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Theorem cdlemn11b 40017
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 1139 . 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 40016 . 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 3982 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   βŠ† wss 3947  βŸ¨cop 4633   class class class wbr 5147   ↦ cmpt 5230   I cid 5572   β†Ύ cres 5677  β€˜cfv 6540  β„©crio 7359  (class class class)co 7404  Basecbs 17140  +gcplusg 17193  lecple 17200  occoc 17201  joincjn 18260  LSSumclsm 19495  Atomscatm 38071  HLchlt 38158  LHypclh 38793  LTrncltrn 38910  trLctrl 38967  TEndoctendo 39561  DVecHcdvh 39887  DIsoBcdib 39947  DIsoCcdic 39981
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 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7720  ax-riotaBAD 37761
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  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 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-iin 4999  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-1st 7970  df-2nd 7971  df-undef 8253  df-map 8818  df-proset 18244  df-poset 18262  df-plt 18279  df-lub 18295  df-glb 18296  df-join 18297  df-meet 18298  df-p0 18374  df-p1 18375  df-lat 18381  df-clat 18448  df-oposet 37984  df-ol 37986  df-oml 37987  df-covers 38074  df-ats 38075  df-atl 38106  df-cvlat 38130  df-hlat 38159  df-llines 38307  df-lplanes 38308  df-lvols 38309  df-lines 38310  df-psubsp 38312  df-pmap 38313  df-padd 38605  df-lhyp 38797  df-laut 38798  df-ldil 38913  df-ltrn 38914  df-trl 38968  df-tendo 39564  df-dic 39982
This theorem is referenced by:  cdlemn11c  40018
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