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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > tendoinvcl | Structured version Visualization version GIF version |
Description: Closure of multiplicative inverse for endomorphism. We use the scalar inverse of the vector space since it is much simpler than the direct inverse of cdleml8 40158. (Contributed by NM, 10-Apr-2014.) (Revised by Mario Carneiro, 23-Jun-2014.) |
Ref | Expression |
---|---|
tendoinv.b | β’ π΅ = (BaseβπΎ) |
tendoinv.h | β’ π» = (LHypβπΎ) |
tendoinv.t | β’ π = ((LTrnβπΎ)βπ) |
tendoinv.e | β’ πΈ = ((TEndoβπΎ)βπ) |
tendoinv.o | β’ π = (β β π β¦ ( I βΎ π΅)) |
tendoinv.u | β’ π = ((DVecHβπΎ)βπ) |
tendoinv.f | β’ πΉ = (Scalarβπ) |
tendoinv.n | β’ π = (invrβπΉ) |
Ref | Expression |
---|---|
tendoinvcl | β’ (((πΎ β HL β§ π β π») β§ π β πΈ β§ π β π) β ((πβπ) β πΈ β§ (πβπ) β π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tendoinv.h | . . . . . . 7 β’ π» = (LHypβπΎ) | |
2 | eqid 2731 | . . . . . . 7 β’ ((EDRingβπΎ)βπ) = ((EDRingβπΎ)βπ) | |
3 | tendoinv.u | . . . . . . 7 β’ π = ((DVecHβπΎ)βπ) | |
4 | tendoinv.f | . . . . . . 7 β’ πΉ = (Scalarβπ) | |
5 | 1, 2, 3, 4 | dvhsca 40257 | . . . . . 6 β’ ((πΎ β HL β§ π β π») β πΉ = ((EDRingβπΎ)βπ)) |
6 | 1, 2 | erngdv 40168 | . . . . . 6 β’ ((πΎ β HL β§ π β π») β ((EDRingβπΎ)βπ) β DivRing) |
7 | 5, 6 | eqeltrd 2832 | . . . . 5 β’ ((πΎ β HL β§ π β π») β πΉ β DivRing) |
8 | 7 | 3ad2ant1 1132 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ π β πΈ β§ π β π) β πΉ β DivRing) |
9 | simp2 1136 | . . . . 5 β’ (((πΎ β HL β§ π β π») β§ π β πΈ β§ π β π) β π β πΈ) | |
10 | tendoinv.e | . . . . . . 7 β’ πΈ = ((TEndoβπΎ)βπ) | |
11 | eqid 2731 | . . . . . . 7 β’ (BaseβπΉ) = (BaseβπΉ) | |
12 | 1, 10, 3, 4, 11 | dvhbase 40258 | . . . . . 6 β’ ((πΎ β HL β§ π β π») β (BaseβπΉ) = πΈ) |
13 | 12 | 3ad2ant1 1132 | . . . . 5 β’ (((πΎ β HL β§ π β π») β§ π β πΈ β§ π β π) β (BaseβπΉ) = πΈ) |
14 | 9, 13 | eleqtrrd 2835 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ π β πΈ β§ π β π) β π β (BaseβπΉ)) |
15 | simp3 1137 | . . . . 5 β’ (((πΎ β HL β§ π β π») β§ π β πΈ β§ π β π) β π β π) | |
16 | 5 | fveq2d 6896 | . . . . . . 7 β’ ((πΎ β HL β§ π β π») β (0gβπΉ) = (0gβ((EDRingβπΎ)βπ))) |
17 | tendoinv.b | . . . . . . . 8 β’ π΅ = (BaseβπΎ) | |
18 | tendoinv.t | . . . . . . . 8 β’ π = ((LTrnβπΎ)βπ) | |
19 | tendoinv.o | . . . . . . . 8 β’ π = (β β π β¦ ( I βΎ π΅)) | |
20 | eqid 2731 | . . . . . . . 8 β’ (0gβ((EDRingβπΎ)βπ)) = (0gβ((EDRingβπΎ)βπ)) | |
21 | 17, 1, 18, 2, 19, 20 | erng0g 40169 | . . . . . . 7 β’ ((πΎ β HL β§ π β π») β (0gβ((EDRingβπΎ)βπ)) = π) |
22 | 16, 21 | eqtrd 2771 | . . . . . 6 β’ ((πΎ β HL β§ π β π») β (0gβπΉ) = π) |
23 | 22 | 3ad2ant1 1132 | . . . . 5 β’ (((πΎ β HL β§ π β π») β§ π β πΈ β§ π β π) β (0gβπΉ) = π) |
24 | 15, 23 | neeqtrrd 3014 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ π β πΈ β§ π β π) β π β (0gβπΉ)) |
25 | eqid 2731 | . . . . 5 β’ (0gβπΉ) = (0gβπΉ) | |
26 | tendoinv.n | . . . . 5 β’ π = (invrβπΉ) | |
27 | 11, 25, 26 | drnginvrcl 20523 | . . . 4 β’ ((πΉ β DivRing β§ π β (BaseβπΉ) β§ π β (0gβπΉ)) β (πβπ) β (BaseβπΉ)) |
28 | 8, 14, 24, 27 | syl3anc 1370 | . . 3 β’ (((πΎ β HL β§ π β π») β§ π β πΈ β§ π β π) β (πβπ) β (BaseβπΉ)) |
29 | 28, 13 | eleqtrd 2834 | . 2 β’ (((πΎ β HL β§ π β π») β§ π β πΈ β§ π β π) β (πβπ) β πΈ) |
30 | 11, 25, 26 | drnginvrn0 20524 | . . . 4 β’ ((πΉ β DivRing β§ π β (BaseβπΉ) β§ π β (0gβπΉ)) β (πβπ) β (0gβπΉ)) |
31 | 8, 14, 24, 30 | syl3anc 1370 | . . 3 β’ (((πΎ β HL β§ π β π») β§ π β πΈ β§ π β π) β (πβπ) β (0gβπΉ)) |
32 | 31, 23 | neeqtrd 3009 | . 2 β’ (((πΎ β HL β§ π β π») β§ π β πΈ β§ π β π) β (πβπ) β π) |
33 | 29, 32 | jca 511 | 1 β’ (((πΎ β HL β§ π β π») β§ π β πΈ β§ π β π) β ((πβπ) β πΈ β§ (πβπ) β π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 β§ w3a 1086 = wceq 1540 β wcel 2105 β wne 2939 β¦ cmpt 5232 I cid 5574 βΎ cres 5679 βcfv 6544 Basecbs 17149 Scalarcsca 17205 0gc0g 17390 invrcinvr 20279 DivRingcdr 20501 HLchlt 38524 LHypclh 39159 LTrncltrn 39276 TEndoctendo 39927 EDRingcedring 39928 DVecHcdvh 40253 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7728 ax-cnex 11169 ax-resscn 11170 ax-1cn 11171 ax-icn 11172 ax-addcl 11173 ax-addrcl 11174 ax-mulcl 11175 ax-mulrcl 11176 ax-mulcom 11177 ax-addass 11178 ax-mulass 11179 ax-distr 11180 ax-i2m1 11181 ax-1ne0 11182 ax-1rid 11183 ax-rnegex 11184 ax-rrecex 11185 ax-cnre 11186 ax-pre-lttri 11187 ax-pre-lttrn 11188 ax-pre-ltadd 11189 ax-pre-mulgt0 11190 ax-riotaBAD 38127 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7859 df-1st 7978 df-2nd 7979 df-tpos 8214 df-undef 8261 df-frecs 8269 df-wrecs 8300 df-recs 8374 df-rdg 8413 df-1o 8469 df-er 8706 df-map 8825 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-nn 12218 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-n0 12478 df-z 12564 df-uz 12828 df-fz 13490 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-sca 17218 df-vsca 17219 df-0g 17392 df-proset 18253 df-poset 18271 df-plt 18288 df-lub 18304 df-glb 18305 df-join 18306 df-meet 18307 df-p0 18383 df-p1 18384 df-lat 18390 df-clat 18457 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-grp 18859 df-minusg 18860 df-cmn 19692 df-abl 19693 df-mgp 20030 df-rng 20048 df-ur 20077 df-ring 20130 df-oppr 20226 df-dvdsr 20249 df-unit 20250 df-invr 20280 df-dvr 20293 df-drng 20503 df-oposet 38350 df-ol 38352 df-oml 38353 df-covers 38440 df-ats 38441 df-atl 38472 df-cvlat 38496 df-hlat 38525 df-llines 38673 df-lplanes 38674 df-lvols 38675 df-lines 38676 df-psubsp 38678 df-pmap 38679 df-padd 38971 df-lhyp 39163 df-laut 39164 df-ldil 39279 df-ltrn 39280 df-trl 39334 df-tendo 39930 df-edring 39932 df-dvech 40254 |
This theorem is referenced by: tendolinv 40280 tendorinv 40281 dih1dimatlem0 40503 dih1dimatlem 40504 |
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