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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 2738 | . . . 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 2737 | . . . . . . 7 ⊢ (Base‘𝐻) = (Base‘𝐻) | |
| 6 | 4, 5 | mhmf 18802 | . . . . . 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 6967 | . . . . 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 7149 | . . 3 ⊢ (𝜑 → (𝐾 ∘ (𝑥 ∈ 𝐴 ↦ 𝐶)) = (𝑥 ∈ 𝐴 ↦ (𝐾‘𝐶))) |
| 13 | 12 | oveq2d 7447 | . 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 7135 | . . 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 19956 | . 2 ⊢ (𝜑 → (𝐻 Σg (𝐾 ∘ (𝑥 ∈ 𝐴 ↦ 𝐶))) = (𝐾‘(𝐺 Σg (𝑥 ∈ 𝐴 ↦ 𝐶)))) |
| 21 | 13, 20 | eqtr3d 2779 | 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 1540 ∈ wcel 2108 class class class wbr 5143 ↦ cmpt 5225 ∘ ccom 5689 Fn wfn 6556 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 finSupp cfsupp 9401 Basecbs 17247 0gc0g 17484 Σg cgsu 17485 Mndcmnd 18747 MndHom cmhm 18794 CMndccmn 19798 |
| 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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-fzo 13695 df-seq 14043 df-hash 14370 df-0g 17486 df-gsum 17487 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-mhm 18796 df-cntz 19335 df-cmn 19800 |
| This theorem is referenced by: evlsgsumadd 22115 evlsgsummul 22116 evls1gsumadd 22328 evls1gsummul 22329 evl1gsummul 22364 rhmcomulmpl 22386 mat2pmatmul 22737 pm2mp 22831 cayhamlem4 22894 rhmcomulpsr 42561 |
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