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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 2735 | . . . 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 2734 | . . . . . . 7 ⊢ (Base‘𝐻) = (Base‘𝐻) | |
6 | 4, 5 | mhmf 18814 | . . . . . 6 ⊢ (𝐾 ∈ (𝐺 MndHom 𝐻) → 𝐾:𝐵⟶(Base‘𝐻)) |
7 | ffn 6736 | . . . . . 6 ⊢ (𝐾:𝐵⟶(Base‘𝐻) → 𝐾 Fn 𝐵) | |
8 | 3, 6, 7 | 3syl 18 | . . . . 5 ⊢ (𝜑 → 𝐾 Fn 𝐵) |
9 | dffn5 6966 | . . . . 5 ⊢ (𝐾 Fn 𝐵 ↔ 𝐾 = (𝑦 ∈ 𝐵 ↦ (𝐾‘𝑦))) | |
10 | 8, 9 | sylib 218 | . . . 4 ⊢ (𝜑 → 𝐾 = (𝑦 ∈ 𝐵 ↦ (𝐾‘𝑦))) |
11 | fveq2 6906 | . . . 4 ⊢ (𝑦 = 𝐶 → (𝐾‘𝑦) = (𝐾‘𝐶)) | |
12 | 1, 2, 10, 11 | fmptco 7148 | . . 3 ⊢ (𝜑 → (𝐾 ∘ (𝑥 ∈ 𝐴 ↦ 𝐶)) = (𝑥 ∈ 𝐴 ↦ (𝐾‘𝐶))) |
13 | 12 | oveq2d 7446 | . 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 7134 | . . 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 19970 | . 2 ⊢ (𝜑 → (𝐻 Σg (𝐾 ∘ (𝑥 ∈ 𝐴 ↦ 𝐶))) = (𝐾‘(𝐺 Σg (𝑥 ∈ 𝐴 ↦ 𝐶)))) |
21 | 13, 20 | eqtr3d 2776 | 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 395 = wceq 1536 ∈ wcel 2105 class class class wbr 5147 ↦ cmpt 5230 ∘ ccom 5692 Fn wfn 6557 ⟶wf 6558 ‘cfv 6562 (class class class)co 7430 finSupp cfsupp 9398 Basecbs 17244 0gc0g 17485 Σg cgsu 17486 Mndcmnd 18759 MndHom cmhm 18806 CMndccmn 19812 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-supp 8184 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-er 8743 df-map 8866 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-fsupp 9399 df-oi 9547 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-n0 12524 df-z 12611 df-uz 12876 df-fz 13544 df-fzo 13691 df-seq 14039 df-hash 14366 df-0g 17487 df-gsum 17488 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-mhm 18808 df-cntz 19347 df-cmn 19814 |
This theorem is referenced by: evlsgsumadd 22132 evlsgsummul 22133 evls1gsumadd 22343 evls1gsummul 22344 evl1gsummul 22379 rhmcomulmpl 22401 mat2pmatmul 22752 pm2mp 22846 cayhamlem4 22909 rhmcomulpsr 42537 |
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