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Mirrors > Home > MPE Home > Th. List > gsumadd | Structured version Visualization version GIF version |
Description: The sum of two group sums. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 5-Jun-2019.) |
Ref | Expression |
---|---|
gsumadd.b | β’ π΅ = (BaseβπΊ) |
gsumadd.z | β’ 0 = (0gβπΊ) |
gsumadd.p | β’ + = (+gβπΊ) |
gsumadd.g | β’ (π β πΊ β CMnd) |
gsumadd.a | β’ (π β π΄ β π) |
gsumadd.f | β’ (π β πΉ:π΄βΆπ΅) |
gsumadd.h | β’ (π β π»:π΄βΆπ΅) |
gsumadd.fn | β’ (π β πΉ finSupp 0 ) |
gsumadd.hn | β’ (π β π» finSupp 0 ) |
Ref | Expression |
---|---|
gsumadd | β’ (π β (πΊ Ξ£g (πΉ βf + π»)) = ((πΊ Ξ£g πΉ) + (πΊ Ξ£g π»))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gsumadd.b | . 2 β’ π΅ = (BaseβπΊ) | |
2 | gsumadd.z | . 2 β’ 0 = (0gβπΊ) | |
3 | gsumadd.p | . 2 β’ + = (+gβπΊ) | |
4 | eqid 2728 | . 2 β’ (CntzβπΊ) = (CntzβπΊ) | |
5 | gsumadd.g | . . 3 β’ (π β πΊ β CMnd) | |
6 | cmnmnd 19745 | . . 3 β’ (πΊ β CMnd β πΊ β Mnd) | |
7 | 5, 6 | syl 17 | . 2 β’ (π β πΊ β Mnd) |
8 | gsumadd.a | . 2 β’ (π β π΄ β π) | |
9 | gsumadd.fn | . 2 β’ (π β πΉ finSupp 0 ) | |
10 | gsumadd.hn | . 2 β’ (π β π» finSupp 0 ) | |
11 | 1 | submid 18755 | . . 3 β’ (πΊ β Mnd β π΅ β (SubMndβπΊ)) |
12 | 7, 11 | syl 17 | . 2 β’ (π β π΅ β (SubMndβπΊ)) |
13 | ssid 4000 | . . 3 β’ π΅ β π΅ | |
14 | 1, 4 | cntzcmn 19788 | . . . 4 β’ ((πΊ β CMnd β§ π΅ β π΅) β ((CntzβπΊ)βπ΅) = π΅) |
15 | 5, 13, 14 | sylancl 585 | . . 3 β’ (π β ((CntzβπΊ)βπ΅) = π΅) |
16 | 13, 15 | sseqtrrid 4031 | . 2 β’ (π β π΅ β ((CntzβπΊ)βπ΅)) |
17 | gsumadd.f | . 2 β’ (π β πΉ:π΄βΆπ΅) | |
18 | gsumadd.h | . 2 β’ (π β π»:π΄βΆπ΅) | |
19 | 1, 2, 3, 4, 7, 8, 9, 10, 12, 16, 17, 18 | gsumzadd 19870 | 1 β’ (π β (πΊ Ξ£g (πΉ βf + π»)) = ((πΊ Ξ£g πΉ) + (πΊ Ξ£g π»))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1534 β wcel 2099 β wss 3945 class class class wbr 5142 βΆwf 6538 βcfv 6542 (class class class)co 7414 βf cof 7677 finSupp cfsupp 9379 Basecbs 17173 +gcplusg 17226 0gc0g 17414 Ξ£g cgsu 17415 Mndcmnd 18687 SubMndcsubmnd 18732 Cntzccntz 19259 CMndccmn 19728 |
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 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 |
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 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 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-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 7679 df-om 7865 df-1st 7987 df-2nd 7988 df-supp 8160 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 9380 df-oi 9527 df-card 9956 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-nn 12237 df-2 12299 df-n0 12497 df-z 12583 df-uz 12847 df-fz 13511 df-fzo 13654 df-seq 13993 df-hash 14316 df-sets 17126 df-slot 17144 df-ndx 17156 df-base 17174 df-ress 17203 df-plusg 17239 df-0g 17416 df-gsum 17417 df-mgm 18593 df-sgrp 18672 df-mnd 18688 df-submnd 18734 df-cntz 19261 df-cmn 19730 |
This theorem is referenced by: gsummptfsadd 19872 gsumsub 19896 frlmup1 21725 evlslem1 22021 mhpmulcl 22066 tsmsadd 24044 tdeglem3 25986 tdeglem3OLD 25987 |
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