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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 | ⊢ 𝐵 = (Base‘𝐺) |
gsumadd.z | ⊢ 0 = (0g‘𝐺) |
gsumadd.p | ⊢ + = (+g‘𝐺) |
gsumadd.g | ⊢ (𝜑 → 𝐺 ∈ CMnd) |
gsumadd.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
gsumadd.f | ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
gsumadd.h | ⊢ (𝜑 → 𝐻:𝐴⟶𝐵) |
gsumadd.fn | ⊢ (𝜑 → 𝐹 finSupp 0 ) |
gsumadd.hn | ⊢ (𝜑 → 𝐻 finSupp 0 ) |
Ref | Expression |
---|---|
gsumadd | ⊢ (𝜑 → (𝐺 Σg (𝐹 ∘𝑓 + 𝐻)) = ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gsumadd.b | . 2 ⊢ 𝐵 = (Base‘𝐺) | |
2 | gsumadd.z | . 2 ⊢ 0 = (0g‘𝐺) | |
3 | gsumadd.p | . 2 ⊢ + = (+g‘𝐺) | |
4 | eqid 2824 | . 2 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺) | |
5 | gsumadd.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ CMnd) | |
6 | cmnmnd 18560 | . . 3 ⊢ (𝐺 ∈ CMnd → 𝐺 ∈ Mnd) | |
7 | 5, 6 | syl 17 | . 2 ⊢ (𝜑 → 𝐺 ∈ Mnd) |
8 | gsumadd.a | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
9 | gsumadd.fn | . 2 ⊢ (𝜑 → 𝐹 finSupp 0 ) | |
10 | gsumadd.hn | . 2 ⊢ (𝜑 → 𝐻 finSupp 0 ) | |
11 | 1 | submid 17703 | . . 3 ⊢ (𝐺 ∈ Mnd → 𝐵 ∈ (SubMnd‘𝐺)) |
12 | 7, 11 | syl 17 | . 2 ⊢ (𝜑 → 𝐵 ∈ (SubMnd‘𝐺)) |
13 | ssid 3847 | . . 3 ⊢ 𝐵 ⊆ 𝐵 | |
14 | 1, 4 | cntzcmn 18597 | . . . 4 ⊢ ((𝐺 ∈ CMnd ∧ 𝐵 ⊆ 𝐵) → ((Cntz‘𝐺)‘𝐵) = 𝐵) |
15 | 5, 13, 14 | sylancl 582 | . . 3 ⊢ (𝜑 → ((Cntz‘𝐺)‘𝐵) = 𝐵) |
16 | 13, 15 | syl5sseqr 3878 | . 2 ⊢ (𝜑 → 𝐵 ⊆ ((Cntz‘𝐺)‘𝐵)) |
17 | gsumadd.f | . 2 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
18 | gsumadd.h | . 2 ⊢ (𝜑 → 𝐻:𝐴⟶𝐵) | |
19 | 1, 2, 3, 4, 7, 8, 9, 10, 12, 16, 17, 18 | gsumzadd 18674 | 1 ⊢ (𝜑 → (𝐺 Σg (𝐹 ∘𝑓 + 𝐻)) = ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1658 ∈ wcel 2166 ⊆ wss 3797 class class class wbr 4872 ⟶wf 6118 ‘cfv 6122 (class class class)co 6904 ∘𝑓 cof 7154 finSupp cfsupp 8543 Basecbs 16221 +gcplusg 16304 0gc0g 16452 Σg cgsu 16453 Mndcmnd 17646 SubMndcsubmnd 17686 Cntzccntz 18097 CMndccmn 18545 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 ax-rep 4993 ax-sep 5004 ax-nul 5012 ax-pow 5064 ax-pr 5126 ax-un 7208 ax-cnex 10307 ax-resscn 10308 ax-1cn 10309 ax-icn 10310 ax-addcl 10311 ax-addrcl 10312 ax-mulcl 10313 ax-mulrcl 10314 ax-mulcom 10315 ax-addass 10316 ax-mulass 10317 ax-distr 10318 ax-i2m1 10319 ax-1ne0 10320 ax-1rid 10321 ax-rnegex 10322 ax-rrecex 10323 ax-cnre 10324 ax-pre-lttri 10325 ax-pre-lttrn 10326 ax-pre-ltadd 10327 ax-pre-mulgt0 10328 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2604 df-eu 2639 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-ne 2999 df-nel 3102 df-ral 3121 df-rex 3122 df-reu 3123 df-rmo 3124 df-rab 3125 df-v 3415 df-sbc 3662 df-csb 3757 df-dif 3800 df-un 3802 df-in 3804 df-ss 3811 df-pss 3813 df-nul 4144 df-if 4306 df-pw 4379 df-sn 4397 df-pr 4399 df-tp 4401 df-op 4403 df-uni 4658 df-int 4697 df-iun 4741 df-br 4873 df-opab 4935 df-mpt 4952 df-tr 4975 df-id 5249 df-eprel 5254 df-po 5262 df-so 5263 df-fr 5300 df-se 5301 df-we 5302 df-xp 5347 df-rel 5348 df-cnv 5349 df-co 5350 df-dm 5351 df-rn 5352 df-res 5353 df-ima 5354 df-pred 5919 df-ord 5965 df-on 5966 df-lim 5967 df-suc 5968 df-iota 6085 df-fun 6124 df-fn 6125 df-f 6126 df-f1 6127 df-fo 6128 df-f1o 6129 df-fv 6130 df-isom 6131 df-riota 6865 df-ov 6907 df-oprab 6908 df-mpt2 6909 df-of 7156 df-om 7326 df-1st 7427 df-2nd 7428 df-supp 7559 df-wrecs 7671 df-recs 7733 df-rdg 7771 df-1o 7825 df-oadd 7829 df-er 8008 df-en 8222 df-dom 8223 df-sdom 8224 df-fin 8225 df-fsupp 8544 df-oi 8683 df-card 9077 df-pnf 10392 df-mnf 10393 df-xr 10394 df-ltxr 10395 df-le 10396 df-sub 10586 df-neg 10587 df-nn 11350 df-2 11413 df-n0 11618 df-z 11704 df-uz 11968 df-fz 12619 df-fzo 12760 df-seq 13095 df-hash 13410 df-ndx 16224 df-slot 16225 df-base 16227 df-sets 16228 df-ress 16229 df-plusg 16317 df-0g 16454 df-gsum 16455 df-mgm 17594 df-sgrp 17636 df-mnd 17647 df-submnd 17688 df-cntz 18099 df-cmn 18547 |
This theorem is referenced by: gsummptfsadd 18676 gsumsub 18700 evlslem1 19874 frlmup1 20503 tsmsadd 22319 tdeglem3 24217 |
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