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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 38187 | . . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (𝑋 ∈ dom 𝐽 ↔ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊))) |
6 | 5 | biimpar 480 | . 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 38293 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐽) → (𝐼‘𝑋) = ((𝐽‘𝑋) × { 0 })) |
11 | 6, 10 | syldan 593 | 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 398 = wceq 1537 ∈ wcel 2114 {csn 4567 class class class wbr 5066 ↦ cmpt 5146 I cid 5459 × cxp 5553 dom cdm 5555 ↾ cres 5557 ‘cfv 6355 Basecbs 16483 lecple 16572 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-disoa 38180 df-dib 38290 |
This theorem is referenced by: dibopelval2 38296 dibval3N 38297 dibelval3 38298 dibelval1st 38300 dibelval2nd 38303 dibn0 38304 dibord 38310 dib0 38315 dib1dim 38316 dibss 38320 diblss 38321 dihwN 38440 |
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