Theorem dicn0 42515

Description: The value of the partial isomorphism C is not empty. (Contributed by NM, 15-Feb-2014.)

Hypotheses

Ref	Expression
dicn0.l	⊢ ≤ = (le‘𝐾)
dicn0.a	⊢ 𝐴 = (Atoms‘𝐾)
dicn0.h	⊢ 𝐻 = (LHyp‘𝐾)
dicn0.i	⊢ 𝐼 = ((DIsoC‘𝐾)‘𝑊)

Assertion

Ref	Expression
dicn0	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐼‘𝑄) ≠ ∅)

Proof of Theorem dicn0

Dummy variables 𝑔 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 488	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		dicn0.l	. . . . . . . 8 ⊢ ≤ = (le‘𝐾)
3		eqid 2763	. . . . . . . 8 ⊢ (oc‘𝐾) = (oc‘𝐾)
4		dicn0.a	. . . . . . . 8 ⊢ 𝐴 = (Atoms‘𝐾)
5		dicn0.h	. . . . . . . 8 ⊢ 𝐻 = (LHyp‘𝐾)
6	2, 3, 4, 5	lhpocnel 41341	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (((oc‘𝐾)‘𝑊) ∈ 𝐴 ∧ ¬ ((oc‘𝐾)‘𝑊) ≤ 𝑊))
7	6	adantr 486	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (((oc‘𝐾)‘𝑊) ∈ 𝐴 ∧ ¬ ((oc‘𝐾)‘𝑊) ≤ 𝑊))
8		simpr 490	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
9		eqid 2763	. . . . . . 7 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
10		eqid 2763	. . . . . . 7 ⊢ (℩𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄) = (℩𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄)
11	2, 4, 5, 9, 10	ltrniotacl 41902	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (((oc‘𝐾)‘𝑊) ∈ 𝐴 ∧ ¬ ((oc‘𝐾)‘𝑊) ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (℩𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄) ∈ ((LTrn‘𝐾)‘𝑊))
12	1, 7, 8, 11	syl3anc 1399	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (℩𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄) ∈ ((LTrn‘𝐾)‘𝑊))
13		eqid 2763	. . . . . 6 ⊢ (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾))) = (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))
14		eqid 2763	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
15	13, 14	tendo02 42110	. . . . 5 ⊢ ((℩𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄) ∈ ((LTrn‘𝐾)‘𝑊) → ((𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))‘(℩𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄)) = ( I ↾ (Base‘𝐾)))
16	12, 15	syl 18	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → ((𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))‘(℩𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄)) = ( I ↾ (Base‘𝐾)))
17	16	eqcomd 2769	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → ( I ↾ (Base‘𝐾)) = ((𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))‘(℩𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄)))
18		eqid 2763	. . . . 5 ⊢ ((TEndo‘𝐾)‘𝑊) = ((TEndo‘𝐾)‘𝑊)
19	14, 5, 9, 18, 13	tendo0cl 42113	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾))) ∈ ((TEndo‘𝐾)‘𝑊))
20	19	adantr 486	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾))) ∈ ((TEndo‘𝐾)‘𝑊))
21		eqid 2763	. . . 4 ⊢ ((oc‘𝐾)‘𝑊) = ((oc‘𝐾)‘𝑊)
22		dicn0.i	. . . 4 ⊢ 𝐼 = ((DIsoC‘𝐾)‘𝑊)
23		fvex 6929	. . . . 5 ⊢ (Base‘𝐾) ∈ V
24		resiexg 7958	. . . . 5 ⊢ ((Base‘𝐾) ∈ V → ( I ↾ (Base‘𝐾)) ∈ V)
25	23, 24	ax-mp 5	. . . 4 ⊢ ( I ↾ (Base‘𝐾)) ∈ V
26		fvex 6929	. . . . 5 ⊢ ((LTrn‘𝐾)‘𝑊) ∈ V
27	26	mptex 7260	. . . 4 ⊢ (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾))) ∈ V
28	2, 4, 5, 21, 9, 18, 22, 25, 27	dicopelval 42500	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (⟨( I ↾ (Base‘𝐾)), (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))⟩ ∈ (𝐼‘𝑄) ↔ (( I ↾ (Base‘𝐾)) = ((𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))‘(℩𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄)) ∧ (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾))) ∈ ((TEndo‘𝐾)‘𝑊))))
29	17, 20, 28	mpbir2and 726	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → ⟨( I ↾ (Base‘𝐾)), (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))⟩ ∈ (𝐼‘𝑄))
30	29	ne0d 4291	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐼‘𝑄) ≠ ∅)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  Vcvv 3454  ∅c0 4281  ⟨cop 4840   class class class wbr 5108   ↦ cmpt 5192   I cid 5552   ↾ cres 5660  ‘cfv 6566  ℩crio 7407  Basecbs 17431  lecple 17479  occoc 17480  Atomscatm 40586  HLchlt 40673  LHypclh 41307  LTrncltrn 41424  TEndoctendo 42075  DIsoCcdic 42495

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-riotaBAD 40276

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-iin 4959  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-1st 8035  df-2nd 8036  df-undef 8328  df-map 8887  df-proset 18518  df-poset 18537  df-plt 18552  df-lub 18568  df-glb 18569  df-join 18570  df-meet 18571  df-p0 18647  df-p1 18648  df-lat 18656  df-clat 18723  df-oposet 40499  df-ol 40501  df-oml 40502  df-covers 40589  df-ats 40590  df-atl 40621  df-cvlat 40645  df-hlat 40674  df-llines 40821  df-lplanes 40822  df-lvols 40823  df-lines 40824  df-psubsp 40826  df-pmap 40827  df-padd 41119  df-lhyp 41311  df-laut 41312  df-ldil 41427  df-ltrn 41428  df-trl 41482  df-tendo 42078  df-dic 42496

This theorem is used by:  diclss  42516