| 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 2737 | . . 3 ⊢ ((trL‘𝐾)‘𝑊) = ((trL‘𝐾)‘𝑊) | |
| 6 | diass.i | . . 3 ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊) | |
| 7 | 1, 2, 3, 4, 5, 6 | diaval 41034 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) = {𝑓 ∈ 𝑇 ∣ (((trL‘𝐾)‘𝑊)‘𝑓) ≤ 𝑋}) | 
| 8 | ssrab2 4080 | . 2 ⊢ {𝑓 ∈ 𝑇 ∣ (((trL‘𝐾)‘𝑊)‘𝑓) ≤ 𝑋} ⊆ 𝑇 | |
| 9 | 7, 8 | eqsstrdi 4028 | 1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) ⊆ 𝑇) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {crab 3436 ⊆ wss 3951 class class class wbr 5143 ‘cfv 6561 Basecbs 17247 lecple 17304 LHypclh 39986 LTrncltrn 40103 trLctrl 40160 DIsoAcdia 41030 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-disoa 41031 | 
| This theorem is referenced by: diael 41045 diaelrnN 41047 dialss 41048 dia2dimlem12 41077 diaocN 41127 dibss 41171 | 
| Copyright terms: Public domain | W3C validator |