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Mirrors > Home > MPE Home > Th. List > Mathboxes > diadmclN | Structured version Visualization version GIF version |
Description: A member of domain of the partial isomorphism A is a lattice element. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
diadmcl.b | ⊢ 𝐵 = (Base‘𝐾) |
diadmcl.h | ⊢ 𝐻 = (LHyp‘𝐾) |
diadmcl.i | ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
diadmclN | ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → 𝑋 ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | diadmcl.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
2 | eqid 2736 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
3 | diadmcl.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
4 | diadmcl.i | . . 3 ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊) | |
5 | 1, 2, 3, 4 | diaeldm 38736 | . 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (𝑋 ∈ dom 𝐼 ↔ (𝑋 ∈ 𝐵 ∧ 𝑋(le‘𝐾)𝑊))) |
6 | 5 | simprbda 502 | 1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → 𝑋 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 class class class wbr 5039 dom cdm 5536 ‘cfv 6358 Basecbs 16666 lecple 16756 LHypclh 37684 DIsoAcdia 38728 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-disoa 38729 |
This theorem is referenced by: diameetN 38756 docaclN 38824 diaocN 38825 doca2N 38826 djajN 38837 |
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