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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 2728 | . 2 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺) | |
5 | gsumadd.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ CMnd) | |
6 | cmnmnd 19752 | . . 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 18762 | . . 3 ⊢ (𝐺 ∈ Mnd → 𝐵 ∈ (SubMnd‘𝐺)) |
12 | 7, 11 | syl 17 | . 2 ⊢ (𝜑 → 𝐵 ∈ (SubMnd‘𝐺)) |
13 | ssid 4002 | . . 3 ⊢ 𝐵 ⊆ 𝐵 | |
14 | 1, 4 | cntzcmn 19795 | . . . 4 ⊢ ((𝐺 ∈ CMnd ∧ 𝐵 ⊆ 𝐵) → ((Cntz‘𝐺)‘𝐵) = 𝐵) |
15 | 5, 13, 14 | sylancl 585 | . . 3 ⊢ (𝜑 → ((Cntz‘𝐺)‘𝐵) = 𝐵) |
16 | 13, 15 | sseqtrrid 4033 | . 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 19877 | 1 ⊢ (𝜑 → (𝐺 Σg (𝐹 ∘f + 𝐻)) = ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ⊆ wss 3947 class class class wbr 5148 ⟶wf 6544 ‘cfv 6548 (class class class)co 7420 ∘f cof 7683 finSupp cfsupp 9386 Basecbs 17180 +gcplusg 17233 0gc0g 17421 Σg cgsu 17422 Mndcmnd 18694 SubMndcsubmnd 18739 Cntzccntz 19266 CMndccmn 19735 |
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 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 |
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 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7685 df-om 7871 df-1st 7993 df-2nd 7994 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9387 df-oi 9534 df-card 9963 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-2 12306 df-n0 12504 df-z 12590 df-uz 12854 df-fz 13518 df-fzo 13661 df-seq 14000 df-hash 14323 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-ress 17210 df-plusg 17246 df-0g 17423 df-gsum 17424 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-submnd 18741 df-cntz 19268 df-cmn 19737 |
This theorem is referenced by: gsummptfsadd 19879 gsumsub 19903 frlmup1 21732 evlslem1 22028 mhpmulcl 22073 tsmsadd 24064 tdeglem3 26006 tdeglem3OLD 26007 |
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