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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 21743. (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 6882 | . . . . 5 β’ ( I βΎ π):πβ1-1-ontoβπ | |
8 | f1of 6844 | . . . . 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 6745 | . . 3 β’ (π β ( I βΎ π):πβΆπ΅) |
12 | ellspds.m | . . 3 β’ (π β π β LMod) | |
13 | 2 | fvexi 6916 | . . . . 5 β’ π΅ β V |
14 | 13 | a1i 11 | . . . 4 β’ (π β π΅ β V) |
15 | 14, 10 | ssexd 5328 | . . 3 β’ (π β π β V) |
16 | 1, 2, 3, 4, 5, 6, 11, 12, 15 | ellspd 21743 | . 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 4004 | . . . . 5 β’ π β π | |
18 | resiima 6084 | . . . . 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 6906 | . . 3 β’ (π β (πβ(( I βΎ π) β π)) = (πβπ)) |
21 | 20 | eleq2d 2815 | . 2 β’ (π β (π β (πβ(( I βΎ π) β π)) β π β (πβπ))) |
22 | elmapfn 8890 | . . . . . . . 8 β’ (π β (πΎ βm π) β π Fn π) | |
23 | 22 | adantl 480 | . . . . . . 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 6728 | . . . . . . 7 β’ ((π β§ π β (πΎ βm π)) β ( I βΎ π) Fn π) |
26 | 15 | adantr 479 | . . . . . . 7 β’ ((π β§ π β (πΎ βm π)) β π β V) |
27 | inidm 4221 | . . . . . . 7 β’ (π β© π) = π | |
28 | eqidd 2729 | . . . . . . 7 β’ (((π β§ π β (πΎ βm π)) β§ π£ β π) β (πβπ£) = (πβπ£)) | |
29 | fvresi 7188 | . . . . . . . 8 β’ (π£ β π β (( I βΎ π)βπ£) = π£) | |
30 | 29 | adantl 480 | . . . . . . 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 7700 | . . . . . 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 7442 | . . . . 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 2739 | . . . 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 3174 | . 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 308 | 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 394 = wceq 1533 β wcel 2098 βwrex 3067 Vcvv 3473 β wss 3949 class class class wbr 5152 β¦ cmpt 5235 I cid 5579 βΎ cres 5684 β cima 5685 Fn wfn 6548 βΆwf 6549 β1-1-ontoβwf1o 6552 βcfv 6553 (class class class)co 7426 βf cof 7689 βm cmap 8851 finSupp cfsupp 9393 Basecbs 17187 Scalarcsca 17243 Β·π cvsca 17244 0gc0g 17428 Ξ£g cgsu 17429 LModclmod 20750 LSpanclspn 20862 |
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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 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 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7691 df-om 7877 df-1st 7999 df-2nd 8000 df-supp 8172 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-map 8853 df-ixp 8923 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-fsupp 9394 df-sup 9473 df-oi 9541 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-fz 13525 df-fzo 13668 df-seq 14007 df-hash 14330 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-mulr 17254 df-sca 17256 df-vsca 17257 df-ip 17258 df-tset 17259 df-ple 17260 df-ds 17262 df-hom 17264 df-cco 17265 df-0g 17430 df-gsum 17431 df-prds 17436 df-pws 17438 df-mre 17573 df-mrc 17574 df-acs 17576 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-mhm 18747 df-submnd 18748 df-grp 18900 df-minusg 18901 df-sbg 18902 df-mulg 19031 df-subg 19085 df-ghm 19175 df-cntz 19275 df-cmn 19744 df-abl 19745 df-mgp 20082 df-rng 20100 df-ur 20129 df-ring 20182 df-nzr 20459 df-subrg 20515 df-lmod 20752 df-lss 20823 df-lsp 20863 df-lmhm 20914 df-lbs 20967 df-sra 21065 df-rgmod 21066 df-dsmm 21673 df-frlm 21688 df-uvc 21724 |
This theorem is referenced by: elrsp 33109 lbslsp 33117 lbsdiflsp0 33357 fedgmul 33362 |
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