![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
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 2825 | . . 3 ⊢ ((trL‘𝐾)‘𝑊) = ((trL‘𝐾)‘𝑊) | |
6 | diass.i | . . 3 ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊) | |
7 | 1, 2, 3, 4, 5, 6 | diaval 37107 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) = {𝑓 ∈ 𝑇 ∣ (((trL‘𝐾)‘𝑊)‘𝑓) ≤ 𝑋}) |
8 | ssrab2 3912 | . 2 ⊢ {𝑓 ∈ 𝑇 ∣ (((trL‘𝐾)‘𝑊)‘𝑓) ≤ 𝑋} ⊆ 𝑇 | |
9 | 7, 8 | syl6eqss 3880 | 1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) ⊆ 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1658 ∈ wcel 2166 {crab 3121 ⊆ wss 3798 class class class wbr 4873 ‘cfv 6123 Basecbs 16222 lecple 16312 LHypclh 36059 LTrncltrn 36176 trLctrl 36233 DIsoAcdia 37103 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pr 5127 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-id 5250 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-disoa 37104 |
This theorem is referenced by: diael 37118 diaelrnN 37120 dialss 37121 dia2dimlem12 37150 diaocN 37200 dibss 37244 |
Copyright terms: Public domain | W3C validator |