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Mirrors > Home > MPE Home > Th. List > Mathboxes > dia2dimlem13 | Structured version Visualization version GIF version |
Description: Lemma for dia2dim 40487. Eliminate π β π condition. (Contributed by NM, 8-Sep-2014.) |
Ref | Expression |
---|---|
dia2dimlem12.l | β’ β€ = (leβπΎ) |
dia2dimlem12.j | β’ β¨ = (joinβπΎ) |
dia2dimlem12.m | β’ β§ = (meetβπΎ) |
dia2dimlem12.a | β’ π΄ = (AtomsβπΎ) |
dia2dimlem12.h | β’ π» = (LHypβπΎ) |
dia2dimlem12.t | β’ π = ((LTrnβπΎ)βπ) |
dia2dimlem12.r | β’ π = ((trLβπΎ)βπ) |
dia2dimlem12.y | β’ π = ((DVecAβπΎ)βπ) |
dia2dimlem12.s | β’ π = (LSubSpβπ) |
dia2dimlem12.pl | β’ β = (LSSumβπ) |
dia2dimlem12.n | β’ π = (LSpanβπ) |
dia2dimlem12.i | β’ πΌ = ((DIsoAβπΎ)βπ) |
dia2dimlem12.k | β’ (π β (πΎ β HL β§ π β π»)) |
dia2dimlem12.u | β’ (π β (π β π΄ β§ π β€ π)) |
dia2dimlem12.v | β’ (π β (π β π΄ β§ π β€ π)) |
Ref | Expression |
---|---|
dia2dimlem13 | β’ (π β (πΌβ(π β¨ π)) β ((πΌβπ) β (πΌβπ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 7422 | . . . . . . 7 β’ (π = π β (π β¨ π) = (π β¨ π)) | |
2 | 1 | adantl 481 | . . . . . 6 β’ ((π β§ π = π) β (π β¨ π) = (π β¨ π)) |
3 | dia2dimlem12.k | . . . . . . . . 9 β’ (π β (πΎ β HL β§ π β π»)) | |
4 | 3 | simpld 494 | . . . . . . . 8 β’ (π β πΎ β HL) |
5 | dia2dimlem12.u | . . . . . . . . 9 β’ (π β (π β π΄ β§ π β€ π)) | |
6 | 5 | simpld 494 | . . . . . . . 8 β’ (π β π β π΄) |
7 | dia2dimlem12.j | . . . . . . . . 9 β’ β¨ = (joinβπΎ) | |
8 | dia2dimlem12.a | . . . . . . . . 9 β’ π΄ = (AtomsβπΎ) | |
9 | 7, 8 | hlatjidm 38778 | . . . . . . . 8 β’ ((πΎ β HL β§ π β π΄) β (π β¨ π) = π) |
10 | 4, 6, 9 | syl2anc 583 | . . . . . . 7 β’ (π β (π β¨ π) = π) |
11 | 10 | adantr 480 | . . . . . 6 β’ ((π β§ π = π) β (π β¨ π) = π) |
12 | 2, 11 | eqtr3d 2769 | . . . . 5 β’ ((π β§ π = π) β (π β¨ π) = π) |
13 | 12 | fveq2d 6895 | . . . 4 β’ ((π β§ π = π) β (πΌβ(π β¨ π)) = (πΌβπ)) |
14 | ssid 4000 | . . . 4 β’ (πΌβπ) β (πΌβπ) | |
15 | 13, 14 | eqsstrdi 4032 | . . 3 β’ ((π β§ π = π) β (πΌβ(π β¨ π)) β (πΌβπ)) |
16 | dia2dimlem12.h | . . . . . . . 8 β’ π» = (LHypβπΎ) | |
17 | dia2dimlem12.y | . . . . . . . 8 β’ π = ((DVecAβπΎ)βπ) | |
18 | 16, 17 | dvalvec 40436 | . . . . . . 7 β’ ((πΎ β HL β§ π β π») β π β LVec) |
19 | lveclmod 20980 | . . . . . . 7 β’ (π β LVec β π β LMod) | |
20 | 3, 18, 19 | 3syl 18 | . . . . . 6 β’ (π β π β LMod) |
21 | eqid 2727 | . . . . . . . . 9 β’ (BaseβπΎ) = (BaseβπΎ) | |
22 | 21, 8 | atbase 38698 | . . . . . . . 8 β’ (π β π΄ β π β (BaseβπΎ)) |
23 | 6, 22 | syl 17 | . . . . . . 7 β’ (π β π β (BaseβπΎ)) |
24 | 5 | simprd 495 | . . . . . . 7 β’ (π β π β€ π) |
25 | dia2dimlem12.l | . . . . . . . 8 β’ β€ = (leβπΎ) | |
26 | dia2dimlem12.i | . . . . . . . 8 β’ πΌ = ((DIsoAβπΎ)βπ) | |
27 | dia2dimlem12.s | . . . . . . . 8 β’ π = (LSubSpβπ) | |
28 | 21, 25, 16, 17, 26, 27 | dialss 40456 | . . . . . . 7 β’ (((πΎ β HL β§ π β π») β§ (π β (BaseβπΎ) β§ π β€ π)) β (πΌβπ) β π) |
29 | 3, 23, 24, 28 | syl12anc 836 | . . . . . 6 β’ (π β (πΌβπ) β π) |
30 | 27 | lsssubg 20830 | . . . . . 6 β’ ((π β LMod β§ (πΌβπ) β π) β (πΌβπ) β (SubGrpβπ)) |
31 | 20, 29, 30 | syl2anc 583 | . . . . 5 β’ (π β (πΌβπ) β (SubGrpβπ)) |
32 | dia2dimlem12.pl | . . . . . 6 β’ β = (LSSumβπ) | |
33 | 32 | lsmidm 19609 | . . . . 5 β’ ((πΌβπ) β (SubGrpβπ) β ((πΌβπ) β (πΌβπ)) = (πΌβπ)) |
34 | 31, 33 | syl 17 | . . . 4 β’ (π β ((πΌβπ) β (πΌβπ)) = (πΌβπ)) |
35 | fveq2 6891 | . . . . 5 β’ (π = π β (πΌβπ) = (πΌβπ)) | |
36 | 35 | oveq2d 7430 | . . . 4 β’ (π = π β ((πΌβπ) β (πΌβπ)) = ((πΌβπ) β (πΌβπ))) |
37 | 34, 36 | sylan9req 2788 | . . 3 β’ ((π β§ π = π) β (πΌβπ) = ((πΌβπ) β (πΌβπ))) |
38 | 15, 37 | sseqtrd 4018 | . 2 β’ ((π β§ π = π) β (πΌβ(π β¨ π)) β ((πΌβπ) β (πΌβπ))) |
39 | dia2dimlem12.m | . . 3 β’ β§ = (meetβπΎ) | |
40 | dia2dimlem12.t | . . 3 β’ π = ((LTrnβπΎ)βπ) | |
41 | dia2dimlem12.r | . . 3 β’ π = ((trLβπΎ)βπ) | |
42 | dia2dimlem12.n | . . 3 β’ π = (LSpanβπ) | |
43 | 3 | adantr 480 | . . 3 β’ ((π β§ π β π) β (πΎ β HL β§ π β π»)) |
44 | 5 | adantr 480 | . . 3 β’ ((π β§ π β π) β (π β π΄ β§ π β€ π)) |
45 | dia2dimlem12.v | . . . 4 β’ (π β (π β π΄ β§ π β€ π)) | |
46 | 45 | adantr 480 | . . 3 β’ ((π β§ π β π) β (π β π΄ β§ π β€ π)) |
47 | simpr 484 | . . 3 β’ ((π β§ π β π) β π β π) | |
48 | 25, 7, 39, 8, 16, 40, 41, 17, 27, 32, 42, 26, 43, 44, 46, 47 | dia2dimlem12 40485 | . 2 β’ ((π β§ π β π) β (πΌβ(π β¨ π)) β ((πΌβπ) β (πΌβπ))) |
49 | 38, 48 | pm2.61dane 3024 | 1 β’ (π β (πΌβ(π β¨ π)) β ((πΌβπ) β (πΌβπ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1534 β wcel 2099 β wne 2935 β wss 3944 class class class wbr 5142 βcfv 6542 (class class class)co 7414 Basecbs 17171 lecple 17231 joincjn 18294 meetcmee 18295 SubGrpcsubg 19066 LSSumclsm 19580 LModclmod 20732 LSubSpclss 20804 LSpanclspn 20844 LVecclvec 20976 Atomscatm 38672 HLchlt 38759 LHypclh 39394 LTrncltrn 39511 trLctrl 39568 DVecAcdveca 40412 DIsoAcdia 40438 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 ax-riotaBAD 38362 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 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 5143 df-opab 5205 df-mpt 5226 df-tr 5260 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 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-tpos 8225 df-undef 8272 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-map 8838 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-nn 12235 df-2 12297 df-3 12298 df-4 12299 df-5 12300 df-6 12301 df-n0 12495 df-z 12581 df-uz 12845 df-fz 13509 df-struct 17107 df-sets 17124 df-slot 17142 df-ndx 17154 df-base 17172 df-ress 17201 df-plusg 17237 df-mulr 17238 df-sca 17240 df-vsca 17241 df-0g 17414 df-proset 18278 df-poset 18296 df-plt 18313 df-lub 18329 df-glb 18330 df-join 18331 df-meet 18332 df-p0 18408 df-p1 18409 df-lat 18415 df-clat 18482 df-mgm 18591 df-sgrp 18670 df-mnd 18686 df-submnd 18732 df-grp 18884 df-minusg 18885 df-sbg 18886 df-subg 19069 df-cntz 19259 df-lsm 19582 df-cmn 19728 df-abl 19729 df-mgp 20066 df-rng 20084 df-ur 20113 df-ring 20166 df-oppr 20262 df-dvdsr 20285 df-unit 20286 df-invr 20316 df-dvr 20329 df-drng 20615 df-lmod 20734 df-lss 20805 df-lsp 20845 df-lvec 20977 df-oposet 38585 df-ol 38587 df-oml 38588 df-covers 38675 df-ats 38676 df-atl 38707 df-cvlat 38731 df-hlat 38760 df-llines 38908 df-lplanes 38909 df-lvols 38910 df-lines 38911 df-psubsp 38913 df-pmap 38914 df-padd 39206 df-lhyp 39398 df-laut 39399 df-ldil 39514 df-ltrn 39515 df-trl 39569 df-tgrp 40153 df-tendo 40165 df-edring 40167 df-dveca 40413 df-disoa 40439 |
This theorem is referenced by: dia2dim 40487 |
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