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Mirrors > Home > MPE Home > Th. List > Mathboxes > diass | Structured version Visualization version GIF version |
Description: The value of the partial isomorphism A is a set of translations, i.e., a set of vectors. (Contributed by NM, 26-Nov-2013.) |
Ref | Expression |
---|---|
diass.b | ⊢ 𝐵 = (Base‘𝐾) |
diass.l | ⊢ ≤ = (le‘𝐾) |
diass.h | ⊢ 𝐻 = (LHyp‘𝐾) |
diass.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
diass.i | ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
diass | ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) ⊆ 𝑇) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | diass.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
2 | diass.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
3 | diass.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
4 | diass.t | . . 3 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
5 | eqid 2738 | . . 3 ⊢ ((trL‘𝐾)‘𝑊) = ((trL‘𝐾)‘𝑊) | |
6 | diass.i | . . 3 ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊) | |
7 | 1, 2, 3, 4, 5, 6 | diaval 39054 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) = {𝑓 ∈ 𝑇 ∣ (((trL‘𝐾)‘𝑊)‘𝑓) ≤ 𝑋}) |
8 | ssrab2 4012 | . 2 ⊢ {𝑓 ∈ 𝑇 ∣ (((trL‘𝐾)‘𝑊)‘𝑓) ≤ 𝑋} ⊆ 𝑇 | |
9 | 7, 8 | eqsstrdi 3974 | 1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) ⊆ 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 {crab 3068 ⊆ wss 3886 class class class wbr 5073 ‘cfv 6426 Basecbs 16922 lecple 16979 LHypclh 38006 LTrncltrn 38123 trLctrl 38180 DIsoAcdia 39050 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5208 ax-sep 5221 ax-nul 5228 ax-pr 5350 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3071 df-rab 3073 df-v 3431 df-sbc 3716 df-csb 3832 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5074 df-opab 5136 df-mpt 5157 df-id 5484 df-xp 5590 df-rel 5591 df-cnv 5592 df-co 5593 df-dm 5594 df-rn 5595 df-res 5596 df-ima 5597 df-iota 6384 df-fun 6428 df-fn 6429 df-f 6430 df-f1 6431 df-fo 6432 df-f1o 6433 df-fv 6434 df-disoa 39051 |
This theorem is referenced by: diael 39065 diaelrnN 39067 dialss 39068 dia2dimlem12 39097 diaocN 39147 dibss 39191 |
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