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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ellspds | Structured version Visualization version GIF version |
Description: Variation on ellspd 21692. (Contributed by Thierry Arnoux, 18-May-2023.) |
Ref | Expression |
---|---|
ellspds.n | β’ π = (LSpanβπ) |
ellspds.v | β’ π΅ = (Baseβπ) |
ellspds.k | β’ πΎ = (Baseβπ) |
ellspds.s | β’ π = (Scalarβπ) |
ellspds.z | β’ 0 = (0gβπ) |
ellspds.t | β’ Β· = ( Β·π βπ) |
ellspds.m | β’ (π β π β LMod) |
ellspds.1 | β’ (π β π β π΅) |
Ref | Expression |
---|---|
ellspds | β’ (π β (π β (πβπ) β βπ β (πΎ βm π)(π finSupp 0 β§ π = (π Ξ£g (π£ β π β¦ ((πβπ£) Β· π£)))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ellspds.n | . . 3 β’ π = (LSpanβπ) | |
2 | ellspds.v | . . 3 β’ π΅ = (Baseβπ) | |
3 | ellspds.k | . . 3 β’ πΎ = (Baseβπ) | |
4 | ellspds.s | . . 3 β’ π = (Scalarβπ) | |
5 | ellspds.z | . . 3 β’ 0 = (0gβπ) | |
6 | ellspds.t | . . 3 β’ Β· = ( Β·π βπ) | |
7 | f1oi 6864 | . . . . 5 β’ ( I βΎ π):πβ1-1-ontoβπ | |
8 | f1of 6826 | . . . . 5 β’ (( I βΎ π):πβ1-1-ontoβπ β ( I βΎ π):πβΆπ) | |
9 | 7, 8 | mp1i 13 | . . . 4 β’ (π β ( I βΎ π):πβΆπ) |
10 | ellspds.1 | . . . 4 β’ (π β π β π΅) | |
11 | 9, 10 | fssd 6728 | . . 3 β’ (π β ( I βΎ π):πβΆπ΅) |
12 | ellspds.m | . . 3 β’ (π β π β LMod) | |
13 | 2 | fvexi 6898 | . . . . 5 β’ π΅ β V |
14 | 13 | a1i 11 | . . . 4 β’ (π β π΅ β V) |
15 | 14, 10 | ssexd 5317 | . . 3 β’ (π β π β V) |
16 | 1, 2, 3, 4, 5, 6, 11, 12, 15 | ellspd 21692 | . 2 β’ (π β (π β (πβ(( I βΎ π) β π)) β βπ β (πΎ βm π)(π finSupp 0 β§ π = (π Ξ£g (π βf Β· ( I βΎ π)))))) |
17 | ssid 3999 | . . . . 5 β’ π β π | |
18 | resiima 6068 | . . . . 5 β’ (π β π β (( I βΎ π) β π) = π) | |
19 | 17, 18 | mp1i 13 | . . . 4 β’ (π β (( I βΎ π) β π) = π) |
20 | 19 | fveq2d 6888 | . . 3 β’ (π β (πβ(( I βΎ π) β π)) = (πβπ)) |
21 | 20 | eleq2d 2813 | . 2 β’ (π β (π β (πβ(( I βΎ π) β π)) β π β (πβπ))) |
22 | elmapfn 8858 | . . . . . . . 8 β’ (π β (πΎ βm π) β π Fn π) | |
23 | 22 | adantl 481 | . . . . . . 7 β’ ((π β§ π β (πΎ βm π)) β π Fn π) |
24 | 7, 8 | mp1i 13 | . . . . . . . 8 β’ ((π β§ π β (πΎ βm π)) β ( I βΎ π):πβΆπ) |
25 | 24 | ffnd 6711 | . . . . . . 7 β’ ((π β§ π β (πΎ βm π)) β ( I βΎ π) Fn π) |
26 | 15 | adantr 480 | . . . . . . 7 β’ ((π β§ π β (πΎ βm π)) β π β V) |
27 | inidm 4213 | . . . . . . 7 β’ (π β© π) = π | |
28 | eqidd 2727 | . . . . . . 7 β’ (((π β§ π β (πΎ βm π)) β§ π£ β π) β (πβπ£) = (πβπ£)) | |
29 | fvresi 7166 | . . . . . . . 8 β’ (π£ β π β (( I βΎ π)βπ£) = π£) | |
30 | 29 | adantl 481 | . . . . . . 7 β’ (((π β§ π β (πΎ βm π)) β§ π£ β π) β (( I βΎ π)βπ£) = π£) |
31 | 23, 25, 26, 26, 27, 28, 30 | offval 7675 | . . . . . 6 β’ ((π β§ π β (πΎ βm π)) β (π βf Β· ( I βΎ π)) = (π£ β π β¦ ((πβπ£) Β· π£))) |
32 | 31 | oveq2d 7420 | . . . . 5 β’ ((π β§ π β (πΎ βm π)) β (π Ξ£g (π βf Β· ( I βΎ π))) = (π Ξ£g (π£ β π β¦ ((πβπ£) Β· π£)))) |
33 | 32 | eqeq2d 2737 | . . . 4 β’ ((π β§ π β (πΎ βm π)) β (π = (π Ξ£g (π βf Β· ( I βΎ π))) β π = (π Ξ£g (π£ β π β¦ ((πβπ£) Β· π£))))) |
34 | 33 | anbi2d 628 | . . 3 β’ ((π β§ π β (πΎ βm π)) β ((π finSupp 0 β§ π = (π Ξ£g (π βf Β· ( I βΎ π)))) β (π finSupp 0 β§ π = (π Ξ£g (π£ β π β¦ ((πβπ£) Β· π£)))))) |
35 | 34 | rexbidva 3170 | . 2 β’ (π β (βπ β (πΎ βm π)(π finSupp 0 β§ π = (π Ξ£g (π βf Β· ( I βΎ π)))) β βπ β (πΎ βm π)(π finSupp 0 β§ π = (π Ξ£g (π£ β π β¦ ((πβπ£) Β· π£)))))) |
36 | 16, 21, 35 | 3bitr3d 309 | 1 β’ (π β (π β (πβπ) β βπ β (πΎ βm π)(π finSupp 0 β§ π = (π Ξ£g (π£ β π β¦ ((πβπ£) Β· 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 Vcvv 3468 β wss 3943 class class class wbr 5141 β¦ cmpt 5224 I cid 5566 βΎ cres 5671 β cima 5672 Fn wfn 6531 βΆwf 6532 β1-1-ontoβwf1o 6535 βcfv 6536 (class class class)co 7404 βf cof 7664 βm cmap 8819 finSupp cfsupp 9360 Basecbs 17150 Scalarcsca 17206 Β·π cvsca 17207 0gc0g 17391 Ξ£g cgsu 17392 LModclmod 20703 LSpanclspn 20815 |
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 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
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-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 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-se 5625 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 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8144 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-er 8702 df-map 8821 df-ixp 8891 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fsupp 9361 df-sup 9436 df-oi 9504 df-card 9933 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 df-n0 12474 df-z 12560 df-dec 12679 df-uz 12824 df-fz 13488 df-fzo 13631 df-seq 13970 df-hash 14293 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17151 df-ress 17180 df-plusg 17216 df-mulr 17217 df-sca 17219 df-vsca 17220 df-ip 17221 df-tset 17222 df-ple 17223 df-ds 17225 df-hom 17227 df-cco 17228 df-0g 17393 df-gsum 17394 df-prds 17399 df-pws 17401 df-mre 17536 df-mrc 17537 df-acs 17539 df-mgm 18570 df-sgrp 18649 df-mnd 18665 df-mhm 18710 df-submnd 18711 df-grp 18863 df-minusg 18864 df-sbg 18865 df-mulg 18993 df-subg 19047 df-ghm 19136 df-cntz 19230 df-cmn 19699 df-abl 19700 df-mgp 20037 df-rng 20055 df-ur 20084 df-ring 20137 df-nzr 20412 df-subrg 20468 df-lmod 20705 df-lss 20776 df-lsp 20816 df-lmhm 20867 df-lbs 20920 df-sra 21018 df-rgmod 21019 df-dsmm 21622 df-frlm 21637 df-uvc 21673 |
This theorem is referenced by: elrsp 32991 lbslsp 32998 lbsdiflsp0 33228 fedgmul 33233 |
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