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Theorem cdlemn11a 40073
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 38885 . . . . . 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 39445 . . . . 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 7170 . . . 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 39635 . . 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 7368 . . . . 5 (β„©β„Ž ∈ 𝑇 (β„Žβ€˜π‘ƒ) = 𝑁) ∈ V
2110, 20eqeltri 2829 . . . 4 𝐺 ∈ V
229fvexi 6905 . . . . 5 𝑇 ∈ V
23 resiexg 7904 . . . . 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 40047 . . 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 7363  (class class class)co 7408  Basecbs 17143  +gcplusg 17196  lecple 17203  occoc 17204  joincjn 18263  LSSumclsm 19501  Atomscatm 38128  HLchlt 38215  LHypclh 38850  LTrncltrn 38967  trLctrl 39024  TEndoctendo 39618  DVecHcdvh 39944  DIsoBcdib 40004  DIsoCcdic 40038
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 7724  ax-riotaBAD 37818
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 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-1st 7974  df-2nd 7975  df-undef 8257  df-map 8821  df-proset 18247  df-poset 18265  df-plt 18282  df-lub 18298  df-glb 18299  df-join 18300  df-meet 18301  df-p0 18377  df-p1 18378  df-lat 18384  df-clat 18451  df-oposet 38041  df-ol 38043  df-oml 38044  df-covers 38131  df-ats 38132  df-atl 38163  df-cvlat 38187  df-hlat 38216  df-llines 38364  df-lplanes 38365  df-lvols 38366  df-lines 38367  df-psubsp 38369  df-pmap 38370  df-padd 38662  df-lhyp 38854  df-laut 38855  df-ldil 38970  df-ltrn 38971  df-trl 39025  df-tendo 39621  df-dic 40039
This theorem is referenced by:  cdlemn11b  40074
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