Theorem dibval 40615

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 40614	. . . 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 6899	. 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐽) → (𝐼‘𝑋) = ((𝑥 ∈ dom 𝐽 ↦ ((𝐽‘𝑥) × { 0 }))‘𝑋))
10		fveq2 6897	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝐽‘𝑥) = (𝐽‘𝑋))
11	10	xpeq1d 5707	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝐽‘𝑥) × { 0 }) = ((𝐽‘𝑋) × { 0 }))
12		eqid 2728	. . . 4 ⊢ (𝑥 ∈ dom 𝐽 ↦ ((𝐽‘𝑥) × { 0 })) = (𝑥 ∈ dom 𝐽 ↦ ((𝐽‘𝑥) × { 0 }))
13		fvex 6910	. . . . 5 ⊢ (𝐽‘𝑋) ∈ V
14		snex 5433	. . . . 5 ⊢ { 0 } ∈ V
15	13, 14	xpex 7755	. . . 4 ⊢ ((𝐽‘𝑋) × { 0 }) ∈ V
16	11, 12, 15	fvmpt 7005	. . 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 2768	1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐽) → (𝐼‘𝑋) = ((𝐽‘𝑋) × { 0 }))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1534   ∈ wcel 2099  {csn 4629   ↦ cmpt 5231   I cid 5575   × cxp 5676  dom cdm 5678   ↾ cres 5680  ‘cfv 6548  Basecbs 17180  LHypclh 39457  LTrncltrn 39574  DIsoAcdia 40501  DIsoBcdib 40611

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-dib 40612

This theorem is referenced by:  dibopelvalN  40616  dibval2  40617  dibvalrel  40636