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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 2733 | . . 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 5647 | . . . . 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 4770 | . . . . . . . 8 β’ (π β {π} β πΆ) |
13 | 10, 12 | unssd 4147 | . . . . . . 7 β’ (π β (π΄ βͺ {π}) β πΆ) |
14 | 9, 13 | feqresmpt 6912 | . . . . . 6 β’ (π β (πΉ βΎ (π΄ βͺ {π})) = (π₯ β (π΄ βͺ {π}) β¦ (πΉβπ₯))) |
15 | 14 | rneqd 5894 | . . . . 5 β’ (π β ran (πΉ βΎ (π΄ βͺ {π})) = ran (π₯ β (π΄ βͺ {π}) β¦ (πΉβπ₯))) |
16 | 8, 15 | eqtrid 2785 | . . . 4 β’ (π β (πΉ β (π΄ βͺ {π})) = ran (π₯ β (π΄ βͺ {π}) β¦ (πΉβπ₯))) |
17 | 16 | fveq2d 6847 | . . . 4 β’ (π β (πβ(πΉ β (π΄ βͺ {π}))) = (πβran (π₯ β (π΄ βͺ {π}) β¦ (πΉβπ₯)))) |
18 | 7, 16, 17 | 3sstr3d 3991 | . . 3 β’ (π β ran (π₯ β (π΄ βͺ {π}) β¦ (πΉβπ₯)) β (πβran (π₯ β (π΄ βͺ {π}) β¦ (πΉβπ₯)))) |
19 | 9 | adantr 482 | . . . 4 β’ ((π β§ π₯ β π΄) β πΉ:πΆβΆπ΅) |
20 | 10 | sselda 3945 | . . . 4 β’ ((π β§ π₯ β π΄) β π₯ β πΆ) |
21 | 19, 20 | ffvelcdmd 7037 | . . 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 486 | . . . . 5 β’ ((π β§ π₯ = π) β π₯ = π) | |
25 | 24 | fveq2d 6847 | . . . 4 β’ ((π β§ π₯ = π) β (πΉβπ₯) = (πΉβπ)) |
26 | gsumzresunsn.y | . . . 4 β’ π = (πΉβπ) | |
27 | 25, 26 | eqtr4di 2791 | . . 3 β’ ((π β§ π₯ = π) β (πΉβπ₯) = π) |
28 | 1, 2, 3, 4, 5, 6, 18, 21, 11, 22, 23, 27 | gsumzunsnd 19738 | . 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 7374 | . 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 6912 | . . . 4 β’ (π β (πΉ βΎ π΄) = (π₯ β π΄ β¦ (πΉβπ₯))) |
31 | 30 | oveq2d 7374 | . . 3 β’ (π β (πΊ Ξ£g (πΉ βΎ π΄)) = (πΊ Ξ£g (π₯ β π΄ β¦ (πΉβπ₯)))) |
32 | 31 | oveq1d 7373 | . 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 2783 | 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 397 = wceq 1542 β wcel 2107 βͺ cun 3909 β wss 3911 {csn 4587 β¦ cmpt 5189 ran crn 5635 βΎ cres 5636 β cima 5637 βΆwf 6493 βcfv 6497 (class class class)co 7358 Fincfn 8886 Basecbs 17088 +gcplusg 17138 Ξ£g cgsu 17327 Mndcmnd 18561 Cntzccntz 19100 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-iin 4958 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-se 5590 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-isom 6506 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7618 df-om 7804 df-1st 7922 df-2nd 7923 df-supp 8094 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-fsupp 9309 df-oi 9451 df-card 9880 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-n0 12419 df-z 12505 df-uz 12769 df-fz 13431 df-fzo 13574 df-seq 13913 df-hash 14237 df-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-ress 17118 df-plusg 17151 df-0g 17328 df-gsum 17329 df-mre 17471 df-mrc 17472 df-acs 17474 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-submnd 18607 df-mulg 18878 df-cntz 19102 df-cmn 19569 |
This theorem is referenced by: (None) |
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