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Theorem diaord 39913
Description: The partial isomorphism A for a lattice 𝐾 is order-preserving in the region under co-atom π‘Š. Part of Lemma M of [Crawley] p. 120 line 28. (Contributed by NM, 26-Nov-2013.)
Hypotheses
Ref Expression
dia11.b 𝐡 = (Baseβ€˜πΎ)
dia11.l ≀ = (leβ€˜πΎ)
dia11.h 𝐻 = (LHypβ€˜πΎ)
dia11.i 𝐼 = ((DIsoAβ€˜πΎ)β€˜π‘Š)
Assertion
Ref Expression
diaord (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š) ∧ (π‘Œ ∈ 𝐡 ∧ π‘Œ ≀ π‘Š)) β†’ ((πΌβ€˜π‘‹) βŠ† (πΌβ€˜π‘Œ) ↔ 𝑋 ≀ π‘Œ))

Proof of Theorem diaord
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 dia11.b . . . . 5 𝐡 = (Baseβ€˜πΎ)
2 dia11.l . . . . 5 ≀ = (leβ€˜πΎ)
3 dia11.h . . . . 5 𝐻 = (LHypβ€˜πΎ)
4 eqid 2732 . . . . 5 ((LTrnβ€˜πΎ)β€˜π‘Š) = ((LTrnβ€˜πΎ)β€˜π‘Š)
5 eqid 2732 . . . . 5 ((trLβ€˜πΎ)β€˜π‘Š) = ((trLβ€˜πΎ)β€˜π‘Š)
6 dia11.i . . . . 5 𝐼 = ((DIsoAβ€˜πΎ)β€˜π‘Š)
71, 2, 3, 4, 5, 6diaval 39898 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š)) β†’ (πΌβ€˜π‘‹) = {𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ∣ (((trLβ€˜πΎ)β€˜π‘Š)β€˜π‘“) ≀ 𝑋})
873adant3 1132 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š) ∧ (π‘Œ ∈ 𝐡 ∧ π‘Œ ≀ π‘Š)) β†’ (πΌβ€˜π‘‹) = {𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ∣ (((trLβ€˜πΎ)β€˜π‘Š)β€˜π‘“) ≀ 𝑋})
91, 2, 3, 4, 5, 6diaval 39898 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (π‘Œ ∈ 𝐡 ∧ π‘Œ ≀ π‘Š)) β†’ (πΌβ€˜π‘Œ) = {𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ∣ (((trLβ€˜πΎ)β€˜π‘Š)β€˜π‘“) ≀ π‘Œ})
1093adant2 1131 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š) ∧ (π‘Œ ∈ 𝐡 ∧ π‘Œ ≀ π‘Š)) β†’ (πΌβ€˜π‘Œ) = {𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ∣ (((trLβ€˜πΎ)β€˜π‘Š)β€˜π‘“) ≀ π‘Œ})
118, 10sseq12d 4015 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š) ∧ (π‘Œ ∈ 𝐡 ∧ π‘Œ ≀ π‘Š)) β†’ ((πΌβ€˜π‘‹) βŠ† (πΌβ€˜π‘Œ) ↔ {𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ∣ (((trLβ€˜πΎ)β€˜π‘Š)β€˜π‘“) ≀ 𝑋} βŠ† {𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ∣ (((trLβ€˜πΎ)β€˜π‘Š)β€˜π‘“) ≀ π‘Œ}))
12 ss2rab 4068 . . 3 ({𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ∣ (((trLβ€˜πΎ)β€˜π‘Š)β€˜π‘“) ≀ 𝑋} βŠ† {𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ∣ (((trLβ€˜πΎ)β€˜π‘Š)β€˜π‘“) ≀ π‘Œ} ↔ βˆ€π‘“ ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)((((trLβ€˜πΎ)β€˜π‘Š)β€˜π‘“) ≀ 𝑋 β†’ (((trLβ€˜πΎ)β€˜π‘Š)β€˜π‘“) ≀ π‘Œ))
13 eqid 2732 . . . 4 (Atomsβ€˜πΎ) = (Atomsβ€˜πΎ)
141, 2, 13, 3, 4, 5trlord 39435 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š) ∧ (π‘Œ ∈ 𝐡 ∧ π‘Œ ≀ π‘Š)) β†’ (𝑋 ≀ π‘Œ ↔ βˆ€π‘“ ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)((((trLβ€˜πΎ)β€˜π‘Š)β€˜π‘“) ≀ 𝑋 β†’ (((trLβ€˜πΎ)β€˜π‘Š)β€˜π‘“) ≀ π‘Œ)))
1512, 14bitr4id 289 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š) ∧ (π‘Œ ∈ 𝐡 ∧ π‘Œ ≀ π‘Š)) β†’ ({𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ∣ (((trLβ€˜πΎ)β€˜π‘Š)β€˜π‘“) ≀ 𝑋} βŠ† {𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ∣ (((trLβ€˜πΎ)β€˜π‘Š)β€˜π‘“) ≀ π‘Œ} ↔ 𝑋 ≀ π‘Œ))
1611, 15bitrd 278 1 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š) ∧ (π‘Œ ∈ 𝐡 ∧ π‘Œ ≀ π‘Š)) β†’ ((πΌβ€˜π‘‹) βŠ† (πΌβ€˜π‘Œ) ↔ 𝑋 ≀ π‘Œ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  {crab 3432   βŠ† wss 3948   class class class wbr 5148  β€˜cfv 6543  Basecbs 17143  lecple 17203  Atomscatm 38128  HLchlt 38215  LHypclh 38850  LTrncltrn 38967  trLctrl 39024  DIsoAcdia 39894
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-disoa 39895
This theorem is referenced by:  dia11N  39914  dia2dimlem10  39939  dibord  40025
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