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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 36430 | . . . 4 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐼 = (𝑥 ∈ dom 𝐽 ↦ ((𝐽‘𝑥) × { 0 }))) |
8 | 7 | adantr 481 | . . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐽) → 𝐼 = (𝑥 ∈ dom 𝐽 ↦ ((𝐽‘𝑥) × { 0 }))) |
9 | 8 | fveq1d 6193 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐽) → (𝐼‘𝑋) = ((𝑥 ∈ dom 𝐽 ↦ ((𝐽‘𝑥) × { 0 }))‘𝑋)) |
10 | fveq2 6191 | . . . . 5 ⊢ (𝑥 = 𝑋 → (𝐽‘𝑥) = (𝐽‘𝑋)) | |
11 | 10 | xpeq1d 5138 | . . . 4 ⊢ (𝑥 = 𝑋 → ((𝐽‘𝑥) × { 0 }) = ((𝐽‘𝑋) × { 0 })) |
12 | eqid 2622 | . . . 4 ⊢ (𝑥 ∈ dom 𝐽 ↦ ((𝐽‘𝑥) × { 0 })) = (𝑥 ∈ dom 𝐽 ↦ ((𝐽‘𝑥) × { 0 })) | |
13 | fvex 6201 | . . . . 5 ⊢ (𝐽‘𝑋) ∈ V | |
14 | snex 4908 | . . . . 5 ⊢ { 0 } ∈ V | |
15 | 13, 14 | xpex 6962 | . . . 4 ⊢ ((𝐽‘𝑋) × { 0 }) ∈ V |
16 | 11, 12, 15 | fvmpt 6282 | . . 3 ⊢ (𝑋 ∈ dom 𝐽 → ((𝑥 ∈ dom 𝐽 ↦ ((𝐽‘𝑥) × { 0 }))‘𝑋) = ((𝐽‘𝑋) × { 0 })) |
17 | 16 | adantl 482 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐽) → ((𝑥 ∈ dom 𝐽 ↦ ((𝐽‘𝑥) × { 0 }))‘𝑋) = ((𝐽‘𝑋) × { 0 })) |
18 | 9, 17 | eqtrd 2656 | 1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐽) → (𝐼‘𝑋) = ((𝐽‘𝑋) × { 0 })) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 {csn 4177 ↦ cmpt 4729 I cid 5023 × cxp 5112 dom cdm 5114 ↾ cres 5116 ‘cfv 5888 Basecbs 15857 LHypclh 35270 LTrncltrn 35387 DIsoAcdia 36317 DIsoBcdib 36427 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-dib 36428 |
This theorem is referenced by: dibopelvalN 36432 dibval2 36433 dibvalrel 36452 |
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