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Theorem cdlemn11a 40164
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
cdlemn11a (((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‘))
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 cdlemn11a
StepHypRef Expression
1 simp1 1136 . . . . 5 (((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†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
2 cdlemn11a.l . . . . . . 7 ≀ = (leβ€˜πΎ)
3 cdlemn11a.a . . . . . . 7 𝐴 = (Atomsβ€˜πΎ)
4 cdlemn11a.h . . . . . . 7 𝐻 = (LHypβ€˜πΎ)
5 cdlemn11a.p . . . . . . 7 𝑃 = ((ocβ€˜πΎ)β€˜π‘Š)
62, 3, 4, 5lhpocnel2 38976 . . . . . 6 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š))
763ad2ant1 1133 . . . . 5 (((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†’ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š))
8 simp22 1207 . . . . 5 (((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†’ (𝑁 ∈ 𝐴 ∧ Β¬ 𝑁 ≀ π‘Š))
9 cdlemn11a.t . . . . . 6 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
10 cdlemn11a.g . . . . . 6 𝐺 = (β„©β„Ž ∈ 𝑇 (β„Žβ€˜π‘ƒ) = 𝑁)
112, 3, 4, 9, 10ltrniotacl 39536 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑁 ∈ 𝐴 ∧ Β¬ 𝑁 ≀ π‘Š)) β†’ 𝐺 ∈ 𝑇)
121, 7, 8, 11syl3anc 1371 . . . 4 (((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 ∈ 𝑇)
13 fvresi 7173 . . . 4 (𝐺 ∈ 𝑇 β†’ (( I β†Ύ 𝑇)β€˜πΊ) = 𝐺)
1412, 13syl 17 . . 3 (((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†’ (( I β†Ύ 𝑇)β€˜πΊ) = 𝐺)
1514eqcomd 2738 . 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)β€˜πΊ))
16 cdlemn11a.e . . . 4 𝐸 = ((TEndoβ€˜πΎ)β€˜π‘Š)
174, 9, 16tendoidcl 39726 . . 3 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ ( I β†Ύ 𝑇) ∈ 𝐸)
18173ad2ant1 1133 . 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†’ ( I β†Ύ 𝑇) ∈ 𝐸)
19 cdlemn11a.J . . . 4 𝐽 = ((DIsoCβ€˜πΎ)β€˜π‘Š)
20 riotaex 7371 . . . . 5 (β„©β„Ž ∈ 𝑇 (β„Žβ€˜π‘ƒ) = 𝑁) ∈ V
2110, 20eqeltri 2829 . . . 4 𝐺 ∈ V
229fvexi 6905 . . . . 5 𝑇 ∈ V
23 resiexg 7907 . . . . 5 (𝑇 ∈ V β†’ ( I β†Ύ 𝑇) ∈ V)
2422, 23ax-mp 5 . . . 4 ( I β†Ύ 𝑇) ∈ V
252, 3, 4, 5, 9, 16, 19, 10, 21, 24dicopelval2 40138 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑁 ∈ 𝐴 ∧ Β¬ 𝑁 ≀ π‘Š)) β†’ (⟨𝐺, ( I β†Ύ 𝑇)⟩ ∈ (π½β€˜π‘) ↔ (𝐺 = (( I β†Ύ 𝑇)β€˜πΊ) ∧ ( I β†Ύ 𝑇) ∈ 𝐸)))
261, 8, 25syl2anc 584 . 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‘) ↔ (𝐺 = (( I β†Ύ 𝑇)β€˜πΊ) ∧ ( I β†Ύ 𝑇) ∈ 𝐸)))
2715, 18, 26mpbir2and 711 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‘))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  Vcvv 3474   βŠ† wss 3948  βŸ¨cop 4634   class class class wbr 5148   ↦ cmpt 5231   I cid 5573   β†Ύ cres 5678  β€˜cfv 6543  β„©crio 7366  (class class class)co 7411  Basecbs 17146  +gcplusg 17199  lecple 17206  occoc 17207  joincjn 18266  LSSumclsm 19504  Atomscatm 38219  HLchlt 38306  LHypclh 38941  LTrncltrn 39058  trLctrl 39115  TEndoctendo 39709  DVecHcdvh 40035  DIsoBcdib 40095  DIsoCcdic 40129
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-riotaBAD 37909
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7977  df-2nd 7978  df-undef 8260  df-map 8824  df-proset 18250  df-poset 18268  df-plt 18285  df-lub 18301  df-glb 18302  df-join 18303  df-meet 18304  df-p0 18380  df-p1 18381  df-lat 18387  df-clat 18454  df-oposet 38132  df-ol 38134  df-oml 38135  df-covers 38222  df-ats 38223  df-atl 38254  df-cvlat 38278  df-hlat 38307  df-llines 38455  df-lplanes 38456  df-lvols 38457  df-lines 38458  df-psubsp 38460  df-pmap 38461  df-padd 38753  df-lhyp 38945  df-laut 38946  df-ldil 39061  df-ltrn 39062  df-trl 39116  df-tendo 39712  df-dic 40130
This theorem is referenced by:  cdlemn11b  40165
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