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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 2736 | . 2 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺) | |
| 5 | gsumadd.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ CMnd) | |
| 6 | cmnmnd 19816 | . . 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 18824 | . . 3 ⊢ (𝐺 ∈ Mnd → 𝐵 ∈ (SubMnd‘𝐺)) | 
| 12 | 7, 11 | syl 17 | . 2 ⊢ (𝜑 → 𝐵 ∈ (SubMnd‘𝐺)) | 
| 13 | ssid 4005 | . . 3 ⊢ 𝐵 ⊆ 𝐵 | |
| 14 | 1, 4 | cntzcmn 19859 | . . . 4 ⊢ ((𝐺 ∈ CMnd ∧ 𝐵 ⊆ 𝐵) → ((Cntz‘𝐺)‘𝐵) = 𝐵) | 
| 15 | 5, 13, 14 | sylancl 586 | . . 3 ⊢ (𝜑 → ((Cntz‘𝐺)‘𝐵) = 𝐵) | 
| 16 | 13, 15 | sseqtrrid 4026 | . 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 19941 | 1 ⊢ (𝜑 → (𝐺 Σg (𝐹 ∘f + 𝐻)) = ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 ⊆ wss 3950 class class class wbr 5142 ⟶wf 6556 ‘cfv 6560 (class class class)co 7432 ∘f cof 7696 finSupp cfsupp 9402 Basecbs 17248 +gcplusg 17298 0gc0g 17485 Σg cgsu 17486 Mndcmnd 18748 SubMndcsubmnd 18796 Cntzccntz 19334 CMndccmn 19799 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-se 5637 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-isom 6569 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-of 7698 df-om 7889 df-1st 8015 df-2nd 8016 df-supp 8187 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-fsupp 9403 df-oi 9551 df-card 9980 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-2 12330 df-n0 12529 df-z 12616 df-uz 12880 df-fz 13549 df-fzo 13696 df-seq 14044 df-hash 14371 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 df-plusg 17311 df-0g 17487 df-gsum 17488 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-submnd 18798 df-cntz 19336 df-cmn 19801 | 
| This theorem is referenced by: gsummptfsadd 19943 gsumsub 19967 frlmup1 21819 evlslem1 22107 mhpmulcl 22154 tsmsadd 24156 tdeglem3 26099 | 
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