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Theorem dicn0 40063
Description: The value of the partial isomorphism C is not empty. (Contributed by NM, 15-Feb-2014.)
Hypotheses
Ref Expression
dicn0.l ≀ = (leβ€˜πΎ)
dicn0.a 𝐴 = (Atomsβ€˜πΎ)
dicn0.h 𝐻 = (LHypβ€˜πΎ)
dicn0.i 𝐼 = ((DIsoCβ€˜πΎ)β€˜π‘Š)
Assertion
Ref Expression
dicn0 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) β†’ (πΌβ€˜π‘„) β‰  βˆ…)

Proof of Theorem dicn0
Dummy variables 𝑔 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 484 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
2 dicn0.l . . . . . . . 8 ≀ = (leβ€˜πΎ)
3 eqid 2733 . . . . . . . 8 (ocβ€˜πΎ) = (ocβ€˜πΎ)
4 dicn0.a . . . . . . . 8 𝐴 = (Atomsβ€˜πΎ)
5 dicn0.h . . . . . . . 8 𝐻 = (LHypβ€˜πΎ)
62, 3, 4, 5lhpocnel 38889 . . . . . . 7 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ (((ocβ€˜πΎ)β€˜π‘Š) ∈ 𝐴 ∧ Β¬ ((ocβ€˜πΎ)β€˜π‘Š) ≀ π‘Š))
76adantr 482 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) β†’ (((ocβ€˜πΎ)β€˜π‘Š) ∈ 𝐴 ∧ Β¬ ((ocβ€˜πΎ)β€˜π‘Š) ≀ π‘Š))
8 simpr 486 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) β†’ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š))
9 eqid 2733 . . . . . . 7 ((LTrnβ€˜πΎ)β€˜π‘Š) = ((LTrnβ€˜πΎ)β€˜π‘Š)
10 eqid 2733 . . . . . . 7 (℩𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)(π‘”β€˜((ocβ€˜πΎ)β€˜π‘Š)) = 𝑄) = (℩𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)(π‘”β€˜((ocβ€˜πΎ)β€˜π‘Š)) = 𝑄)
112, 4, 5, 9, 10ltrniotacl 39450 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (((ocβ€˜πΎ)β€˜π‘Š) ∈ 𝐴 ∧ Β¬ ((ocβ€˜πΎ)β€˜π‘Š) ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) β†’ (℩𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)(π‘”β€˜((ocβ€˜πΎ)β€˜π‘Š)) = 𝑄) ∈ ((LTrnβ€˜πΎ)β€˜π‘Š))
121, 7, 8, 11syl3anc 1372 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) β†’ (℩𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)(π‘”β€˜((ocβ€˜πΎ)β€˜π‘Š)) = 𝑄) ∈ ((LTrnβ€˜πΎ)β€˜π‘Š))
13 eqid 2733 . . . . . 6 (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ ( I β†Ύ (Baseβ€˜πΎ))) = (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ ( I β†Ύ (Baseβ€˜πΎ)))
14 eqid 2733 . . . . . 6 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
1513, 14tendo02 39658 . . . . 5 ((℩𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)(π‘”β€˜((ocβ€˜πΎ)β€˜π‘Š)) = 𝑄) ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) β†’ ((𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ ( I β†Ύ (Baseβ€˜πΎ)))β€˜(℩𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)(π‘”β€˜((ocβ€˜πΎ)β€˜π‘Š)) = 𝑄)) = ( I β†Ύ (Baseβ€˜πΎ)))
1612, 15syl 17 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) β†’ ((𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ ( I β†Ύ (Baseβ€˜πΎ)))β€˜(℩𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)(π‘”β€˜((ocβ€˜πΎ)β€˜π‘Š)) = 𝑄)) = ( I β†Ύ (Baseβ€˜πΎ)))
1716eqcomd 2739 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) β†’ ( I β†Ύ (Baseβ€˜πΎ)) = ((𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ ( I β†Ύ (Baseβ€˜πΎ)))β€˜(℩𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)(π‘”β€˜((ocβ€˜πΎ)β€˜π‘Š)) = 𝑄)))
18 eqid 2733 . . . . 5 ((TEndoβ€˜πΎ)β€˜π‘Š) = ((TEndoβ€˜πΎ)β€˜π‘Š)
1914, 5, 9, 18, 13tendo0cl 39661 . . . 4 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ ( I β†Ύ (Baseβ€˜πΎ))) ∈ ((TEndoβ€˜πΎ)β€˜π‘Š))
2019adantr 482 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) β†’ (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ ( I β†Ύ (Baseβ€˜πΎ))) ∈ ((TEndoβ€˜πΎ)β€˜π‘Š))
21 eqid 2733 . . . 4 ((ocβ€˜πΎ)β€˜π‘Š) = ((ocβ€˜πΎ)β€˜π‘Š)
22 dicn0.i . . . 4 𝐼 = ((DIsoCβ€˜πΎ)β€˜π‘Š)
23 fvex 6905 . . . . 5 (Baseβ€˜πΎ) ∈ V
24 resiexg 7905 . . . . 5 ((Baseβ€˜πΎ) ∈ V β†’ ( I β†Ύ (Baseβ€˜πΎ)) ∈ V)
2523, 24ax-mp 5 . . . 4 ( I β†Ύ (Baseβ€˜πΎ)) ∈ V
26 fvex 6905 . . . . 5 ((LTrnβ€˜πΎ)β€˜π‘Š) ∈ V
2726mptex 7225 . . . 4 (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ ( I β†Ύ (Baseβ€˜πΎ))) ∈ V
282, 4, 5, 21, 9, 18, 22, 25, 27dicopelval 40048 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) β†’ (⟨( I β†Ύ (Baseβ€˜πΎ)), (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ ( I β†Ύ (Baseβ€˜πΎ)))⟩ ∈ (πΌβ€˜π‘„) ↔ (( I β†Ύ (Baseβ€˜πΎ)) = ((𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ ( I β†Ύ (Baseβ€˜πΎ)))β€˜(℩𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)(π‘”β€˜((ocβ€˜πΎ)β€˜π‘Š)) = 𝑄)) ∧ (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ ( I β†Ύ (Baseβ€˜πΎ))) ∈ ((TEndoβ€˜πΎ)β€˜π‘Š))))
2917, 20, 28mpbir2and 712 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) β†’ ⟨( I β†Ύ (Baseβ€˜πΎ)), (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ ( I β†Ύ (Baseβ€˜πΎ)))⟩ ∈ (πΌβ€˜π‘„))
3029ne0d 4336 1 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) β†’ (πΌβ€˜π‘„) β‰  βˆ…)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107   β‰  wne 2941  Vcvv 3475  βˆ…c0 4323  βŸ¨cop 4635   class class class wbr 5149   ↦ cmpt 5232   I cid 5574   β†Ύ cres 5679  β€˜cfv 6544  β„©crio 7364  Basecbs 17144  lecple 17204  occoc 17205  Atomscatm 38133  HLchlt 38220  LHypclh 38855  LTrncltrn 38972  TEndoctendo 39623  DIsoCcdic 40043
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 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-riotaBAD 37823
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 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-iin 5001  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-undef 8258  df-map 8822  df-proset 18248  df-poset 18266  df-plt 18283  df-lub 18299  df-glb 18300  df-join 18301  df-meet 18302  df-p0 18378  df-p1 18379  df-lat 18385  df-clat 18452  df-oposet 38046  df-ol 38048  df-oml 38049  df-covers 38136  df-ats 38137  df-atl 38168  df-cvlat 38192  df-hlat 38221  df-llines 38369  df-lplanes 38370  df-lvols 38371  df-lines 38372  df-psubsp 38374  df-pmap 38375  df-padd 38667  df-lhyp 38859  df-laut 38860  df-ldil 38975  df-ltrn 38976  df-trl 39030  df-tendo 39626  df-dic 40044
This theorem is referenced by:  diclss  40064
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