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| Mirrors > Home > MPE Home > Th. List > Mathboxes > lcfrlem41 | Structured version Visualization version GIF version | ||
| Description: Lemma for lcfr 40760. Eliminate span condition. (Contributed by NM, 11-Mar-2015.) |
| Ref | Expression |
|---|---|
| lcfrlem38.h | β’ π» = (LHypβπΎ) |
| lcfrlem38.o | β’ β₯ = ((ocHβπΎ)βπ) |
| lcfrlem38.u | β’ π = ((DVecHβπΎ)βπ) |
| lcfrlem38.p | β’ + = (+gβπ) |
| lcfrlem38.f | β’ πΉ = (LFnlβπ) |
| lcfrlem38.l | β’ πΏ = (LKerβπ) |
| lcfrlem38.d | β’ π· = (LDualβπ) |
| lcfrlem38.q | β’ π = (LSubSpβπ·) |
| lcfrlem38.c | β’ πΆ = {π β (LFnlβπ) β£ ( β₯ β( β₯ β(πΏβπ))) = (πΏβπ)} |
| lcfrlem38.e | β’ πΈ = βͺ π β πΊ ( β₯ β(πΏβπ)) |
| lcfrlem38.k | β’ (π β (πΎ β HL β§ π β π»)) |
| lcfrlem38.g | β’ (π β πΊ β π) |
| lcfrlem38.gs | β’ (π β πΊ β πΆ) |
| lcfrlem38.xe | β’ (π β π β πΈ) |
| lcfrlem38.ye | β’ (π β π β πΈ) |
| lcfrlem38.z | β’ 0 = (0gβπ) |
| lcfrlem38.x | β’ (π β π β 0 ) |
| lcfrlem38.y | β’ (π β π β 0 ) |
| Ref | Expression |
|---|---|
| lcfrlem41 | β’ (π β (π + π) β πΈ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | lcfrlem38.h | . . 3 β’ π» = (LHypβπΎ) | |
| 2 | lcfrlem38.o | . . 3 β’ β₯ = ((ocHβπΎ)βπ) | |
| 3 | lcfrlem38.u | . . 3 β’ π = ((DVecHβπΎ)βπ) | |
| 4 | lcfrlem38.p | . . 3 β’ + = (+gβπ) | |
| 5 | eqid 2731 | . . 3 β’ (LSpanβπ) = (LSpanβπ) | |
| 6 | lcfrlem38.l | . . 3 β’ πΏ = (LKerβπ) | |
| 7 | lcfrlem38.d | . . 3 β’ π· = (LDualβπ) | |
| 8 | lcfrlem38.q | . . 3 β’ π = (LSubSpβπ·) | |
| 9 | lcfrlem38.k | . . . 4 β’ (π β (πΎ β HL β§ π β π»)) | |
| 10 | 9 | adantr 480 | . . 3 β’ ((π β§ ((LSpanβπ)β{π}) = ((LSpanβπ)β{π})) β (πΎ β HL β§ π β π»)) |
| 11 | lcfrlem38.g | . . . 4 β’ (π β πΊ β π) | |
| 12 | 11 | adantr 480 | . . 3 β’ ((π β§ ((LSpanβπ)β{π}) = ((LSpanβπ)β{π})) β πΊ β π) |
| 13 | lcfrlem38.e | . . 3 β’ πΈ = βͺ π β πΊ ( β₯ β(πΏβπ)) | |
| 14 | lcfrlem38.xe | . . . 4 β’ (π β π β πΈ) | |
| 15 | 14 | adantr 480 | . . 3 β’ ((π β§ ((LSpanβπ)β{π}) = ((LSpanβπ)β{π})) β π β πΈ) |
| 16 | lcfrlem38.ye | . . . 4 β’ (π β π β πΈ) | |
| 17 | 16 | adantr 480 | . . 3 β’ ((π β§ ((LSpanβπ)β{π}) = ((LSpanβπ)β{π})) β π β πΈ) |
| 18 | simpr 484 | . . 3 β’ ((π β§ ((LSpanβπ)β{π}) = ((LSpanβπ)β{π})) β ((LSpanβπ)β{π}) = ((LSpanβπ)β{π})) | |
| 19 | 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 13, 15, 17, 18 | lcfrlem6 40722 | . 2 β’ ((π β§ ((LSpanβπ)β{π}) = ((LSpanβπ)β{π})) β (π + π) β πΈ) |
| 20 | lcfrlem38.f | . . 3 β’ πΉ = (LFnlβπ) | |
| 21 | lcfrlem38.c | . . 3 β’ πΆ = {π β (LFnlβπ) β£ ( β₯ β( β₯ β(πΏβπ))) = (πΏβπ)} | |
| 22 | 9 | adantr 480 | . . 3 β’ ((π β§ ((LSpanβπ)β{π}) β ((LSpanβπ)β{π})) β (πΎ β HL β§ π β π»)) |
| 23 | 11 | adantr 480 | . . 3 β’ ((π β§ ((LSpanβπ)β{π}) β ((LSpanβπ)β{π})) β πΊ β π) |
| 24 | lcfrlem38.gs | . . . 4 β’ (π β πΊ β πΆ) | |
| 25 | 24 | adantr 480 | . . 3 β’ ((π β§ ((LSpanβπ)β{π}) β ((LSpanβπ)β{π})) β πΊ β πΆ) |
| 26 | 14 | adantr 480 | . . 3 β’ ((π β§ ((LSpanβπ)β{π}) β ((LSpanβπ)β{π})) β π β πΈ) |
| 27 | 16 | adantr 480 | . . 3 β’ ((π β§ ((LSpanβπ)β{π}) β ((LSpanβπ)β{π})) β π β πΈ) |
| 28 | lcfrlem38.z | . . 3 β’ 0 = (0gβπ) | |
| 29 | lcfrlem38.x | . . . 4 β’ (π β π β 0 ) | |
| 30 | 29 | adantr 480 | . . 3 β’ ((π β§ ((LSpanβπ)β{π}) β ((LSpanβπ)β{π})) β π β 0 ) |
| 31 | lcfrlem38.y | . . . 4 β’ (π β π β 0 ) | |
| 32 | 31 | adantr 480 | . . 3 β’ ((π β§ ((LSpanβπ)β{π}) β ((LSpanβπ)β{π})) β π β 0 ) |
| 33 | simpr 484 | . . 3 β’ ((π β§ ((LSpanβπ)β{π}) β ((LSpanβπ)β{π})) β ((LSpanβπ)β{π}) β ((LSpanβπ)β{π})) | |
| 34 | 1, 2, 3, 4, 20, 6, 7, 8, 21, 13, 22, 23, 25, 26, 27, 28, 30, 32, 5, 33 | lcfrlem40 40757 | . 2 β’ ((π β§ ((LSpanβπ)β{π}) β ((LSpanβπ)β{π})) β (π + π) β πΈ) |
| 35 | 19, 34 | pm2.61dane 3028 | 1 β’ (π β (π + π) β πΈ) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 395 = wceq 1540 β wcel 2105 β wne 2939 {crab 3431 β wss 3948 {csn 4628 βͺ ciun 4997 βcfv 6543 (class class class)co 7412 +gcplusg 17202 0gc0g 17390 LSubSpclss 20687 LSpanclspn 20727 LFnlclfn 38231 LKerclk 38259 LDualcld 38297 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 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 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 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 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 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7674 df-om 7860 df-1st 7979 df-2nd 7980 df-tpos 8215 df-undef 8262 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-er 8707 df-map 8826 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 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-lfl 38232 df-lkr 38260 df-ldual 38298 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: lcfrlem42 40759 |
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