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Mirrors > Home > MPE Home > Th. List > Mathboxes > diaf1oN | Structured version Visualization version GIF version |
Description: The partial isomorphism A for a lattice πΎ is a one-to-one, onto function. Part of Lemma M of [Crawley] p. 121 line 12, with closed subspaces rather than subspaces. See diadm 40418 for the domain. (Contributed by NM, 17-Jan-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dvadia.h | β’ π» = (LHypβπΎ) |
dvadia.u | β’ π = ((DVecAβπΎ)βπ) |
dvadia.i | β’ πΌ = ((DIsoAβπΎ)βπ) |
dvadia.n | β’ β₯ = ((ocAβπΎ)βπ) |
dvadia.s | β’ π = (LSubSpβπ) |
Ref | Expression |
---|---|
diaf1oN | β’ ((πΎ β HL β§ π β π») β πΌ:dom πΌβ1-1-ontoβ{π₯ β π β£ ( β₯ β( β₯ βπ₯)) = π₯}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvadia.h | . . . . 5 β’ π» = (LHypβπΎ) | |
2 | dvadia.i | . . . . 5 β’ πΌ = ((DIsoAβπΎ)βπ) | |
3 | 1, 2 | diaf11N 40432 | . . . 4 β’ ((πΎ β HL β§ π β π») β πΌ:dom πΌβ1-1-ontoβran πΌ) |
4 | f1of1 6825 | . . . 4 β’ (πΌ:dom πΌβ1-1-ontoβran πΌ β πΌ:dom πΌβ1-1βran πΌ) | |
5 | 3, 4 | syl 17 | . . 3 β’ ((πΎ β HL β§ π β π») β πΌ:dom πΌβ1-1βran πΌ) |
6 | dvadia.u | . . . . 5 β’ π = ((DVecAβπΎ)βπ) | |
7 | dvadia.n | . . . . 5 β’ β₯ = ((ocAβπΎ)βπ) | |
8 | dvadia.s | . . . . 5 β’ π = (LSubSpβπ) | |
9 | 1, 6, 2, 7, 8 | diarnN 40512 | . . . 4 β’ ((πΎ β HL β§ π β π») β ran πΌ = {π₯ β π β£ ( β₯ β( β₯ βπ₯)) = π₯}) |
10 | f1eq3 6777 | . . . 4 β’ (ran πΌ = {π₯ β π β£ ( β₯ β( β₯ βπ₯)) = π₯} β (πΌ:dom πΌβ1-1βran πΌ β πΌ:dom πΌβ1-1β{π₯ β π β£ ( β₯ β( β₯ βπ₯)) = π₯})) | |
11 | 9, 10 | syl 17 | . . 3 β’ ((πΎ β HL β§ π β π») β (πΌ:dom πΌβ1-1βran πΌ β πΌ:dom πΌβ1-1β{π₯ β π β£ ( β₯ β( β₯ βπ₯)) = π₯})) |
12 | 5, 11 | mpbid 231 | . 2 β’ ((πΎ β HL β§ π β π») β πΌ:dom πΌβ1-1β{π₯ β π β£ ( β₯ β( β₯ βπ₯)) = π₯}) |
13 | dff1o5 6835 | . 2 β’ (πΌ:dom πΌβ1-1-ontoβ{π₯ β π β£ ( β₯ β( β₯ βπ₯)) = π₯} β (πΌ:dom πΌβ1-1β{π₯ β π β£ ( β₯ β( β₯ βπ₯)) = π₯} β§ ran πΌ = {π₯ β π β£ ( β₯ β( β₯ βπ₯)) = π₯})) | |
14 | 12, 9, 13 | sylanbrc 582 | 1 β’ ((πΎ β HL β§ π β π») β πΌ:dom πΌβ1-1-ontoβ{π₯ β π β£ ( β₯ β( β₯ βπ₯)) = π₯}) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 = wceq 1533 β wcel 2098 {crab 3426 dom cdm 5669 ran crn 5670 β1-1βwf1 6533 β1-1-ontoβwf1o 6535 βcfv 6536 LSubSpclss 20775 HLchlt 38732 LHypclh 39367 DVecAcdveca 40385 DIsoAcdia 40411 ocAcocaN 40502 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-riotaBAD 38335 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-undef 8256 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-n0 12474 df-z 12560 df-uz 12824 df-fz 13488 df-struct 17086 df-slot 17121 df-ndx 17133 df-base 17151 df-plusg 17216 df-mulr 17217 df-sca 17219 df-vsca 17220 df-proset 18257 df-poset 18275 df-plt 18292 df-lub 18308 df-glb 18309 df-join 18310 df-meet 18311 df-p0 18387 df-p1 18388 df-lat 18394 df-clat 18461 df-lss 20776 df-oposet 38558 df-cmtN 38559 df-ol 38560 df-oml 38561 df-covers 38648 df-ats 38649 df-atl 38680 df-cvlat 38704 df-hlat 38733 df-llines 38881 df-lplanes 38882 df-lvols 38883 df-lines 38884 df-psubsp 38886 df-pmap 38887 df-padd 39179 df-lhyp 39371 df-laut 39372 df-ldil 39487 df-ltrn 39488 df-trl 39542 df-tendo 40138 df-edring 40140 df-dveca 40386 df-disoa 40412 df-docaN 40503 |
This theorem is referenced by: (None) |
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