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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dia1elN | Structured version Visualization version GIF version |
Description: The largest subspace in the range of partial isomorphism A. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dia1.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dia1.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
dia1.i | ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
dia1elN | ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑇 ∈ ran 𝐼) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dia1.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | dia1.t | . . 3 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
3 | dia1.i | . . 3 ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊) | |
4 | 1, 2, 3 | dia1N 37207 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝐼‘𝑊) = 𝑇) |
5 | 1, 3 | diaf11N 37203 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐼:dom 𝐼–1-1-onto→ran 𝐼) |
6 | f1ofun 6393 | . . . 4 ⊢ (𝐼:dom 𝐼–1-1-onto→ran 𝐼 → Fun 𝐼) | |
7 | 5, 6 | syl 17 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → Fun 𝐼) |
8 | 1, 3 | dia1eldmN 37195 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑊 ∈ dom 𝐼) |
9 | fvelrn 6616 | . . 3 ⊢ ((Fun 𝐼 ∧ 𝑊 ∈ dom 𝐼) → (𝐼‘𝑊) ∈ ran 𝐼) | |
10 | 7, 8, 9 | syl2anc 579 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝐼‘𝑊) ∈ ran 𝐼) |
11 | 4, 10 | eqeltrrd 2860 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑇 ∈ ran 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 dom cdm 5355 ran crn 5356 Fun wfun 6129 –1-1-onto→wf1o 6134 ‘cfv 6135 HLchlt 35504 LHypclh 36138 LTrncltrn 36255 DIsoAcdia 37182 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-riotaBAD 35107 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-iun 4755 df-iin 4756 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-1st 7445 df-2nd 7446 df-undef 7681 df-map 8142 df-proset 17314 df-poset 17332 df-plt 17344 df-lub 17360 df-glb 17361 df-join 17362 df-meet 17363 df-p0 17425 df-p1 17426 df-lat 17432 df-clat 17494 df-oposet 35330 df-ol 35332 df-oml 35333 df-covers 35420 df-ats 35421 df-atl 35452 df-cvlat 35476 df-hlat 35505 df-llines 35652 df-lplanes 35653 df-lvols 35654 df-lines 35655 df-psubsp 35657 df-pmap 35658 df-padd 35950 df-lhyp 36142 df-laut 36143 df-ldil 36258 df-ltrn 36259 df-trl 36313 df-disoa 37183 |
This theorem is referenced by: docaclN 37278 |
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