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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > gsumzresunsn | Structured version Visualization version GIF version |
Description: Append an element to a finite group sum expressed as a function restriction. (Contributed by Thierry Arnoux, 20-Nov-2023.) |
Ref | Expression |
---|---|
gsumzresunsn.b | β’ π΅ = (BaseβπΊ) |
gsumzresunsn.p | β’ + = (+gβπΊ) |
gsumzresunsn.z | β’ π = (CntzβπΊ) |
gsumzresunsn.y | β’ π = (πΉβπ) |
gsumzresunsn.f | β’ (π β πΉ:πΆβΆπ΅) |
gsumzresunsn.1 | β’ (π β π΄ β πΆ) |
gsumzresunsn.g | β’ (π β πΊ β Mnd) |
gsumzresunsn.a | β’ (π β π΄ β Fin) |
gsumzresunsn.2 | β’ (π β Β¬ π β π΄) |
gsumzresunsn.3 | β’ (π β π β πΆ) |
gsumzresunsn.4 | β’ (π β π β π΅) |
gsumzresunsn.5 | β’ (π β (πΉ β (π΄ βͺ {π})) β (πβ(πΉ β (π΄ βͺ {π})))) |
Ref | Expression |
---|---|
gsumzresunsn | β’ (π β (πΊ Ξ£g (πΉ βΎ (π΄ βͺ {π}))) = ((πΊ Ξ£g (πΉ βΎ π΄)) + π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gsumzresunsn.b | . . 3 β’ π΅ = (BaseβπΊ) | |
2 | gsumzresunsn.p | . . 3 β’ + = (+gβπΊ) | |
3 | gsumzresunsn.z | . . 3 β’ π = (CntzβπΊ) | |
4 | eqid 2727 | . . 3 β’ (π₯ β (π΄ βͺ {π}) β¦ (πΉβπ₯)) = (π₯ β (π΄ βͺ {π}) β¦ (πΉβπ₯)) | |
5 | gsumzresunsn.g | . . 3 β’ (π β πΊ β Mnd) | |
6 | gsumzresunsn.a | . . 3 β’ (π β π΄ β Fin) | |
7 | gsumzresunsn.5 | . . . 4 β’ (π β (πΉ β (π΄ βͺ {π})) β (πβ(πΉ β (π΄ βͺ {π})))) | |
8 | df-ima 5685 | . . . . 5 β’ (πΉ β (π΄ βͺ {π})) = ran (πΉ βΎ (π΄ βͺ {π})) | |
9 | gsumzresunsn.f | . . . . . . 7 β’ (π β πΉ:πΆβΆπ΅) | |
10 | gsumzresunsn.1 | . . . . . . . 8 β’ (π β π΄ β πΆ) | |
11 | gsumzresunsn.3 | . . . . . . . . 9 β’ (π β π β πΆ) | |
12 | 11 | snssd 4808 | . . . . . . . 8 β’ (π β {π} β πΆ) |
13 | 10, 12 | unssd 4182 | . . . . . . 7 β’ (π β (π΄ βͺ {π}) β πΆ) |
14 | 9, 13 | feqresmpt 6962 | . . . . . 6 β’ (π β (πΉ βΎ (π΄ βͺ {π})) = (π₯ β (π΄ βͺ {π}) β¦ (πΉβπ₯))) |
15 | 14 | rneqd 5934 | . . . . 5 β’ (π β ran (πΉ βΎ (π΄ βͺ {π})) = ran (π₯ β (π΄ βͺ {π}) β¦ (πΉβπ₯))) |
16 | 8, 15 | eqtrid 2779 | . . . 4 β’ (π β (πΉ β (π΄ βͺ {π})) = ran (π₯ β (π΄ βͺ {π}) β¦ (πΉβπ₯))) |
17 | 16 | fveq2d 6895 | . . . 4 β’ (π β (πβ(πΉ β (π΄ βͺ {π}))) = (πβran (π₯ β (π΄ βͺ {π}) β¦ (πΉβπ₯)))) |
18 | 7, 16, 17 | 3sstr3d 4024 | . . 3 β’ (π β ran (π₯ β (π΄ βͺ {π}) β¦ (πΉβπ₯)) β (πβran (π₯ β (π΄ βͺ {π}) β¦ (πΉβπ₯)))) |
19 | 9 | adantr 480 | . . . 4 β’ ((π β§ π₯ β π΄) β πΉ:πΆβΆπ΅) |
20 | 10 | sselda 3978 | . . . 4 β’ ((π β§ π₯ β π΄) β π₯ β πΆ) |
21 | 19, 20 | ffvelcdmd 7089 | . . 3 β’ ((π β§ π₯ β π΄) β (πΉβπ₯) β π΅) |
22 | gsumzresunsn.2 | . . 3 β’ (π β Β¬ π β π΄) | |
23 | gsumzresunsn.4 | . . 3 β’ (π β π β π΅) | |
24 | simpr 484 | . . . . 5 β’ ((π β§ π₯ = π) β π₯ = π) | |
25 | 24 | fveq2d 6895 | . . . 4 β’ ((π β§ π₯ = π) β (πΉβπ₯) = (πΉβπ)) |
26 | gsumzresunsn.y | . . . 4 β’ π = (πΉβπ) | |
27 | 25, 26 | eqtr4di 2785 | . . 3 β’ ((π β§ π₯ = π) β (πΉβπ₯) = π) |
28 | 1, 2, 3, 4, 5, 6, 18, 21, 11, 22, 23, 27 | gsumzunsnd 19895 | . 2 β’ (π β (πΊ Ξ£g (π₯ β (π΄ βͺ {π}) β¦ (πΉβπ₯))) = ((πΊ Ξ£g (π₯ β π΄ β¦ (πΉβπ₯))) + π)) |
29 | 14 | oveq2d 7430 | . 2 β’ (π β (πΊ Ξ£g (πΉ βΎ (π΄ βͺ {π}))) = (πΊ Ξ£g (π₯ β (π΄ βͺ {π}) β¦ (πΉβπ₯)))) |
30 | 9, 10 | feqresmpt 6962 | . . . 4 β’ (π β (πΉ βΎ π΄) = (π₯ β π΄ β¦ (πΉβπ₯))) |
31 | 30 | oveq2d 7430 | . . 3 β’ (π β (πΊ Ξ£g (πΉ βΎ π΄)) = (πΊ Ξ£g (π₯ β π΄ β¦ (πΉβπ₯)))) |
32 | 31 | oveq1d 7429 | . 2 β’ (π β ((πΊ Ξ£g (πΉ βΎ π΄)) + π) = ((πΊ Ξ£g (π₯ β π΄ β¦ (πΉβπ₯))) + π)) |
33 | 28, 29, 32 | 3eqtr4d 2777 | 1 β’ (π β (πΊ Ξ£g (πΉ βΎ (π΄ βͺ {π}))) = ((πΊ Ξ£g (πΉ βΎ π΄)) + π)) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 395 = wceq 1534 β wcel 2099 βͺ cun 3942 β wss 3944 {csn 4624 β¦ cmpt 5225 ran crn 5673 βΎ cres 5674 β cima 5675 βΆwf 6538 βcfv 6542 (class class class)co 7414 Fincfn 8953 Basecbs 17165 +gcplusg 17218 Ξ£g cgsu 17407 Mndcmnd 18679 Cntzccntz 19250 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7677 df-om 7863 df-1st 7985 df-2nd 7986 df-supp 8158 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-er 8716 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-fsupp 9376 df-oi 9519 df-card 9948 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-nn 12229 df-2 12291 df-n0 12489 df-z 12575 df-uz 12839 df-fz 13503 df-fzo 13646 df-seq 13985 df-hash 14308 df-sets 17118 df-slot 17136 df-ndx 17148 df-base 17166 df-ress 17195 df-plusg 17231 df-0g 17408 df-gsum 17409 df-mre 17551 df-mrc 17552 df-acs 17554 df-mgm 18585 df-sgrp 18664 df-mnd 18680 df-submnd 18726 df-mulg 19008 df-cntz 19252 df-cmn 19721 |
This theorem is referenced by: (None) |
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