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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 38171 | . . 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 38277 | . 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 1533 ∈ wcel 2110 {csn 4566 class class class wbr 5065 ↦ cmpt 5145 I cid 5458 × cxp 5552 dom cdm 5554 ↾ cres 5556 ‘cfv 6354 Basecbs 16482 lecple 16571 LHypclh 37119 LTrncltrn 37236 DIsoAcdia 38163 DIsoBcdib 38273 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-disoa 38164 df-dib 38274 |
This theorem is referenced by: dibopelval2 38280 dibval3N 38281 dibelval3 38282 dibelval1st 38284 dibelval2nd 38287 dibn0 38288 dibord 38294 dib0 38299 dib1dim 38300 dibss 38304 diblss 38305 dihwN 38424 |
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