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Mirrors > Home > MPE Home > Th. List > gsummhm | Structured version Visualization version GIF version |
Description: Apply a group homomorphism to a group sum. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by Mario Carneiro, 24-Apr-2016.) (Revised by AV, 6-Jun-2019.) |
Ref | Expression |
---|---|
gsummhm.b | ⊢ 𝐵 = (Base‘𝐺) |
gsummhm.z | ⊢ 0 = (0g‘𝐺) |
gsummhm.g | ⊢ (𝜑 → 𝐺 ∈ CMnd) |
gsummhm.h | ⊢ (𝜑 → 𝐻 ∈ Mnd) |
gsummhm.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
gsummhm.k | ⊢ (𝜑 → 𝐾 ∈ (𝐺 MndHom 𝐻)) |
gsummhm.f | ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
gsummhm.w | ⊢ (𝜑 → 𝐹 finSupp 0 ) |
Ref | Expression |
---|---|
gsummhm | ⊢ (𝜑 → (𝐻 Σg (𝐾 ∘ 𝐹)) = (𝐾‘(𝐺 Σg 𝐹))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gsummhm.b | . 2 ⊢ 𝐵 = (Base‘𝐺) | |
2 | eqid 2821 | . 2 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺) | |
3 | gsummhm.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ CMnd) | |
4 | cmnmnd 18921 | . . 3 ⊢ (𝐺 ∈ CMnd → 𝐺 ∈ Mnd) | |
5 | 3, 4 | syl 17 | . 2 ⊢ (𝜑 → 𝐺 ∈ Mnd) |
6 | gsummhm.h | . 2 ⊢ (𝜑 → 𝐻 ∈ Mnd) | |
7 | gsummhm.a | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
8 | gsummhm.k | . 2 ⊢ (𝜑 → 𝐾 ∈ (𝐺 MndHom 𝐻)) | |
9 | gsummhm.f | . 2 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
10 | 1, 2, 3, 9 | cntzcmnf 18964 | . 2 ⊢ (𝜑 → ran 𝐹 ⊆ ((Cntz‘𝐺)‘ran 𝐹)) |
11 | gsummhm.z | . 2 ⊢ 0 = (0g‘𝐺) | |
12 | gsummhm.w | . 2 ⊢ (𝜑 → 𝐹 finSupp 0 ) | |
13 | 1, 2, 5, 6, 7, 8, 9, 10, 11, 12 | gsumzmhm 19056 | 1 ⊢ (𝜑 → (𝐻 Σg (𝐾 ∘ 𝐹)) = (𝐾‘(𝐺 Σg 𝐹))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 class class class wbr 5065 ∘ ccom 5558 ⟶wf 6350 ‘cfv 6354 (class class class)co 7155 finSupp cfsupp 8832 Basecbs 16482 0gc0g 16712 Σg cgsu 16713 Mndcmnd 17910 MndHom cmhm 17953 Cntzccntz 18444 CMndccmn 18905 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-se 5514 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-isom 6363 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-supp 7830 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-oadd 8105 df-er 8288 df-map 8407 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-fsupp 8833 df-oi 8973 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-n0 11897 df-z 11981 df-uz 12243 df-fz 12892 df-fzo 13033 df-seq 13369 df-hash 13690 df-0g 16714 df-gsum 16715 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-mhm 17955 df-cntz 18446 df-cmn 18907 |
This theorem is referenced by: gsummhm2 19058 gsummptmhm 19059 gsuminv 19065 evlslem2 20291 tsmsmhm 22753 plypf1 24801 amgmlem 25566 amgmwlem 44902 amgmlemALT 44903 |
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