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Mirrors > Home > MPE Home > Th. List > Mathboxes > evlsvvvallem | Structured version Visualization version GIF version |
Description: Lemma for evlsvvval 41857 akin to psrbagev2 22025. (Contributed by SN, 6-Mar-2025.) |
Ref | Expression |
---|---|
evlsvvvallem.d | β’ π· = {β β (β0 βm πΌ) β£ (β‘β β β) β Fin} |
evlsvvvallem.k | β’ πΎ = (Baseβπ) |
evlsvvvallem.m | β’ π = (mulGrpβπ) |
evlsvvvallem.w | β’ β = (.gβπ) |
evlsvvvallem.i | β’ (π β πΌ β π) |
evlsvvvallem.s | β’ (π β π β CRing) |
evlsvvvallem.a | β’ (π β π΄ β (πΎ βm πΌ)) |
evlsvvvallem.b | β’ (π β π΅ β π·) |
Ref | Expression |
---|---|
evlsvvvallem | β’ (π β (π Ξ£g (π£ β πΌ β¦ ((π΅βπ£) β (π΄βπ£)))) β πΎ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | evlsvvvallem.m | . . 3 β’ π = (mulGrpβπ) | |
2 | evlsvvvallem.k | . . 3 β’ πΎ = (Baseβπ) | |
3 | 1, 2 | mgpbas 20079 | . 2 β’ πΎ = (Baseβπ) |
4 | eqid 2725 | . . 3 β’ (1rβπ) = (1rβπ) | |
5 | 1, 4 | ringidval 20122 | . 2 β’ (1rβπ) = (0gβπ) |
6 | evlsvvvallem.s | . . 3 β’ (π β π β CRing) | |
7 | 1 | crngmgp 20180 | . . 3 β’ (π β CRing β π β CMnd) |
8 | 6, 7 | syl 17 | . 2 β’ (π β π β CMnd) |
9 | evlsvvvallem.i | . 2 β’ (π β πΌ β π) | |
10 | evlsvvvallem.w | . . . 4 β’ β = (.gβπ) | |
11 | 6 | crngringd 20185 | . . . . . 6 β’ (π β π β Ring) |
12 | 1 | ringmgp 20178 | . . . . . 6 β’ (π β Ring β π β Mnd) |
13 | 11, 12 | syl 17 | . . . . 5 β’ (π β π β Mnd) |
14 | 13 | adantr 479 | . . . 4 β’ ((π β§ π£ β πΌ) β π β Mnd) |
15 | evlsvvvallem.b | . . . . . 6 β’ (π β π΅ β π·) | |
16 | evlsvvvallem.d | . . . . . . 7 β’ π· = {β β (β0 βm πΌ) β£ (β‘β β β) β Fin} | |
17 | 16 | psrbagf 21850 | . . . . . 6 β’ (π΅ β π· β π΅:πΌβΆβ0) |
18 | 15, 17 | syl 17 | . . . . 5 β’ (π β π΅:πΌβΆβ0) |
19 | 18 | ffvelcdmda 7087 | . . . 4 β’ ((π β§ π£ β πΌ) β (π΅βπ£) β β0) |
20 | evlsvvvallem.a | . . . . . 6 β’ (π β π΄ β (πΎ βm πΌ)) | |
21 | elmapi 8861 | . . . . . 6 β’ (π΄ β (πΎ βm πΌ) β π΄:πΌβΆπΎ) | |
22 | 20, 21 | syl 17 | . . . . 5 β’ (π β π΄:πΌβΆπΎ) |
23 | 22 | ffvelcdmda 7087 | . . . 4 β’ ((π β§ π£ β πΌ) β (π΄βπ£) β πΎ) |
24 | 3, 10, 14, 19, 23 | mulgnn0cld 19049 | . . 3 β’ ((π β§ π£ β πΌ) β ((π΅βπ£) β (π΄βπ£)) β πΎ) |
25 | 24 | fmpttd 7118 | . 2 β’ (π β (π£ β πΌ β¦ ((π΅βπ£) β (π΄βπ£))):πΌβΆπΎ) |
26 | 9 | mptexd 7230 | . . 3 β’ (π β (π£ β πΌ β¦ ((π΅βπ£) β (π΄βπ£))) β V) |
27 | fvexd 6905 | . . 3 β’ (π β (1rβπ) β V) | |
28 | 25 | ffund 6721 | . . 3 β’ (π β Fun (π£ β πΌ β¦ ((π΅βπ£) β (π΄βπ£)))) |
29 | 16 | psrbagfsupp 21852 | . . . 4 β’ (π΅ β π· β π΅ finSupp 0) |
30 | 15, 29 | syl 17 | . . 3 β’ (π β π΅ finSupp 0) |
31 | ssidd 3997 | . . . . . . 7 β’ (π β (π΅ supp 0) β (π΅ supp 0)) | |
32 | 0zd 12595 | . . . . . . 7 β’ (π β 0 β β€) | |
33 | 18, 31, 9, 32 | suppssr 8194 | . . . . . 6 β’ ((π β§ π£ β (πΌ β (π΅ supp 0))) β (π΅βπ£) = 0) |
34 | 33 | oveq1d 7428 | . . . . 5 β’ ((π β§ π£ β (πΌ β (π΅ supp 0))) β ((π΅βπ£) β (π΄βπ£)) = (0 β (π΄βπ£))) |
35 | eldifi 4120 | . . . . . . 7 β’ (π£ β (πΌ β (π΅ supp 0)) β π£ β πΌ) | |
36 | 35, 23 | sylan2 591 | . . . . . 6 β’ ((π β§ π£ β (πΌ β (π΅ supp 0))) β (π΄βπ£) β πΎ) |
37 | 3, 5, 10 | mulg0 19029 | . . . . . 6 β’ ((π΄βπ£) β πΎ β (0 β (π΄βπ£)) = (1rβπ)) |
38 | 36, 37 | syl 17 | . . . . 5 β’ ((π β§ π£ β (πΌ β (π΅ supp 0))) β (0 β (π΄βπ£)) = (1rβπ)) |
39 | 34, 38 | eqtrd 2765 | . . . 4 β’ ((π β§ π£ β (πΌ β (π΅ supp 0))) β ((π΅βπ£) β (π΄βπ£)) = (1rβπ)) |
40 | 39, 9 | suppss2 8199 | . . 3 β’ (π β ((π£ β πΌ β¦ ((π΅βπ£) β (π΄βπ£))) supp (1rβπ)) β (π΅ supp 0)) |
41 | 26, 27, 28, 30, 40 | fsuppsssuppgd 9400 | . 2 β’ (π β (π£ β πΌ β¦ ((π΅βπ£) β (π΄βπ£))) finSupp (1rβπ)) |
42 | 3, 5, 8, 9, 25, 41 | gsumcl 19869 | 1 β’ (π β (π Ξ£g (π£ β πΌ β¦ ((π΅βπ£) β (π΄βπ£)))) β πΎ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 {crab 3419 Vcvv 3463 β cdif 3938 class class class wbr 5144 β¦ cmpt 5227 β‘ccnv 5672 β cima 5676 βΆwf 6539 βcfv 6543 (class class class)co 7413 supp csupp 8158 βm cmap 8838 Fincfn 8957 finSupp cfsupp 9380 0cc0 11133 βcn 12237 β0cn0 12497 β€cz 12583 Basecbs 17174 Ξ£g cgsu 17416 Mndcmnd 18688 .gcmg 19022 CMndccmn 19734 mulGrpcmgp 20073 1rcur 20120 Ringcrg 20172 CRingccrg 20173 |
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 2696 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 |
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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-int 4946 df-iun 4994 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-se 5629 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 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-isom 6552 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-1st 7987 df-2nd 7988 df-supp 8159 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-map 8840 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fsupp 9381 df-oi 9528 df-card 9957 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-nn 12238 df-2 12300 df-n0 12498 df-z 12584 df-uz 12848 df-fz 13512 df-fzo 13655 df-seq 13994 df-hash 14317 df-sets 17127 df-slot 17145 df-ndx 17157 df-base 17175 df-plusg 17240 df-0g 17417 df-gsum 17418 df-mgm 18594 df-sgrp 18673 df-mnd 18689 df-mulg 19023 df-cntz 19267 df-cmn 19736 df-mgp 20074 df-ur 20121 df-ring 20174 df-cring 20175 |
This theorem is referenced by: evlsvvvallem2 41856 evlsvvval 41857 evlsbagval 41860 evlselv 41881 evlsmhpvvval 41889 mhphf 41891 |
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