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Mirrors > Home > MPE Home > Th. List > Mathboxes > dibval | Structured version Visualization version GIF version |
Description: The partial isomorphism B for a lattice 𝐾. (Contributed by NM, 8-Dec-2013.) |
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) |
Ref | Expression |
---|---|
dibval | ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐽) → (𝐼‘𝑋) = ((𝐽‘𝑋) × { 0 })) |
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 35448 | . . . 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 6105 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐽) → (𝐼‘𝑋) = ((𝑥 ∈ dom 𝐽 ↦ ((𝐽‘𝑥) × { 0 }))‘𝑋)) |
10 | fveq2 6103 | . . . . 5 ⊢ (𝑥 = 𝑋 → (𝐽‘𝑥) = (𝐽‘𝑋)) | |
11 | 10 | xpeq1d 5062 | . . . 4 ⊢ (𝑥 = 𝑋 → ((𝐽‘𝑥) × { 0 }) = ((𝐽‘𝑋) × { 0 })) |
12 | eqid 2610 | . . . 4 ⊢ (𝑥 ∈ dom 𝐽 ↦ ((𝐽‘𝑥) × { 0 })) = (𝑥 ∈ dom 𝐽 ↦ ((𝐽‘𝑥) × { 0 })) | |
13 | fvex 6113 | . . . . 5 ⊢ (𝐽‘𝑋) ∈ V | |
14 | snex 4835 | . . . . 5 ⊢ { 0 } ∈ V | |
15 | 13, 14 | xpex 6860 | . . . 4 ⊢ ((𝐽‘𝑋) × { 0 }) ∈ V |
16 | 11, 12, 15 | fvmpt 6191 | . . 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 2644 | 1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐽) → (𝐼‘𝑋) = ((𝐽‘𝑋) × { 0 })) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 {csn 4125 ↦ cmpt 4643 I cid 4948 × cxp 5036 dom cdm 5038 ↾ cres 5040 ‘cfv 5804 Basecbs 15695 LHypclh 34288 LTrncltrn 34405 DIsoAcdia 35335 DIsoBcdib 35445 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-dib 35446 |
This theorem is referenced by: dibopelvalN 35450 dibval2 35451 dibvalrel 35470 |
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