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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 | β’ π» = (LHypβπΎ) |
dia1.t | β’ π = ((LTrnβπΎ)βπ) |
dia1.i | β’ πΌ = ((DIsoAβπΎ)βπ) |
Ref | Expression |
---|---|
dia1elN | β’ ((πΎ β HL β§ π β π») β π β ran πΌ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dia1.h | . . 3 β’ π» = (LHypβπΎ) | |
2 | dia1.t | . . 3 β’ π = ((LTrnβπΎ)βπ) | |
3 | dia1.i | . . 3 β’ πΌ = ((DIsoAβπΎ)βπ) | |
4 | 1, 2, 3 | dia1N 40530 | . 2 β’ ((πΎ β HL β§ π β π») β (πΌβπ) = π) |
5 | 1, 3 | diaf11N 40526 | . . . 4 β’ ((πΎ β HL β§ π β π») β πΌ:dom πΌβ1-1-ontoβran πΌ) |
6 | f1ofun 6844 | . . . 4 β’ (πΌ:dom πΌβ1-1-ontoβran πΌ β Fun πΌ) | |
7 | 5, 6 | syl 17 | . . 3 β’ ((πΎ β HL β§ π β π») β Fun πΌ) |
8 | 1, 3 | dia1eldmN 40518 | . . 3 β’ ((πΎ β HL β§ π β π») β π β dom πΌ) |
9 | fvelrn 7089 | . . 3 β’ ((Fun πΌ β§ π β dom πΌ) β (πΌβπ) β ran πΌ) | |
10 | 7, 8, 9 | syl2anc 582 | . 2 β’ ((πΎ β HL β§ π β π») β (πΌβπ) β ran πΌ) |
11 | 4, 10 | eqeltrrd 2829 | 1 β’ ((πΎ β HL β§ π β π») β π β ran πΌ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 dom cdm 5680 ran crn 5681 Fun wfun 6545 β1-1-ontoβwf1o 6550 βcfv 6551 HLchlt 38826 LHypclh 39461 LTrncltrn 39578 DIsoAcdia 40505 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-rep 5287 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-riotaBAD 38429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-iun 5000 df-iin 5001 df-br 5151 df-opab 5213 df-mpt 5234 df-id 5578 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-1st 7997 df-2nd 7998 df-undef 8283 df-map 8851 df-proset 18292 df-poset 18310 df-plt 18327 df-lub 18343 df-glb 18344 df-join 18345 df-meet 18346 df-p0 18422 df-p1 18423 df-lat 18429 df-clat 18496 df-oposet 38652 df-ol 38654 df-oml 38655 df-covers 38742 df-ats 38743 df-atl 38774 df-cvlat 38798 df-hlat 38827 df-llines 38975 df-lplanes 38976 df-lvols 38977 df-lines 38978 df-psubsp 38980 df-pmap 38981 df-padd 39273 df-lhyp 39465 df-laut 39466 df-ldil 39581 df-ltrn 39582 df-trl 39636 df-disoa 40506 |
This theorem is referenced by: docaclN 40601 |
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