Theorem dibval 41131

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
dibval	⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐽) → (𝐼‘𝑋) = ((𝐽‘𝑋) × { 0 }))

Distinct variable groups:   𝑓,𝐾   𝑓,𝑊

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

Proof of Theorem dibval

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dibval.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐾)
2		dibval.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
3		dibval.t	. . . . 5 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
4		dibval.o	. . . . 5 ⊢ 0 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵))
5		dibval.j	. . . . 5 ⊢ 𝐽 = ((DIsoA‘𝐾)‘𝑊)
6		dibval.i	. . . . 5 ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊)
7	1, 2, 3, 4, 5, 6	dibfval 41130	. . . 4 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐼 = (𝑥 ∈ dom 𝐽 ↦ ((𝐽‘𝑥) × { 0 })))
8	7	adantr 480	. . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐽) → 𝐼 = (𝑥 ∈ dom 𝐽 ↦ ((𝐽‘𝑥) × { 0 })))
9	8	fveq1d 6862	. 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐽) → (𝐼‘𝑋) = ((𝑥 ∈ dom 𝐽 ↦ ((𝐽‘𝑥) × { 0 }))‘𝑋))
10		fveq2 6860	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝐽‘𝑥) = (𝐽‘𝑋))
11	10	xpeq1d 5669	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝐽‘𝑥) × { 0 }) = ((𝐽‘𝑋) × { 0 }))
12		eqid 2730	. . . 4 ⊢ (𝑥 ∈ dom 𝐽 ↦ ((𝐽‘𝑥) × { 0 })) = (𝑥 ∈ dom 𝐽 ↦ ((𝐽‘𝑥) × { 0 }))
13		fvex 6873	. . . . 5 ⊢ (𝐽‘𝑋) ∈ V
14		snex 5393	. . . . 5 ⊢ { 0 } ∈ V
15	13, 14	xpex 7731	. . . 4 ⊢ ((𝐽‘𝑋) × { 0 }) ∈ V
16	11, 12, 15	fvmpt 6970	. . 3 ⊢ (𝑋 ∈ dom 𝐽 → ((𝑥 ∈ dom 𝐽 ↦ ((𝐽‘𝑥) × { 0 }))‘𝑋) = ((𝐽‘𝑋) × { 0 }))
17	16	adantl 481	. 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐽) → ((𝑥 ∈ dom 𝐽 ↦ ((𝐽‘𝑥) × { 0 }))‘𝑋) = ((𝐽‘𝑋) × { 0 }))
18	9, 17	eqtrd 2765	1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐽) → (𝐼‘𝑋) = ((𝐽‘𝑋) × { 0 }))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {csn 4591   ↦ cmpt 5190   I cid 5534   × cxp 5638  dom cdm 5640   ↾ cres 5642  ‘cfv 6513  Basecbs 17185  LHypclh 39973  LTrncltrn 40090  DIsoAcdia 41017  DIsoBcdib 41127

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5236  ax-sep 5253  ax-nul 5263  ax-pow 5322  ax-pr 5389  ax-un 7713

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3756  df-csb 3865  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5110  df-opab 5172  df-mpt 5191  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-iota 6466  df-fun 6515  df-fn 6516  df-f 6517  df-f1 6518  df-fo 6519  df-f1o 6520  df-fv 6521  df-dib 41128

This theorem is referenced by:  dibopelvalN  41132  dibval2  41133  dibvalrel  41152