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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapd11 | Structured version Visualization version GIF version |
Description: The map defined by df-mapd 41007 is one-to-one. Property (c) of [Baer] p. 40. (Contributed by NM, 12-Mar-2015.) |
Ref | Expression |
---|---|
mapdord.h | β’ π» = (LHypβπΎ) |
mapdord.u | β’ π = ((DVecHβπΎ)βπ) |
mapdord.s | β’ π = (LSubSpβπ) |
mapdord.m | β’ π = ((mapdβπΎ)βπ) |
mapdord.k | β’ (π β (πΎ β HL β§ π β π»)) |
mapdord.x | β’ (π β π β π) |
mapdord.y | β’ (π β π β π) |
Ref | Expression |
---|---|
mapd11 | β’ (π β ((πβπ) = (πβπ) β π = π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapdord.h | . . . 4 β’ π» = (LHypβπΎ) | |
2 | mapdord.u | . . . 4 β’ π = ((DVecHβπΎ)βπ) | |
3 | mapdord.s | . . . 4 β’ π = (LSubSpβπ) | |
4 | mapdord.m | . . . 4 β’ π = ((mapdβπΎ)βπ) | |
5 | mapdord.k | . . . 4 β’ (π β (πΎ β HL β§ π β π»)) | |
6 | mapdord.x | . . . 4 β’ (π β π β π) | |
7 | mapdord.y | . . . 4 β’ (π β π β π) | |
8 | 1, 2, 3, 4, 5, 6, 7 | mapdord 41020 | . . 3 β’ (π β ((πβπ) β (πβπ) β π β π)) |
9 | 1, 2, 3, 4, 5, 7, 6 | mapdord 41020 | . . 3 β’ (π β ((πβπ) β (πβπ) β π β π)) |
10 | 8, 9 | anbi12d 630 | . 2 β’ (π β (((πβπ) β (πβπ) β§ (πβπ) β (πβπ)) β (π β π β§ π β π))) |
11 | eqss 3992 | . 2 β’ ((πβπ) = (πβπ) β ((πβπ) β (πβπ) β§ (πβπ) β (πβπ))) | |
12 | eqss 3992 | . 2 β’ (π = π β (π β π β§ π β π)) | |
13 | 10, 11, 12 | 3bitr4g 314 | 1 β’ (π β ((πβπ) = (πβπ) β π = π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 = wceq 1533 β wcel 2098 β wss 3943 βcfv 6536 LSubSpclss 20776 HLchlt 38731 LHypclh 39366 DVecHcdvh 40460 mapdcmpd 41006 |
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 38334 |
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-tpos 8209 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 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-sca 17220 df-vsca 17221 df-0g 17394 df-proset 18258 df-poset 18276 df-plt 18293 df-lub 18309 df-glb 18310 df-join 18311 df-meet 18312 df-p0 18388 df-p1 18389 df-lat 18395 df-clat 18462 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-submnd 18712 df-grp 18864 df-minusg 18865 df-sbg 18866 df-subg 19048 df-cntz 19231 df-lsm 19554 df-cmn 19700 df-abl 19701 df-mgp 20038 df-rng 20056 df-ur 20085 df-ring 20138 df-oppr 20234 df-dvdsr 20257 df-unit 20258 df-invr 20288 df-dvr 20301 df-drng 20587 df-lmod 20706 df-lss 20777 df-lsp 20817 df-lvec 20949 df-lsatoms 38357 df-lshyp 38358 df-lfl 38439 df-lkr 38467 df-oposet 38557 df-ol 38559 df-oml 38560 df-covers 38647 df-ats 38648 df-atl 38679 df-cvlat 38703 df-hlat 38732 df-llines 38880 df-lplanes 38881 df-lvols 38882 df-lines 38883 df-psubsp 38885 df-pmap 38886 df-padd 39178 df-lhyp 39370 df-laut 39371 df-ldil 39486 df-ltrn 39487 df-trl 39541 df-tgrp 40125 df-tendo 40137 df-edring 40139 df-dveca 40385 df-disoa 40411 df-dvech 40461 df-dib 40521 df-dic 40555 df-dih 40611 df-doch 40730 df-djh 40777 df-mapd 41007 |
This theorem is referenced by: mapd1o 41030 mapdsord 41037 mapdn0 41051 mapdncol 41052 mapdpglem29 41082 hdmapeq0 41226 hdmaprnlem1N 41231 hdmaprnlem3N 41232 hdmaprnlem9N 41239 hdmap14lem9 41258 |
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