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Mirrors > Home > MPE Home > Th. List > Mathboxes > diarnN | Structured version Visualization version GIF version |
Description: Partial isomorphism A maps onto the set of all closed subspaces of partial vector space A. Part of Lemma M of [Crawley] p. 121 line 12, with closed subspaces rather than subspaces. (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 |
---|---|
diarnN | β’ ((πΎ β HL β§ π β π») β ran πΌ = {π₯ β π β£ ( β₯ β( β₯ βπ₯)) = π₯}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvadia.h | . . . 4 β’ π» = (LHypβπΎ) | |
2 | dvadia.u | . . . 4 β’ π = ((DVecAβπΎ)βπ) | |
3 | dvadia.i | . . . 4 β’ πΌ = ((DIsoAβπΎ)βπ) | |
4 | dvadia.s | . . . 4 β’ π = (LSubSpβπ) | |
5 | 1, 2, 3, 4 | diasslssN 39525 | . . 3 β’ ((πΎ β HL β§ π β π») β ran πΌ β π) |
6 | sseqin2 4176 | . . 3 β’ (ran πΌ β π β (π β© ran πΌ) = ran πΌ) | |
7 | 5, 6 | sylib 217 | . 2 β’ ((πΎ β HL β§ π β π») β (π β© ran πΌ) = ran πΌ) |
8 | dvadia.n | . . . . . . 7 β’ β₯ = ((ocAβπΎ)βπ) | |
9 | 1, 3, 8 | doca3N 39593 | . . . . . 6 β’ (((πΎ β HL β§ π β π») β§ π₯ β ran πΌ) β ( β₯ β( β₯ βπ₯)) = π₯) |
10 | 9 | ex 414 | . . . . 5 β’ ((πΎ β HL β§ π β π») β (π₯ β ran πΌ β ( β₯ β( β₯ βπ₯)) = π₯)) |
11 | 10 | adantr 482 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ π₯ β π) β (π₯ β ran πΌ β ( β₯ β( β₯ βπ₯)) = π₯)) |
12 | 1, 2, 3, 8, 4 | dvadiaN 39594 | . . . . 5 β’ (((πΎ β HL β§ π β π») β§ (π₯ β π β§ ( β₯ β( β₯ βπ₯)) = π₯)) β π₯ β ran πΌ) |
13 | 12 | expr 458 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ π₯ β π) β (( β₯ β( β₯ βπ₯)) = π₯ β π₯ β ran πΌ)) |
14 | 11, 13 | impbid 211 | . . 3 β’ (((πΎ β HL β§ π β π») β§ π₯ β π) β (π₯ β ran πΌ β ( β₯ β( β₯ βπ₯)) = π₯)) |
15 | 14 | rabbi2dva 4178 | . 2 β’ ((πΎ β HL β§ π β π») β (π β© ran πΌ) = {π₯ β π β£ ( β₯ β( β₯ βπ₯)) = π₯}) |
16 | 7, 15 | eqtr3d 2779 | 1 β’ ((πΎ β HL β§ π β π») β ran πΌ = {π₯ β π β£ ( β₯ β( β₯ βπ₯)) = π₯}) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 {crab 3408 β© cin 3910 β wss 3911 ran crn 5635 βcfv 6497 LSubSpclss 20395 HLchlt 37815 LHypclh 38450 DVecAcdveca 39468 DIsoAcdia 39494 ocAcocaN 39585 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 ax-resscn 11109 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-mulrcl 11115 ax-mulcom 11116 ax-addass 11117 ax-mulass 11118 ax-distr 11119 ax-i2m1 11120 ax-1ne0 11121 ax-1rid 11122 ax-rnegex 11123 ax-rrecex 11124 ax-cnre 11125 ax-pre-lttri 11126 ax-pre-lttrn 11127 ax-pre-ltadd 11128 ax-pre-mulgt0 11129 ax-riotaBAD 37418 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3354 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-iin 4958 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-undef 8205 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8649 df-map 8768 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 df-sub 11388 df-neg 11389 df-nn 12155 df-2 12217 df-3 12218 df-4 12219 df-5 12220 df-6 12221 df-n0 12415 df-z 12501 df-uz 12765 df-fz 13426 df-struct 17020 df-slot 17055 df-ndx 17067 df-base 17085 df-plusg 17147 df-mulr 17148 df-sca 17150 df-vsca 17151 df-proset 18185 df-poset 18203 df-plt 18220 df-lub 18236 df-glb 18237 df-join 18238 df-meet 18239 df-p0 18315 df-p1 18316 df-lat 18322 df-clat 18389 df-lss 20396 df-oposet 37641 df-cmtN 37642 df-ol 37643 df-oml 37644 df-covers 37731 df-ats 37732 df-atl 37763 df-cvlat 37787 df-hlat 37816 df-llines 37964 df-lplanes 37965 df-lvols 37966 df-lines 37967 df-psubsp 37969 df-pmap 37970 df-padd 38262 df-lhyp 38454 df-laut 38455 df-ldil 38570 df-ltrn 38571 df-trl 38625 df-tendo 39221 df-edring 39223 df-dveca 39469 df-disoa 39495 df-docaN 39586 |
This theorem is referenced by: diaf1oN 39596 |
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