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Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > islssfg | Structured version Visualization version GIF version |
Description: Property of a finitely generated left (sub)module. (Contributed by Stefan O'Rear, 1-Jan-2015.) |
Ref | Expression |
---|---|
islssfg.x | β’ π = (π βΎs π) |
islssfg.s | β’ π = (LSubSpβπ) |
islssfg.n | β’ π = (LSpanβπ) |
Ref | Expression |
---|---|
islssfg | β’ ((π β LMod β§ π β π) β (π β LFinGen β βπ β π« π(π β Fin β§ (πβπ) = π))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2726 | . . . . . . 7 β’ (Baseβπ) = (Baseβπ) | |
2 | islssfg.s | . . . . . . 7 β’ π = (LSubSpβπ) | |
3 | 1, 2 | lssss 20783 | . . . . . 6 β’ (π β π β π β (Baseβπ)) |
4 | islssfg.x | . . . . . . 7 β’ π = (π βΎs π) | |
5 | 4, 1 | ressbas2 17191 | . . . . . 6 β’ (π β (Baseβπ) β π = (Baseβπ)) |
6 | 3, 5 | syl 17 | . . . . 5 β’ (π β π β π = (Baseβπ)) |
7 | 6 | pweqd 4614 | . . . 4 β’ (π β π β π« π = π« (Baseβπ)) |
8 | 7 | rexeqdv 3320 | . . 3 β’ (π β π β (βπ β π« π(π β Fin β§ ((LSpanβπ)βπ) = (Baseβπ)) β βπ β π« (Baseβπ)(π β Fin β§ ((LSpanβπ)βπ) = (Baseβπ)))) |
9 | 8 | adantl 481 | . 2 β’ ((π β LMod β§ π β π) β (βπ β π« π(π β Fin β§ ((LSpanβπ)βπ) = (Baseβπ)) β βπ β π« (Baseβπ)(π β Fin β§ ((LSpanβπ)βπ) = (Baseβπ)))) |
10 | elpwi 4604 | . . . . . . 7 β’ (π β π« π β π β π) | |
11 | islssfg.n | . . . . . . . . 9 β’ π = (LSpanβπ) | |
12 | eqid 2726 | . . . . . . . . 9 β’ (LSpanβπ) = (LSpanβπ) | |
13 | 4, 11, 12, 2 | lsslsp 20862 | . . . . . . . 8 β’ ((π β LMod β§ π β π β§ π β π) β ((LSpanβπ)βπ) = (πβπ)) |
14 | 13 | 3expa 1115 | . . . . . . 7 β’ (((π β LMod β§ π β π) β§ π β π) β ((LSpanβπ)βπ) = (πβπ)) |
15 | 10, 14 | sylan2 592 | . . . . . 6 β’ (((π β LMod β§ π β π) β§ π β π« π) β ((LSpanβπ)βπ) = (πβπ)) |
16 | 15 | eqcomd 2732 | . . . . 5 β’ (((π β LMod β§ π β π) β§ π β π« π) β (πβπ) = ((LSpanβπ)βπ)) |
17 | 6 | ad2antlr 724 | . . . . 5 β’ (((π β LMod β§ π β π) β§ π β π« π) β π = (Baseβπ)) |
18 | 16, 17 | eqeq12d 2742 | . . . 4 β’ (((π β LMod β§ π β π) β§ π β π« π) β ((πβπ) = π β ((LSpanβπ)βπ) = (Baseβπ))) |
19 | 18 | anbi2d 628 | . . 3 β’ (((π β LMod β§ π β π) β§ π β π« π) β ((π β Fin β§ (πβπ) = π) β (π β Fin β§ ((LSpanβπ)βπ) = (Baseβπ)))) |
20 | 19 | rexbidva 3170 | . 2 β’ ((π β LMod β§ π β π) β (βπ β π« π(π β Fin β§ (πβπ) = π) β βπ β π« π(π β Fin β§ ((LSpanβπ)βπ) = (Baseβπ)))) |
21 | 4, 2 | lsslmod 20807 | . . 3 β’ ((π β LMod β§ π β π) β π β LMod) |
22 | eqid 2726 | . . . 4 β’ (Baseβπ) = (Baseβπ) | |
23 | 22, 12 | islmodfg 42386 | . . 3 β’ (π β LMod β (π β LFinGen β βπ β π« (Baseβπ)(π β Fin β§ ((LSpanβπ)βπ) = (Baseβπ)))) |
24 | 21, 23 | syl 17 | . 2 β’ ((π β LMod β§ π β π) β (π β LFinGen β βπ β π« (Baseβπ)(π β Fin β§ ((LSpanβπ)βπ) = (Baseβπ)))) |
25 | 9, 20, 24 | 3bitr4rd 312 | 1 β’ ((π β LMod β§ π β π) β (π β LFinGen β βπ β π« π(π β Fin β§ (πβπ) = π))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 = wceq 1533 β wcel 2098 βwrex 3064 β wss 3943 π« cpw 4597 βcfv 6537 (class class class)co 7405 Fincfn 8941 Basecbs 17153 βΎs cress 17182 LModclmod 20706 LSubSpclss 20778 LSpanclspn 20818 LFinGenclfig 42384 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-sca 17222 df-vsca 17223 df-0g 17396 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-grp 18866 df-minusg 18867 df-sbg 18868 df-subg 19050 df-mgp 20040 df-ur 20087 df-ring 20140 df-lmod 20708 df-lss 20779 df-lsp 20819 df-lfig 42385 |
This theorem is referenced by: islssfg2 42388 lmhmfgsplit 42403 |
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