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Theorem dibfval 40524
Description: The partial isomorphism B for a lattice 𝐾. (Contributed by NM, 8-Dec-2013.)
Hypotheses
Ref Expression
dibval.b 𝐡 = (Baseβ€˜πΎ)
dibval.h 𝐻 = (LHypβ€˜πΎ)
dibval.t 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
dibval.o 0 = (𝑓 ∈ 𝑇 ↦ ( I β†Ύ 𝐡))
dibval.j 𝐽 = ((DIsoAβ€˜πΎ)β€˜π‘Š)
dibval.i 𝐼 = ((DIsoBβ€˜πΎ)β€˜π‘Š)
Assertion
Ref Expression
dibfval ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ 𝐼 = (π‘₯ ∈ dom 𝐽 ↦ ((π½β€˜π‘₯) Γ— { 0 })))
Distinct variable groups:   π‘₯,𝑓,𝐾   π‘₯,𝐽   𝑓,π‘Š,π‘₯
Allowed substitution hints:   𝐡(π‘₯,𝑓)   𝑇(π‘₯,𝑓)   𝐻(π‘₯,𝑓)   𝐼(π‘₯,𝑓)   𝐽(𝑓)   𝑉(π‘₯,𝑓)   0 (π‘₯,𝑓)

Proof of Theorem dibfval
Dummy variable 𝑀 is distinct from all other variables.
StepHypRef Expression
1 dibval.i . . 3 𝐼 = ((DIsoBβ€˜πΎ)β€˜π‘Š)
2 dibval.b . . . . 5 𝐡 = (Baseβ€˜πΎ)
3 dibval.h . . . . 5 𝐻 = (LHypβ€˜πΎ)
42, 3dibffval 40523 . . . 4 (𝐾 ∈ 𝑉 β†’ (DIsoBβ€˜πΎ) = (𝑀 ∈ 𝐻 ↦ (π‘₯ ∈ dom ((DIsoAβ€˜πΎ)β€˜π‘€) ↦ ((((DIsoAβ€˜πΎ)β€˜π‘€)β€˜π‘₯) Γ— {(𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ ( I β†Ύ 𝐡))}))))
54fveq1d 6886 . . 3 (𝐾 ∈ 𝑉 β†’ ((DIsoBβ€˜πΎ)β€˜π‘Š) = ((𝑀 ∈ 𝐻 ↦ (π‘₯ ∈ dom ((DIsoAβ€˜πΎ)β€˜π‘€) ↦ ((((DIsoAβ€˜πΎ)β€˜π‘€)β€˜π‘₯) Γ— {(𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ ( I β†Ύ 𝐡))})))β€˜π‘Š))
61, 5eqtrid 2778 . 2 (𝐾 ∈ 𝑉 β†’ 𝐼 = ((𝑀 ∈ 𝐻 ↦ (π‘₯ ∈ dom ((DIsoAβ€˜πΎ)β€˜π‘€) ↦ ((((DIsoAβ€˜πΎ)β€˜π‘€)β€˜π‘₯) Γ— {(𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ ( I β†Ύ 𝐡))})))β€˜π‘Š))
7 fveq2 6884 . . . . . 6 (𝑀 = π‘Š β†’ ((DIsoAβ€˜πΎ)β€˜π‘€) = ((DIsoAβ€˜πΎ)β€˜π‘Š))
8 dibval.j . . . . . 6 𝐽 = ((DIsoAβ€˜πΎ)β€˜π‘Š)
97, 8eqtr4di 2784 . . . . 5 (𝑀 = π‘Š β†’ ((DIsoAβ€˜πΎ)β€˜π‘€) = 𝐽)
109dmeqd 5898 . . . 4 (𝑀 = π‘Š β†’ dom ((DIsoAβ€˜πΎ)β€˜π‘€) = dom 𝐽)
119fveq1d 6886 . . . . 5 (𝑀 = π‘Š β†’ (((DIsoAβ€˜πΎ)β€˜π‘€)β€˜π‘₯) = (π½β€˜π‘₯))
12 fveq2 6884 . . . . . . . . 9 (𝑀 = π‘Š β†’ ((LTrnβ€˜πΎ)β€˜π‘€) = ((LTrnβ€˜πΎ)β€˜π‘Š))
13 dibval.t . . . . . . . . 9 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
1412, 13eqtr4di 2784 . . . . . . . 8 (𝑀 = π‘Š β†’ ((LTrnβ€˜πΎ)β€˜π‘€) = 𝑇)
15 eqidd 2727 . . . . . . . 8 (𝑀 = π‘Š β†’ ( I β†Ύ 𝐡) = ( I β†Ύ 𝐡))
1614, 15mpteq12dv 5232 . . . . . . 7 (𝑀 = π‘Š β†’ (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ ( I β†Ύ 𝐡)) = (𝑓 ∈ 𝑇 ↦ ( I β†Ύ 𝐡)))
17 dibval.o . . . . . . 7 0 = (𝑓 ∈ 𝑇 ↦ ( I β†Ύ 𝐡))
1816, 17eqtr4di 2784 . . . . . 6 (𝑀 = π‘Š β†’ (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ ( I β†Ύ 𝐡)) = 0 )
1918sneqd 4635 . . . . 5 (𝑀 = π‘Š β†’ {(𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ ( I β†Ύ 𝐡))} = { 0 })
2011, 19xpeq12d 5700 . . . 4 (𝑀 = π‘Š β†’ ((((DIsoAβ€˜πΎ)β€˜π‘€)β€˜π‘₯) Γ— {(𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ ( I β†Ύ 𝐡))}) = ((π½β€˜π‘₯) Γ— { 0 }))
2110, 20mpteq12dv 5232 . . 3 (𝑀 = π‘Š β†’ (π‘₯ ∈ dom ((DIsoAβ€˜πΎ)β€˜π‘€) ↦ ((((DIsoAβ€˜πΎ)β€˜π‘€)β€˜π‘₯) Γ— {(𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ ( I β†Ύ 𝐡))})) = (π‘₯ ∈ dom 𝐽 ↦ ((π½β€˜π‘₯) Γ— { 0 })))
22 eqid 2726 . . 3 (𝑀 ∈ 𝐻 ↦ (π‘₯ ∈ dom ((DIsoAβ€˜πΎ)β€˜π‘€) ↦ ((((DIsoAβ€˜πΎ)β€˜π‘€)β€˜π‘₯) Γ— {(𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ ( I β†Ύ 𝐡))}))) = (𝑀 ∈ 𝐻 ↦ (π‘₯ ∈ dom ((DIsoAβ€˜πΎ)β€˜π‘€) ↦ ((((DIsoAβ€˜πΎ)β€˜π‘€)β€˜π‘₯) Γ— {(𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ ( I β†Ύ 𝐡))})))
238fvexi 6898 . . . . 5 𝐽 ∈ V
2423dmex 7898 . . . 4 dom 𝐽 ∈ V
2524mptex 7219 . . 3 (π‘₯ ∈ dom 𝐽 ↦ ((π½β€˜π‘₯) Γ— { 0 })) ∈ V
2621, 22, 25fvmpt 6991 . 2 (π‘Š ∈ 𝐻 β†’ ((𝑀 ∈ 𝐻 ↦ (π‘₯ ∈ dom ((DIsoAβ€˜πΎ)β€˜π‘€) ↦ ((((DIsoAβ€˜πΎ)β€˜π‘€)β€˜π‘₯) Γ— {(𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ ( I β†Ύ 𝐡))})))β€˜π‘Š) = (π‘₯ ∈ dom 𝐽 ↦ ((π½β€˜π‘₯) Γ— { 0 })))
276, 26sylan9eq 2786 1 ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ 𝐼 = (π‘₯ ∈ dom 𝐽 ↦ ((π½β€˜π‘₯) Γ— { 0 })))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  {csn 4623   ↦ cmpt 5224   I cid 5566   Γ— cxp 5667  dom cdm 5669   β†Ύ cres 5671  β€˜cfv 6536  Basecbs 17150  LHypclh 39367  LTrncltrn 39484  DIsoAcdia 40411  DIsoBcdib 40521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-dib 40522
This theorem is referenced by:  dibval  40525  dibfna  40537
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