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Mirrors > Home > MPE Home > Th. List > Mathboxes > diaclN | Structured version Visualization version GIF version |
Description: Closure of partial isomorphism A for a lattice 𝐾. (Contributed by NM, 4-Dec-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dia1o.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dia1o.i | ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
diaclN | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → (𝐼‘𝑋) ∈ ran 𝐼) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dia1o.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | dia1o.i | . . . 4 ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊) | |
3 | 1, 2 | diaf11N 38225 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐼:dom 𝐼–1-1-onto→ran 𝐼) |
4 | f1ofun 6610 | . . 3 ⊢ (𝐼:dom 𝐼–1-1-onto→ran 𝐼 → Fun 𝐼) | |
5 | 3, 4 | syl 17 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → Fun 𝐼) |
6 | fvelrn 6837 | . 2 ⊢ ((Fun 𝐼 ∧ 𝑋 ∈ dom 𝐼) → (𝐼‘𝑋) ∈ ran 𝐼) | |
7 | 5, 6 | sylan 582 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → (𝐼‘𝑋) ∈ ran 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 dom cdm 5548 ran crn 5549 Fun wfun 6342 –1-1-onto→wf1o 6347 ‘cfv 6348 HLchlt 36526 LHypclh 37160 DIsoAcdia 38204 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-riotaBAD 36129 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4914 df-iin 4915 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-1st 7682 df-2nd 7683 df-undef 7932 df-map 8401 df-proset 17531 df-poset 17549 df-plt 17561 df-lub 17577 df-glb 17578 df-join 17579 df-meet 17580 df-p0 17642 df-p1 17643 df-lat 17649 df-clat 17711 df-oposet 36352 df-ol 36354 df-oml 36355 df-covers 36442 df-ats 36443 df-atl 36474 df-cvlat 36498 df-hlat 36527 df-llines 36674 df-lplanes 36675 df-lvols 36676 df-lines 36677 df-psubsp 36679 df-pmap 36680 df-padd 36972 df-lhyp 37164 df-laut 37165 df-ldil 37280 df-ltrn 37281 df-trl 37335 df-disoa 38205 |
This theorem is referenced by: diaintclN 38234 diaocN 38301 djajN 38313 |
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