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 2737 | . . . 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 2736 | . . . . . . 7 ⊢ (Base‘𝐻) = (Base‘𝐻) | |
6 | 4, 5 | mhmf 18532 | . . . . . 6 ⊢ (𝐾 ∈ (𝐺 MndHom 𝐻) → 𝐾:𝐵⟶(Base‘𝐻)) |
7 | ffn 6651 | . . . . . 6 ⊢ (𝐾:𝐵⟶(Base‘𝐻) → 𝐾 Fn 𝐵) | |
8 | 3, 6, 7 | 3syl 18 | . . . . 5 ⊢ (𝜑 → 𝐾 Fn 𝐵) |
9 | dffn5 6884 | . . . . 5 ⊢ (𝐾 Fn 𝐵 ↔ 𝐾 = (𝑦 ∈ 𝐵 ↦ (𝐾‘𝑦))) | |
10 | 8, 9 | sylib 217 | . . . 4 ⊢ (𝜑 → 𝐾 = (𝑦 ∈ 𝐵 ↦ (𝐾‘𝑦))) |
11 | fveq2 6825 | . . . 4 ⊢ (𝑦 = 𝐶 → (𝐾‘𝑦) = (𝐾‘𝐶)) | |
12 | 1, 2, 10, 11 | fmptco 7057 | . . 3 ⊢ (𝜑 → (𝐾 ∘ (𝑥 ∈ 𝐴 ↦ 𝐶)) = (𝑥 ∈ 𝐴 ↦ (𝐾‘𝐶))) |
13 | 12 | oveq2d 7353 | . 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 7045 | . . 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 19634 | . 2 ⊢ (𝜑 → (𝐻 Σg (𝐾 ∘ (𝑥 ∈ 𝐴 ↦ 𝐶))) = (𝐾‘(𝐺 Σg (𝑥 ∈ 𝐴 ↦ 𝐶)))) |
21 | 13, 20 | eqtr3d 2778 | 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 396 = wceq 1540 ∈ wcel 2105 class class class wbr 5092 ↦ cmpt 5175 ∘ ccom 5624 Fn wfn 6474 ⟶wf 6475 ‘cfv 6479 (class class class)co 7337 finSupp cfsupp 9226 Basecbs 17009 0gc0g 17247 Σg cgsu 17248 Mndcmnd 18482 MndHom cmhm 18525 CMndccmn 19481 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5229 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 ax-un 7650 ax-cnex 11028 ax-resscn 11029 ax-1cn 11030 ax-icn 11031 ax-addcl 11032 ax-addrcl 11033 ax-mulcl 11034 ax-mulrcl 11035 ax-mulcom 11036 ax-addass 11037 ax-mulass 11038 ax-distr 11039 ax-i2m1 11040 ax-1ne0 11041 ax-1rid 11042 ax-rnegex 11043 ax-rrecex 11044 ax-cnre 11045 ax-pre-lttri 11046 ax-pre-lttrn 11047 ax-pre-ltadd 11048 ax-pre-mulgt0 11049 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3917 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-int 4895 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5176 df-tr 5210 df-id 5518 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5575 df-se 5576 df-we 5577 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6238 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-isom 6488 df-riota 7293 df-ov 7340 df-oprab 7341 df-mpo 7342 df-om 7781 df-1st 7899 df-2nd 7900 df-supp 8048 df-frecs 8167 df-wrecs 8198 df-recs 8272 df-rdg 8311 df-1o 8367 df-er 8569 df-map 8688 df-en 8805 df-dom 8806 df-sdom 8807 df-fin 8808 df-fsupp 9227 df-oi 9367 df-card 9796 df-pnf 11112 df-mnf 11113 df-xr 11114 df-ltxr 11115 df-le 11116 df-sub 11308 df-neg 11309 df-nn 12075 df-n0 12335 df-z 12421 df-uz 12684 df-fz 13341 df-fzo 13484 df-seq 13823 df-hash 14146 df-0g 17249 df-gsum 17250 df-mgm 18423 df-sgrp 18472 df-mnd 18483 df-mhm 18527 df-cntz 19019 df-cmn 19483 |
This theorem is referenced by: evlsgsumadd 21407 evlsgsummul 21408 evls1gsumadd 21596 evls1gsummul 21597 evl1gsummul 21632 mat2pmatmul 21986 pm2mp 22080 cayhamlem4 22143 |
Copyright terms: Public domain | W3C validator |