![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > dibval2 | Structured version Visualization version GIF version |
Description: Value of the partial isomorphism B. (Contributed by NM, 18-Jan-2014.) |
Ref | Expression |
---|---|
dibval2.b | ⊢ 𝐵 = (Base‘𝐾) |
dibval2.l | ⊢ ≤ = (le‘𝐾) |
dibval2.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dibval2.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
dibval2.o | ⊢ 0 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)) |
dibval2.j | ⊢ 𝐽 = ((DIsoA‘𝐾)‘𝑊) |
dibval2.i | ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
dibval2 | ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) = ((𝐽‘𝑋) × { 0 })) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dibval2.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
2 | dibval2.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
3 | dibval2.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
4 | dibval2.j | . . . 4 ⊢ 𝐽 = ((DIsoA‘𝐾)‘𝑊) | |
5 | 1, 2, 3, 4 | diaeldm 40748 | . . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (𝑋 ∈ dom 𝐽 ↔ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊))) |
6 | 5 | biimpar 476 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → 𝑋 ∈ dom 𝐽) |
7 | dibval2.t | . . 3 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
8 | dibval2.o | . . 3 ⊢ 0 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)) | |
9 | dibval2.i | . . 3 ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊) | |
10 | 1, 3, 7, 8, 4, 9 | dibval 40854 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐽) → (𝐼‘𝑋) = ((𝐽‘𝑋) × { 0 })) |
11 | 6, 10 | syldan 589 | 1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) = ((𝐽‘𝑋) × { 0 })) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 {csn 4623 class class class wbr 5145 ↦ cmpt 5228 I cid 5571 × cxp 5672 dom cdm 5674 ↾ cres 5676 ‘cfv 6546 Basecbs 17208 lecple 17268 LHypclh 39696 LTrncltrn 39813 DIsoAcdia 40740 DIsoBcdib 40850 |
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 2697 ax-rep 5282 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-iun 4995 df-br 5146 df-opab 5208 df-mpt 5229 df-id 5572 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-disoa 40741 df-dib 40851 |
This theorem is referenced by: dibopelval2 40857 dibval3N 40858 dibelval3 40859 dibelval1st 40861 dibelval2nd 40864 dibn0 40865 dibord 40871 dib0 40876 dib1dim 40877 dibss 40881 diblss 40882 dihwN 41001 |
Copyright terms: Public domain | W3C validator |