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Mirrors > Home > MPE Home > Th. List > Mathboxes > dia1eldmN | Structured version Visualization version GIF version |
Description: The fiducial hyperplane (the largest allowed lattice element) belongs to the domain of partial isomorphism A. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dia1eldm.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dia1eldm.i | ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
dia1eldmN | ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑊 ∈ dom 𝐼) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2821 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
2 | dia1eldm.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
3 | 1, 2 | lhpbase 37128 | . . 3 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾)) |
4 | 3 | adantl 484 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑊 ∈ (Base‘𝐾)) |
5 | hllat 36493 | . . 3 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) | |
6 | eqid 2821 | . . . 4 ⊢ (le‘𝐾) = (le‘𝐾) | |
7 | 1, 6 | latref 17657 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑊 ∈ (Base‘𝐾)) → 𝑊(le‘𝐾)𝑊) |
8 | 5, 3, 7 | syl2an 597 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑊(le‘𝐾)𝑊) |
9 | dia1eldm.i | . . 3 ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊) | |
10 | 1, 6, 2, 9 | diaeldm 38166 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑊 ∈ dom 𝐼 ↔ (𝑊 ∈ (Base‘𝐾) ∧ 𝑊(le‘𝐾)𝑊))) |
11 | 4, 8, 10 | mpbir2and 711 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑊 ∈ dom 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 class class class wbr 5058 dom cdm 5549 ‘cfv 6349 Basecbs 16477 lecple 16566 Latclat 17649 HLchlt 36480 LHypclh 37114 DIsoAcdia 38158 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-proset 17532 df-poset 17550 df-lat 17650 df-atl 36428 df-cvlat 36452 df-hlat 36481 df-lhyp 37118 df-disoa 38159 |
This theorem is referenced by: dia1elN 38184 |
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