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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > diael | Structured version Visualization version GIF version |
Description: A member of the value of the partial isomorphism A is a translation, i.e., a vector. (Contributed by NM, 17-Jan-2014.) |
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 |
---|---|
diael | ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ 𝐹 ∈ (𝐼‘𝑋)) → 𝐹 ∈ 𝑇) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | diass.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
2 | diass.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
3 | diass.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
4 | diass.t | . . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
5 | diass.i | . . . 4 ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊) | |
6 | 1, 2, 3, 4, 5 | diass 40752 | . . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) ⊆ 𝑇) |
7 | 6 | sseld 3978 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐹 ∈ (𝐼‘𝑋) → 𝐹 ∈ 𝑇)) |
8 | 7 | 3impia 1114 | 1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ 𝐹 ∈ (𝐼‘𝑋)) → 𝐹 ∈ 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 class class class wbr 5144 ‘cfv 6544 Basecbs 17206 lecple 17266 LHypclh 39694 LTrncltrn 39811 DIsoAcdia 40738 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pr 5424 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3366 df-rab 3421 df-v 3465 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4907 df-iun 4996 df-br 5145 df-opab 5207 df-mpt 5228 df-id 5571 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-disoa 40739 |
This theorem is referenced by: dialss 40756 dibelval1st1 40860 diblsmopel 40881 |
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