„Ž)   𝑅(β„Ž)   π‘ˆ(β„Ž)   𝐸(β„Ž)   𝐹(β„Ž)   𝐺(β„Ž)   𝐼(β„Ž)   𝐽(β„Ž)   ∨ (β„Ž)   𝑂(β„Ž)   𝑋(β„Ž)
Proof of Theorem cdlemn11b
StepHypRef Expression
1 simp3 1135 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑁 ∈ 𝐴 ∧ Β¬ 𝑁 ≀ π‘Š) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š)) ∧ (π½β€˜π‘) βŠ† ((π½β€˜π‘„) βŠ• (πΌβ€˜π‘‹))) β†’ (π½β€˜π‘) βŠ† ((π½β€˜π‘„) βŠ• (πΌβ€˜π‘‹)))
2 cdlemn11a.b . . 3 𝐡 = (Baseβ€˜πΎ)
3 cdlemn11a.l . . 3 ≀ = (leβ€˜πΎ)
4 cdlemn11a.j . . 3 ∨ = (joinβ€˜πΎ)
5 cdlemn11a.a . . 3 𝐴 = (Atomsβ€˜πΎ)
6 cdlemn11a.h . . 3 𝐻 = (LHypβ€˜πΎ)
7 cdlemn11a.p . . 3 𝑃 = ((ocβ€˜πΎ)β€˜π‘Š)
8 cdlemn11a.o . . 3 𝑂 = (β„Ž ∈ 𝑇 ↦ ( I β†Ύ 𝐡))
9 cdlemn11a.t . . 3 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
10 cdlemn11a.r . . 3 𝑅 = ((trLβ€˜πΎ)β€˜π‘Š)
11 cdlemn11a.e . . 3 𝐸 = ((TEndoβ€˜πΎ)β€˜π‘Š)
12 cdlemn11a.i . . 3 𝐼 = ((DIsoBβ€˜πΎ)β€˜π‘Š)
13 cdlemn11a.J . . 3 𝐽 = ((DIsoCβ€˜πΎ)β€˜π‘Š)
14 cdlemn11a.u . . 3 π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)
15 cdlemn11a.d . . 3 + = (+gβ€˜π‘ˆ)
16 cdlemn11a.s . . 3 βŠ• = (LSSumβ€˜π‘ˆ)
17 cdlemn11a.f . . 3 𝐹 = (β„©β„Ž ∈ 𝑇 (β„Žβ€˜π‘ƒ) = 𝑄)
18 cdlemn11a.g . . 3 𝐺 = (β„©β„Ž ∈ 𝑇 (β„Žβ€˜π‘ƒ) = 𝑁)
192, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18cdlemn11a 40534 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑁 ∈ 𝐴 ∧ Β¬ 𝑁 ≀ π‘Š) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š)) ∧ (π½β€˜π‘) βŠ† ((π½β€˜π‘„) βŠ• (πΌβ€˜π‘‹))) β†’ ⟨𝐺, ( I β†Ύ 𝑇)⟩ ∈ (π½β€˜π‘))
201, 19sseldd 3975 1 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑁 ∈ 𝐴 ∧ Β¬ 𝑁 ≀ π‘Š) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š)) ∧ (π½β€˜π‘) βŠ† ((π½β€˜π‘„) βŠ• (πΌβ€˜π‘‹))) β†’ ⟨𝐺, ( I β†Ύ 𝑇)⟩ ∈ ((π½β€˜π‘„) βŠ• (πΌβ€˜π‘‹)))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   βŠ† wss 3940  βŸ¨cop 4626   class class class wbr 5138   ↦ cmpt 5221   I cid 5563   β†Ύ cres 5668  β€˜cfv 6533  β„©crio 7356  (class class class)co 7401  Basecbs 17142  +gcplusg 17195  lecple 17202  occoc 17203  joincjn 18265  LSSumclsm 19543  Atomscatm 38589  HLchlt 38676  LHypclh 39311  LTrncltrn 39428  trLctrl 39485  TEndoctendo 40079  DVecHcdvh 40405  DIsoBcdib 40465  DIsoCcdic 40499
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-riotaBAD 38279
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-iin 4990  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-1st 7968  df-2nd 7969  df-undef 8253  df-map 8817  df-proset 18249  df-poset 18267  df-plt 18284  df-lub 18300  df-glb 18301  df-join 18302  df-meet 18303  df-p0 18379  df-p1 18380  df-lat 18386  df-clat 18453  df-oposet 38502  df-ol 38504  df-oml 38505  df-covers 38592  df-ats 38593  df-atl 38624  df-cvlat 38648  df-hlat 38677  df-llines 38825  df-lplanes 38826  df-lvols 38827  df-lines 38828  df-psubsp 38830  df-pmap 38831  df-padd 39123  df-lhyp 39315  df-laut 39316  df-ldil 39431  df-ltrn 39432  df-trl 39486  df-tendo 40082  df-dic 40500
This theorem is referenced by:  cdlemn11c  40536
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