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Theorem dibfval 39607
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 39606 . . . 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 6845 . . 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 2789 . 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 6843 . . . . . 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 2795 . . . . 5 (𝑀 = π‘Š β†’ ((DIsoAβ€˜πΎ)β€˜π‘€) = 𝐽)
109dmeqd 5862 . . . 4 (𝑀 = π‘Š β†’ dom ((DIsoAβ€˜πΎ)β€˜π‘€) = dom 𝐽)
119fveq1d 6845 . . . . 5 (𝑀 = π‘Š β†’ (((DIsoAβ€˜πΎ)β€˜π‘€)β€˜π‘₯) = (π½β€˜π‘₯))
12 fveq2 6843 . . . . . . . . 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 2795 . . . . . . . 8 (𝑀 = π‘Š β†’ ((LTrnβ€˜πΎ)β€˜π‘€) = 𝑇)
15 eqidd 2738 . . . . . . . 8 (𝑀 = π‘Š β†’ ( I β†Ύ 𝐡) = ( I β†Ύ 𝐡))
1614, 15mpteq12dv 5197 . . . . . . 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 2795 . . . . . 6 (𝑀 = π‘Š β†’ (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ ( I β†Ύ 𝐡)) = 0 )
1918sneqd 4599 . . . . 5 (𝑀 = π‘Š β†’ {(𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ ( I β†Ύ 𝐡))} = { 0 })
2011, 19xpeq12d 5665 . . . 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 5197 . . 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 2737 . . 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 6857 . . . . 5 𝐽 ∈ V
2423dmex 7849 . . . 4 dom 𝐽 ∈ V
2524mptex 7174 . . 3 (π‘₯ ∈ dom 𝐽 ↦ ((π½β€˜π‘₯) Γ— { 0 })) ∈ V
2621, 22, 25fvmpt 6949 . 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 2797 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 397   = wceq 1542   ∈ wcel 2107  {csn 4587   ↦ cmpt 5189   I cid 5531   Γ— cxp 5632  dom cdm 5634   β†Ύ cres 5636  β€˜cfv 6497  Basecbs 17084  LHypclh 38450  LTrncltrn 38567  DIsoAcdia 39494  DIsoBcdib 39604
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 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-dib 39605
This theorem is referenced by:  dibval  39608  dibfna  39620
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