![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > dih1dimb2 | Structured version Visualization version GIF version |
Description: Isomorphism H at an atom under π. (Contributed by NM, 27-Apr-2014.) |
Ref | Expression |
---|---|
dih1dimb2.b | β’ π΅ = (BaseβπΎ) |
dih1dimb2.l | β’ β€ = (leβπΎ) |
dih1dimb2.a | β’ π΄ = (AtomsβπΎ) |
dih1dimb2.h | β’ π» = (LHypβπΎ) |
dih1dimb2.t | β’ π = ((LTrnβπΎ)βπ) |
dih1dimb2.o | β’ π = (β β π β¦ ( I βΎ π΅)) |
dih1dimb2.u | β’ π = ((DVecHβπΎ)βπ) |
dih1dimb2.i | β’ πΌ = ((DIsoHβπΎ)βπ) |
dih1dimb2.n | β’ π = (LSpanβπ) |
Ref | Expression |
---|---|
dih1dimb2 | β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ π β€ π)) β βπ β π (π β ( I βΎ π΅) β§ (πΌβπ) = (πβ{β¨π, πβ©}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dih1dimb2.l | . . 3 β’ β€ = (leβπΎ) | |
2 | dih1dimb2.a | . . 3 β’ π΄ = (AtomsβπΎ) | |
3 | dih1dimb2.h | . . 3 β’ π» = (LHypβπΎ) | |
4 | dih1dimb2.t | . . 3 β’ π = ((LTrnβπΎ)βπ) | |
5 | eqid 2725 | . . 3 β’ ((trLβπΎ)βπ) = ((trLβπΎ)βπ) | |
6 | 1, 2, 3, 4, 5 | cdlemf 40092 | . 2 β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ π β€ π)) β βπ β π (((trLβπΎ)βπ)βπ) = π) |
7 | simp3 1135 | . . . . . . 7 β’ ((((πΎ β HL β§ π β π») β§ (π β π΄ β§ π β€ π)) β§ π β π β§ (((trLβπΎ)βπ)βπ) = π) β (((trLβπΎ)βπ)βπ) = π) | |
8 | simp1rl 1235 | . . . . . . 7 β’ ((((πΎ β HL β§ π β π») β§ (π β π΄ β§ π β€ π)) β§ π β π β§ (((trLβπΎ)βπ)βπ) = π) β π β π΄) | |
9 | 7, 8 | eqeltrd 2825 | . . . . . 6 β’ ((((πΎ β HL β§ π β π») β§ (π β π΄ β§ π β€ π)) β§ π β π β§ (((trLβπΎ)βπ)βπ) = π) β (((trLβπΎ)βπ)βπ) β π΄) |
10 | simp1l 1194 | . . . . . . 7 β’ ((((πΎ β HL β§ π β π») β§ (π β π΄ β§ π β€ π)) β§ π β π β§ (((trLβπΎ)βπ)βπ) = π) β (πΎ β HL β§ π β π»)) | |
11 | simp2 1134 | . . . . . . 7 β’ ((((πΎ β HL β§ π β π») β§ (π β π΄ β§ π β€ π)) β§ π β π β§ (((trLβπΎ)βπ)βπ) = π) β π β π) | |
12 | dih1dimb2.b | . . . . . . . 8 β’ π΅ = (BaseβπΎ) | |
13 | 12, 2, 3, 4, 5 | trlnidatb 39706 | . . . . . . 7 β’ (((πΎ β HL β§ π β π») β§ π β π) β (π β ( I βΎ π΅) β (((trLβπΎ)βπ)βπ) β π΄)) |
14 | 10, 11, 13 | syl2anc 582 | . . . . . 6 β’ ((((πΎ β HL β§ π β π») β§ (π β π΄ β§ π β€ π)) β§ π β π β§ (((trLβπΎ)βπ)βπ) = π) β (π β ( I βΎ π΅) β (((trLβπΎ)βπ)βπ) β π΄)) |
15 | 9, 14 | mpbird 256 | . . . . 5 β’ ((((πΎ β HL β§ π β π») β§ (π β π΄ β§ π β€ π)) β§ π β π β§ (((trLβπΎ)βπ)βπ) = π) β π β ( I βΎ π΅)) |
16 | 7 | fveq2d 6896 | . . . . . 6 β’ ((((πΎ β HL β§ π β π») β§ (π β π΄ β§ π β€ π)) β§ π β π β§ (((trLβπΎ)βπ)βπ) = π) β (πΌβ(((trLβπΎ)βπ)βπ)) = (πΌβπ)) |
17 | dih1dimb2.o | . . . . . . . 8 β’ π = (β β π β¦ ( I βΎ π΅)) | |
18 | dih1dimb2.u | . . . . . . . 8 β’ π = ((DVecHβπΎ)βπ) | |
19 | dih1dimb2.i | . . . . . . . 8 β’ πΌ = ((DIsoHβπΎ)βπ) | |
20 | dih1dimb2.n | . . . . . . . 8 β’ π = (LSpanβπ) | |
21 | 12, 3, 4, 5, 17, 18, 19, 20 | dih1dimb 40769 | . . . . . . 7 β’ (((πΎ β HL β§ π β π») β§ π β π) β (πΌβ(((trLβπΎ)βπ)βπ)) = (πβ{β¨π, πβ©})) |
22 | 10, 11, 21 | syl2anc 582 | . . . . . 6 β’ ((((πΎ β HL β§ π β π») β§ (π β π΄ β§ π β€ π)) β§ π β π β§ (((trLβπΎ)βπ)βπ) = π) β (πΌβ(((trLβπΎ)βπ)βπ)) = (πβ{β¨π, πβ©})) |
23 | 16, 22 | eqtr3d 2767 | . . . . 5 β’ ((((πΎ β HL β§ π β π») β§ (π β π΄ β§ π β€ π)) β§ π β π β§ (((trLβπΎ)βπ)βπ) = π) β (πΌβπ) = (πβ{β¨π, πβ©})) |
24 | 15, 23 | jca 510 | . . . 4 β’ ((((πΎ β HL β§ π β π») β§ (π β π΄ β§ π β€ π)) β§ π β π β§ (((trLβπΎ)βπ)βπ) = π) β (π β ( I βΎ π΅) β§ (πΌβπ) = (πβ{β¨π, πβ©}))) |
25 | 24 | 3expia 1118 | . . 3 β’ ((((πΎ β HL β§ π β π») β§ (π β π΄ β§ π β€ π)) β§ π β π) β ((((trLβπΎ)βπ)βπ) = π β (π β ( I βΎ π΅) β§ (πΌβπ) = (πβ{β¨π, πβ©})))) |
26 | 25 | reximdva 3158 | . 2 β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ π β€ π)) β (βπ β π (((trLβπΎ)βπ)βπ) = π β βπ β π (π β ( I βΎ π΅) β§ (πΌβπ) = (πβ{β¨π, πβ©})))) |
27 | 6, 26 | mpd 15 | 1 β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ π β€ π)) β βπ β π (π β ( I βΎ π΅) β§ (πΌβπ) = (πβ{β¨π, πβ©}))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 β wne 2930 βwrex 3060 {csn 4624 β¨cop 4630 class class class wbr 5143 β¦ cmpt 5226 I cid 5569 βΎ cres 5674 βcfv 6543 Basecbs 17179 lecple 17239 LSpanclspn 20859 Atomscatm 38791 HLchlt 38878 LHypclh 39513 LTrncltrn 39630 trLctrl 39687 DVecHcdvh 40607 DIsoHcdih 40757 |
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 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-riotaBAD 38481 |
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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-tpos 8230 df-undef 8277 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-map 8845 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-n0 12503 df-z 12589 df-uz 12853 df-fz 13517 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-mulr 17246 df-sca 17248 df-vsca 17249 df-0g 17422 df-proset 18286 df-poset 18304 df-plt 18321 df-lub 18337 df-glb 18338 df-join 18339 df-meet 18340 df-p0 18416 df-p1 18417 df-lat 18423 df-clat 18490 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-grp 18897 df-minusg 18898 df-sbg 18899 df-cmn 19741 df-abl 19742 df-mgp 20079 df-rng 20097 df-ur 20126 df-ring 20179 df-oppr 20277 df-dvdsr 20300 df-unit 20301 df-invr 20331 df-dvr 20344 df-drng 20630 df-lmod 20749 df-lss 20820 df-lsp 20860 df-lvec 20992 df-oposet 38704 df-ol 38706 df-oml 38707 df-covers 38794 df-ats 38795 df-atl 38826 df-cvlat 38850 df-hlat 38879 df-llines 39027 df-lplanes 39028 df-lvols 39029 df-lines 39030 df-psubsp 39032 df-pmap 39033 df-padd 39325 df-lhyp 39517 df-laut 39518 df-ldil 39633 df-ltrn 39634 df-trl 39688 df-tendo 40284 df-edring 40286 df-disoa 40558 df-dvech 40608 df-dib 40668 df-dih 40758 |
This theorem is referenced by: dihatlat 40863 dihatexv 40867 |
Copyright terms: Public domain | W3C validator |