![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > gsummptmhm | Structured version Visualization version GIF version |
Description: Apply a group homomorphism to a group sum expressed with a mapping. (Contributed by Thierry Arnoux, 7-Sep-2018.) (Revised by AV, 8-Sep-2019.) |
Ref | Expression |
---|---|
gsummptmhm.b | ⊢ 𝐵 = (Base‘𝐺) |
gsummptmhm.z | ⊢ 0 = (0g‘𝐺) |
gsummptmhm.g | ⊢ (𝜑 → 𝐺 ∈ CMnd) |
gsummptmhm.h | ⊢ (𝜑 → 𝐻 ∈ Mnd) |
gsummptmhm.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
gsummptmhm.k | ⊢ (𝜑 → 𝐾 ∈ (𝐺 MndHom 𝐻)) |
gsummptmhm.c | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵) |
gsummptmhm.w | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) finSupp 0 ) |
Ref | Expression |
---|---|
gsummptmhm | ⊢ (𝜑 → (𝐻 Σg (𝑥 ∈ 𝐴 ↦ (𝐾‘𝐶))) = (𝐾‘(𝐺 Σg (𝑥 ∈ 𝐴 ↦ 𝐶)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gsummptmhm.c | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵) | |
2 | eqidd 2731 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐴 ↦ 𝐶)) | |
3 | gsummptmhm.k | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (𝐺 MndHom 𝐻)) | |
4 | gsummptmhm.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝐺) | |
5 | eqid 2730 | . . . . . . 7 ⊢ (Base‘𝐻) = (Base‘𝐻) | |
6 | 4, 5 | mhmf 18711 | . . . . . 6 ⊢ (𝐾 ∈ (𝐺 MndHom 𝐻) → 𝐾:𝐵⟶(Base‘𝐻)) |
7 | ffn 6716 | . . . . . 6 ⊢ (𝐾:𝐵⟶(Base‘𝐻) → 𝐾 Fn 𝐵) | |
8 | 3, 6, 7 | 3syl 18 | . . . . 5 ⊢ (𝜑 → 𝐾 Fn 𝐵) |
9 | dffn5 6949 | . . . . 5 ⊢ (𝐾 Fn 𝐵 ↔ 𝐾 = (𝑦 ∈ 𝐵 ↦ (𝐾‘𝑦))) | |
10 | 8, 9 | sylib 217 | . . . 4 ⊢ (𝜑 → 𝐾 = (𝑦 ∈ 𝐵 ↦ (𝐾‘𝑦))) |
11 | fveq2 6890 | . . . 4 ⊢ (𝑦 = 𝐶 → (𝐾‘𝑦) = (𝐾‘𝐶)) | |
12 | 1, 2, 10, 11 | fmptco 7128 | . . 3 ⊢ (𝜑 → (𝐾 ∘ (𝑥 ∈ 𝐴 ↦ 𝐶)) = (𝑥 ∈ 𝐴 ↦ (𝐾‘𝐶))) |
13 | 12 | oveq2d 7427 | . 2 ⊢ (𝜑 → (𝐻 Σg (𝐾 ∘ (𝑥 ∈ 𝐴 ↦ 𝐶))) = (𝐻 Σg (𝑥 ∈ 𝐴 ↦ (𝐾‘𝐶)))) |
14 | gsummptmhm.z | . . 3 ⊢ 0 = (0g‘𝐺) | |
15 | gsummptmhm.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ CMnd) | |
16 | gsummptmhm.h | . . 3 ⊢ (𝜑 → 𝐻 ∈ Mnd) | |
17 | gsummptmhm.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
18 | 1 | fmpttd 7115 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶):𝐴⟶𝐵) |
19 | gsummptmhm.w | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) finSupp 0 ) | |
20 | 4, 14, 15, 16, 17, 3, 18, 19 | gsummhm 19847 | . 2 ⊢ (𝜑 → (𝐻 Σg (𝐾 ∘ (𝑥 ∈ 𝐴 ↦ 𝐶))) = (𝐾‘(𝐺 Σg (𝑥 ∈ 𝐴 ↦ 𝐶)))) |
21 | 13, 20 | eqtr3d 2772 | 1 ⊢ (𝜑 → (𝐻 Σg (𝑥 ∈ 𝐴 ↦ (𝐾‘𝐶))) = (𝐾‘(𝐺 Σg (𝑥 ∈ 𝐴 ↦ 𝐶)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1539 ∈ wcel 2104 class class class wbr 5147 ↦ cmpt 5230 ∘ ccom 5679 Fn wfn 6537 ⟶wf 6538 ‘cfv 6542 (class class class)co 7411 finSupp cfsupp 9363 Basecbs 17148 0gc0g 17389 Σg cgsu 17390 Mndcmnd 18659 MndHom cmhm 18703 CMndccmn 19689 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-supp 8149 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-n0 12477 df-z 12563 df-uz 12827 df-fz 13489 df-fzo 13632 df-seq 13971 df-hash 14295 df-0g 17391 df-gsum 17392 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-mhm 18705 df-cntz 19222 df-cmn 19691 |
This theorem is referenced by: evlsgsumadd 21873 evlsgsummul 21874 evls1gsumadd 22063 evls1gsummul 22064 evl1gsummul 22099 mat2pmatmul 22453 pm2mp 22547 cayhamlem4 22610 rhmcomulmpl 41426 |
Copyright terms: Public domain | W3C validator |