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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 38292 | . . . 4 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐼 = (𝑥 ∈ dom 𝐽 ↦ ((𝐽‘𝑥) × { 0 }))) |
8 | 7 | adantr 483 | . . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐽) → 𝐼 = (𝑥 ∈ dom 𝐽 ↦ ((𝐽‘𝑥) × { 0 }))) |
9 | 8 | fveq1d 6672 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐽) → (𝐼‘𝑋) = ((𝑥 ∈ dom 𝐽 ↦ ((𝐽‘𝑥) × { 0 }))‘𝑋)) |
10 | fveq2 6670 | . . . . 5 ⊢ (𝑥 = 𝑋 → (𝐽‘𝑥) = (𝐽‘𝑋)) | |
11 | 10 | xpeq1d 5584 | . . . 4 ⊢ (𝑥 = 𝑋 → ((𝐽‘𝑥) × { 0 }) = ((𝐽‘𝑋) × { 0 })) |
12 | eqid 2821 | . . . 4 ⊢ (𝑥 ∈ dom 𝐽 ↦ ((𝐽‘𝑥) × { 0 })) = (𝑥 ∈ dom 𝐽 ↦ ((𝐽‘𝑥) × { 0 })) | |
13 | fvex 6683 | . . . . 5 ⊢ (𝐽‘𝑋) ∈ V | |
14 | snex 5332 | . . . . 5 ⊢ { 0 } ∈ V | |
15 | 13, 14 | xpex 7476 | . . . 4 ⊢ ((𝐽‘𝑋) × { 0 }) ∈ V |
16 | 11, 12, 15 | fvmpt 6768 | . . 3 ⊢ (𝑋 ∈ dom 𝐽 → ((𝑥 ∈ dom 𝐽 ↦ ((𝐽‘𝑥) × { 0 }))‘𝑋) = ((𝐽‘𝑋) × { 0 })) |
17 | 16 | adantl 484 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐽) → ((𝑥 ∈ dom 𝐽 ↦ ((𝐽‘𝑥) × { 0 }))‘𝑋) = ((𝐽‘𝑋) × { 0 })) |
18 | 9, 17 | eqtrd 2856 | 1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐽) → (𝐼‘𝑋) = ((𝐽‘𝑋) × { 0 })) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 {csn 4567 ↦ cmpt 5146 I cid 5459 × cxp 5553 dom cdm 5555 ↾ cres 5557 ‘cfv 6355 Basecbs 16483 LHypclh 37135 LTrncltrn 37252 DIsoAcdia 38179 DIsoBcdib 38289 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-dib 38290 |
This theorem is referenced by: dibopelvalN 38294 dibval2 38295 dibvalrel 38314 |
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