Theorem dibfval 41640

Description: The partial isomorphism B for a lattice 𝐾. (Contributed by NM, 8-Dec-2013.)

Hypotheses

Ref	Expression
dibval.b	⊢ 𝐵 = (Base‘𝐾)
dibval.h	⊢ 𝐻 = (LHyp‘𝐾)
dibval.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
dibval.o	⊢ 0 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵))
dibval.j	⊢ 𝐽 = ((DIsoA‘𝐾)‘𝑊)
dibval.i	⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊)

Assertion

Ref	Expression
dibfval	⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐼 = (𝑥 ∈ dom 𝐽 ↦ ((𝐽‘𝑥) × { 0 })))

Distinct variable groups:   𝑥,𝑓,𝐾   𝑥,𝐽   𝑓,𝑊,𝑥

Allowed substitution hints:   𝐵(𝑥,𝑓)   𝑇(𝑥,𝑓)   𝐻(𝑥,𝑓)   𝐼(𝑥,𝑓)   𝐽(𝑓)   𝑉(𝑥,𝑓)   0 (𝑥,𝑓)

Proof of Theorem dibfval

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dibval.i	. . 3 ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊)
2		dibval.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐾)
3		dibval.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
4	2, 3	dibffval 41639	. . . 4 ⊢ (𝐾 ∈ 𝑉 → (DIsoB‘𝐾) = (𝑤 ∈ 𝐻 ↦ (𝑥 ∈ dom ((DIsoA‘𝐾)‘𝑤) ↦ ((((DIsoA‘𝐾)‘𝑤)‘𝑥) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ( I ↾ 𝐵))}))))
5	4	fveq1d 6836	. . 3 ⊢ (𝐾 ∈ 𝑉 → ((DIsoB‘𝐾)‘𝑊) = ((𝑤 ∈ 𝐻 ↦ (𝑥 ∈ dom ((DIsoA‘𝐾)‘𝑤) ↦ ((((DIsoA‘𝐾)‘𝑤)‘𝑥) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ( I ↾ 𝐵))})))‘𝑊))
6	1, 5	eqtrid 2787	. 2 ⊢ (𝐾 ∈ 𝑉 → 𝐼 = ((𝑤 ∈ 𝐻 ↦ (𝑥 ∈ dom ((DIsoA‘𝐾)‘𝑤) ↦ ((((DIsoA‘𝐾)‘𝑤)‘𝑥) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ( I ↾ 𝐵))})))‘𝑊))
7		fveq2 6834	. . . . . 6 ⊢ (𝑤 = 𝑊 → ((DIsoA‘𝐾)‘𝑤) = ((DIsoA‘𝐾)‘𝑊))
8		dibval.j	. . . . . 6 ⊢ 𝐽 = ((DIsoA‘𝐾)‘𝑊)
9	7, 8	eqtr4di 2793	. . . . 5 ⊢ (𝑤 = 𝑊 → ((DIsoA‘𝐾)‘𝑤) = 𝐽)
10	9	dmeqd 5854	. . . 4 ⊢ (𝑤 = 𝑊 → dom ((DIsoA‘𝐾)‘𝑤) = dom 𝐽)
11	9	fveq1d 6836	. . . . 5 ⊢ (𝑤 = 𝑊 → (((DIsoA‘𝐾)‘𝑤)‘𝑥) = (𝐽‘𝑥))
12		fveq2 6834	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = ((LTrn‘𝐾)‘𝑊))
13		dibval.t	. . . . . . . . 9 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
14	12, 13	eqtr4di 2793	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = 𝑇)
15		eqidd 2741	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → ( I ↾ 𝐵) = ( I ↾ 𝐵))
16	14, 15	mpteq12dv 5166	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ( I ↾ 𝐵)) = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)))
17		dibval.o	. . . . . . 7 ⊢ 0 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵))
18	16, 17	eqtr4di 2793	. . . . . 6 ⊢ (𝑤 = 𝑊 → (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ( I ↾ 𝐵)) = 0 )
19	18	sneqd 4574	. . . . 5 ⊢ (𝑤 = 𝑊 → {(𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ( I ↾ 𝐵))} = { 0 })
20	11, 19	xpeq12d 5656	. . . 4 ⊢ (𝑤 = 𝑊 → ((((DIsoA‘𝐾)‘𝑤)‘𝑥) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ( I ↾ 𝐵))}) = ((𝐽‘𝑥) × { 0 }))
21	10, 20	mpteq12dv 5166	. . 3 ⊢ (𝑤 = 𝑊 → (𝑥 ∈ dom ((DIsoA‘𝐾)‘𝑤) ↦ ((((DIsoA‘𝐾)‘𝑤)‘𝑥) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ( I ↾ 𝐵))})) = (𝑥 ∈ dom 𝐽 ↦ ((𝐽‘𝑥) × { 0 })))
22		eqid 2740	. . 3 ⊢ (𝑤 ∈ 𝐻 ↦ (𝑥 ∈ dom ((DIsoA‘𝐾)‘𝑤) ↦ ((((DIsoA‘𝐾)‘𝑤)‘𝑥) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ( I ↾ 𝐵))}))) = (𝑤 ∈ 𝐻 ↦ (𝑥 ∈ dom ((DIsoA‘𝐾)‘𝑤) ↦ ((((DIsoA‘𝐾)‘𝑤)‘𝑥) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ( I ↾ 𝐵))})))
23	8	fvexi 6848	. . . . 5 ⊢ 𝐽 ∈ V
24	23	dmex 7856	. . . 4 ⊢ dom 𝐽 ∈ V
25	24	mptex 7174	. . 3 ⊢ (𝑥 ∈ dom 𝐽 ↦ ((𝐽‘𝑥) × { 0 })) ∈ V
26	21, 22, 25	fvmpt 6942	. 2 ⊢ (𝑊 ∈ 𝐻 → ((𝑤 ∈ 𝐻 ↦ (𝑥 ∈ dom ((DIsoA‘𝐾)‘𝑤) ↦ ((((DIsoA‘𝐾)‘𝑤)‘𝑥) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ( I ↾ 𝐵))})))‘𝑊) = (𝑥 ∈ dom 𝐽 ↦ ((𝐽‘𝑥) × { 0 })))
27	6, 26	sylan9eq 2795	1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐼 = (𝑥 ∈ dom 𝐽 ↦ ((𝐽‘𝑥) × { 0 })))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  {csn 4562   ↦ cmpt 5160   I cid 5519   × cxp 5623  dom cdm 5625   ↾ cres 5627  ‘cfv 6492  Basecbs 17177  LHypclh 40483  LTrncltrn 40600  DIsoAcdia 41527  DIsoBcdib 41637

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-dib 41638

This theorem is referenced by:  dibval  41641  dibfna  41653