Nearby theorems
|
|
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
| Mirrors > Home > MPE Home > Th. List > Mathboxes > lcfrlem21 | Structured version Visualization version GIF version | ||
| Description: Lemma for lcfr 40760. (Contributed by NM, 11-Mar-2015.) |
| Ref | Expression |
|---|---|
| lcfrlem17.h | β’ π» = (LHypβπΎ) |
| lcfrlem17.o | β’ β₯ = ((ocHβπΎ)βπ) |
| lcfrlem17.u | β’ π = ((DVecHβπΎ)βπ) |
| lcfrlem17.v | β’ π = (Baseβπ) |
| lcfrlem17.p | β’ + = (+gβπ) |
| lcfrlem17.z | β’ 0 = (0gβπ) |
| lcfrlem17.n | β’ π = (LSpanβπ) |
| lcfrlem17.a | β’ π΄ = (LSAtomsβπ) |
| lcfrlem17.k | β’ (π β (πΎ β HL β§ π β π»)) |
| lcfrlem17.x | β’ (π β π β (π β { 0 })) |
| lcfrlem17.y | β’ (π β π β (π β { 0 })) |
| lcfrlem17.ne | β’ (π β (πβ{π}) β (πβ{π})) |
| Ref | Expression |
|---|---|
| lcfrlem21 | β’ (π β ((πβ{π, π}) β© ( β₯ β{(π + π)})) β π΄) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | lcfrlem17.h | . . 3 β’ π» = (LHypβπΎ) | |
| 2 | lcfrlem17.o | . . 3 β’ β₯ = ((ocHβπΎ)βπ) | |
| 3 | lcfrlem17.u | . . 3 β’ π = ((DVecHβπΎ)βπ) | |
| 4 | lcfrlem17.v | . . 3 β’ π = (Baseβπ) | |
| 5 | lcfrlem17.p | . . 3 β’ + = (+gβπ) | |
| 6 | lcfrlem17.z | . . 3 β’ 0 = (0gβπ) | |
| 7 | lcfrlem17.n | . . 3 β’ π = (LSpanβπ) | |
| 8 | lcfrlem17.a | . . 3 β’ π΄ = (LSAtomsβπ) | |
| 9 | lcfrlem17.k | . . . 4 β’ (π β (πΎ β HL β§ π β π»)) | |
| 10 | 9 | adantr 480 | . . 3 β’ ((π β§ Β¬ π β ( β₯ β{(π + π)})) β (πΎ β HL β§ π β π»)) |
| 11 | lcfrlem17.x | . . . 4 β’ (π β π β (π β { 0 })) | |
| 12 | 11 | adantr 480 | . . 3 β’ ((π β§ Β¬ π β ( β₯ β{(π + π)})) β π β (π β { 0 })) |
| 13 | lcfrlem17.y | . . . 4 β’ (π β π β (π β { 0 })) | |
| 14 | 13 | adantr 480 | . . 3 β’ ((π β§ Β¬ π β ( β₯ β{(π + π)})) β π β (π β { 0 })) |
| 15 | lcfrlem17.ne | . . . 4 β’ (π β (πβ{π}) β (πβ{π})) | |
| 16 | 15 | adantr 480 | . . 3 β’ ((π β§ Β¬ π β ( β₯ β{(π + π)})) β (πβ{π}) β (πβ{π})) |
| 17 | simpr 484 | . . 3 β’ ((π β§ Β¬ π β ( β₯ β{(π + π)})) β Β¬ π β ( β₯ β{(π + π)})) | |
| 18 | 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 16, 17 | lcfrlem20 40737 | . 2 β’ ((π β§ Β¬ π β ( β₯ β{(π + π)})) β ((πβ{π, π}) β© ( β₯ β{(π + π)})) β π΄) |
| 19 | 1, 3, 9 | dvhlmod 40285 | . . . . . . . . 9 β’ (π β π β LMod) |
| 20 | 11 | eldifad 3961 | . . . . . . . . 9 β’ (π β π β π) |
| 21 | 13 | eldifad 3961 | . . . . . . . . 9 β’ (π β π β π) |
| 22 | 4, 5 | lmodcom 20663 | . . . . . . . . 9 β’ ((π β LMod β§ π β π β§ π β π) β (π + π) = (π + π)) |
| 23 | 19, 20, 21, 22 | syl3anc 1370 | . . . . . . . 8 β’ (π β (π + π) = (π + π)) |
| 24 | 23 | sneqd 4641 | . . . . . . 7 β’ (π β {(π + π)} = {(π + π)}) |
| 25 | 24 | fveq2d 6896 | . . . . . 6 β’ (π β ( β₯ β{(π + π)}) = ( β₯ β{(π + π)})) |
| 26 | 25 | eleq2d 2818 | . . . . 5 β’ (π β (π β ( β₯ β{(π + π)}) β π β ( β₯ β{(π + π)}))) |
| 27 | 26 | biimprd 247 | . . . 4 β’ (π β (π β ( β₯ β{(π + π)}) β π β ( β₯ β{(π + π)}))) |
| 28 | 27 | con3dimp 408 | . . 3 β’ ((π β§ Β¬ π β ( β₯ β{(π + π)})) β Β¬ π β ( β₯ β{(π + π)})) |
| 29 | prcom 4737 | . . . . . . . 8 β’ {π, π} = {π, π} | |
| 30 | 29 | fveq2i 6895 | . . . . . . 7 β’ (πβ{π, π}) = (πβ{π, π}) |
| 31 | 30 | a1i 11 | . . . . . 6 β’ (π β (πβ{π, π}) = (πβ{π, π})) |
| 32 | 31, 25 | ineq12d 4214 | . . . . 5 β’ (π β ((πβ{π, π}) β© ( β₯ β{(π + π)})) = ((πβ{π, π}) β© ( β₯ β{(π + π)}))) |
| 33 | 32 | adantr 480 | . . . 4 β’ ((π β§ Β¬ π β ( β₯ β{(π + π)})) β ((πβ{π, π}) β© ( β₯ β{(π + π)})) = ((πβ{π, π}) β© ( β₯ β{(π + π)}))) |
| 34 | 9 | adantr 480 | . . . . 5 β’ ((π β§ Β¬ π β ( β₯ β{(π + π)})) β (πΎ β HL β§ π β π»)) |
| 35 | 13 | adantr 480 | . . . . 5 β’ ((π β§ Β¬ π β ( β₯ β{(π + π)})) β π β (π β { 0 })) |
| 36 | 11 | adantr 480 | . . . . 5 β’ ((π β§ Β¬ π β ( β₯ β{(π + π)})) β π β (π β { 0 })) |
| 37 | 15 | necomd 2995 | . . . . . 6 β’ (π β (πβ{π}) β (πβ{π})) |
| 38 | 37 | adantr 480 | . . . . 5 β’ ((π β§ Β¬ π β ( β₯ β{(π + π)})) β (πβ{π}) β (πβ{π})) |
| 39 | simpr 484 | . . . . 5 β’ ((π β§ Β¬ π β ( β₯ β{(π + π)})) β Β¬ π β ( β₯ β{(π + π)})) | |
| 40 | 1, 2, 3, 4, 5, 6, 7, 8, 34, 35, 36, 38, 39 | lcfrlem20 40737 | . . . 4 β’ ((π β§ Β¬ π β ( β₯ β{(π + π)})) β ((πβ{π, π}) β© ( β₯ β{(π + π)})) β π΄) |
| 41 | 33, 40 | eqeltrd 2832 | . . 3 β’ ((π β§ Β¬ π β ( β₯ β{(π + π)})) β ((πβ{π, π}) β© ( β₯ β{(π + π)})) β π΄) |
| 42 | 28, 41 | syldan 590 | . 2 β’ ((π β§ Β¬ π β ( β₯ β{(π + π)})) β ((πβ{π, π}) β© ( β₯ β{(π + π)})) β π΄) |
| 43 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 15 | lcfrlem19 40736 | . 2 β’ (π β (Β¬ π β ( β₯ β{(π + π)}) β¨ Β¬ π β ( β₯ β{(π + π)}))) |
| 44 | 18, 42, 43 | mpjaodan 956 | 1 β’ (π β ((πβ{π, π}) β© ( β₯ β{(π + π)})) β π΄) |
| Colors of variables: wff setvar class |
| Syntax hints: Β¬ wn 3 β wi 4 β§ wa 395 = wceq 1540 β wcel 2105 β wne 2939 β cdif 3946 β© cin 3948 {csn 4629 {cpr 4631 βcfv 6544 (class class class)co 7412 Basecbs 17149 +gcplusg 17202 0gc0g 17390 LModclmod 20615 LSpanclspn 20727 LSAtomsclsa 38148 HLchlt 38524 LHypclh 39159 DVecHcdvh 40253 ocHcoch 40522 |
| 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-int 4952 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-mre 17535 df-mrc 17536 df-acs 17538 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-submnd 18707 df-grp 18859 df-minusg 18860 df-sbg 18861 df-subg 19040 df-cntz 19223 df-oppg 19252 df-lsm 19546 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-lmod 20617 df-lss 20688 df-lsp 20728 df-lvec 20859 df-lsatoms 38150 df-lshyp 38151 df-lcv 38193 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-tgrp 39918 df-tendo 39930 df-edring 39932 df-dveca 40178 df-disoa 40204 df-dvech 40254 df-dib 40314 df-dic 40348 df-dih 40404 df-doch 40523 df-djh 40570 |
| This theorem is referenced by: lcfrlem22 40739 lcfrlem40 40757 |
| Copyright terms: Public domain | W3C validator |