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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 2725 | . . 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 5686 | . . . . 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 4809 | . . . . . . . 8 β’ (π β {π} β πΆ) |
13 | 10, 12 | unssd 4181 | . . . . . . 7 β’ (π β (π΄ βͺ {π}) β πΆ) |
14 | 9, 13 | feqresmpt 6961 | . . . . . 6 β’ (π β (πΉ βΎ (π΄ βͺ {π})) = (π₯ β (π΄ βͺ {π}) β¦ (πΉβπ₯))) |
15 | 14 | rneqd 5935 | . . . . 5 β’ (π β ran (πΉ βΎ (π΄ βͺ {π})) = ran (π₯ β (π΄ βͺ {π}) β¦ (πΉβπ₯))) |
16 | 8, 15 | eqtrid 2777 | . . . 4 β’ (π β (πΉ β (π΄ βͺ {π})) = ran (π₯ β (π΄ βͺ {π}) β¦ (πΉβπ₯))) |
17 | 16 | fveq2d 6894 | . . . 4 β’ (π β (πβ(πΉ β (π΄ βͺ {π}))) = (πβran (π₯ β (π΄ βͺ {π}) β¦ (πΉβπ₯)))) |
18 | 7, 16, 17 | 3sstr3d 4020 | . . 3 β’ (π β ran (π₯ β (π΄ βͺ {π}) β¦ (πΉβπ₯)) β (πβran (π₯ β (π΄ βͺ {π}) β¦ (πΉβπ₯)))) |
19 | 9 | adantr 479 | . . . 4 β’ ((π β§ π₯ β π΄) β πΉ:πΆβΆπ΅) |
20 | 10 | sselda 3973 | . . . 4 β’ ((π β§ π₯ β π΄) β π₯ β πΆ) |
21 | 19, 20 | ffvelcdmd 7088 | . . 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 483 | . . . . 5 β’ ((π β§ π₯ = π) β π₯ = π) | |
25 | 24 | fveq2d 6894 | . . . 4 β’ ((π β§ π₯ = π) β (πΉβπ₯) = (πΉβπ)) |
26 | gsumzresunsn.y | . . . 4 β’ π = (πΉβπ) | |
27 | 25, 26 | eqtr4di 2783 | . . 3 β’ ((π β§ π₯ = π) β (πΉβπ₯) = π) |
28 | 1, 2, 3, 4, 5, 6, 18, 21, 11, 22, 23, 27 | gsumzunsnd 19910 | . 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 7429 | . 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 6961 | . . . 4 β’ (π β (πΉ βΎ π΄) = (π₯ β π΄ β¦ (πΉβπ₯))) |
31 | 30 | oveq2d 7429 | . . 3 β’ (π β (πΊ Ξ£g (πΉ βΎ π΄)) = (πΊ Ξ£g (π₯ β π΄ β¦ (πΉβπ₯)))) |
32 | 31 | oveq1d 7428 | . 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 2775 | 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 394 = wceq 1533 β wcel 2098 βͺ cun 3939 β wss 3941 {csn 4625 β¦ cmpt 5227 ran crn 5674 βΎ cres 5675 β cima 5676 βΆwf 6539 βcfv 6543 (class class class)co 7413 Fincfn 8957 Basecbs 17174 +gcplusg 17227 Ξ£g cgsu 17416 Mndcmnd 18688 Cntzccntz 19265 |
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-iin 4995 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-of 7679 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-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-ress 17204 df-plusg 17240 df-0g 17417 df-gsum 17418 df-mre 17560 df-mrc 17561 df-acs 17563 df-mgm 18594 df-sgrp 18673 df-mnd 18689 df-submnd 18735 df-mulg 19023 df-cntz 19267 df-cmn 19736 |
This theorem is referenced by: rprmdvdsprod 33287 |
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