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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 2821 | . 2 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺) | |
5 | gsumadd.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ CMnd) | |
6 | cmnmnd 18922 | . . 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 17975 | . . 3 ⊢ (𝐺 ∈ Mnd → 𝐵 ∈ (SubMnd‘𝐺)) |
12 | 7, 11 | syl 17 | . 2 ⊢ (𝜑 → 𝐵 ∈ (SubMnd‘𝐺)) |
13 | ssid 3989 | . . 3 ⊢ 𝐵 ⊆ 𝐵 | |
14 | 1, 4 | cntzcmn 18960 | . . . 4 ⊢ ((𝐺 ∈ CMnd ∧ 𝐵 ⊆ 𝐵) → ((Cntz‘𝐺)‘𝐵) = 𝐵) |
15 | 5, 13, 14 | sylancl 588 | . . 3 ⊢ (𝜑 → ((Cntz‘𝐺)‘𝐵) = 𝐵) |
16 | 13, 15 | sseqtrrid 4020 | . 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 19042 | 1 ⊢ (𝜑 → (𝐺 Σg (𝐹 ∘f + 𝐻)) = ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ⊆ wss 3936 class class class wbr 5066 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 ∘f cof 7407 finSupp cfsupp 8833 Basecbs 16483 +gcplusg 16565 0gc0g 16713 Σg cgsu 16714 Mndcmnd 17911 SubMndcsubmnd 17955 Cntzccntz 18445 CMndccmn 18906 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-fzo 13035 df-seq 13371 df-hash 13692 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-0g 16715 df-gsum 16716 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-cntz 18447 df-cmn 18908 |
This theorem is referenced by: gsummptfsadd 19044 gsumsub 19068 evlslem1 20295 frlmup1 20942 tsmsadd 22755 tdeglem3 24653 |
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