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Definition df-dprd 18315
Description: Define the internal direct product of a family of subgroups. (Contributed by Mario Carneiro, 21-Apr-2016.) (Revised by AV, 11-Jul-2019.)
Assertion
Ref Expression
df-dprd DProd = (𝑔 ∈ Grp, 𝑠 ∈ { ∣ (:dom ⟶(SubGrp‘𝑔) ∧ ∀𝑥 ∈ dom (∀𝑦 ∈ (dom ∖ {𝑥})(𝑥) ⊆ ((Cntz‘𝑔)‘(𝑦)) ∧ ((𝑥) ∩ ((mrCls‘(SubGrp‘𝑔))‘ ( “ (dom ∖ {𝑥})))) = {(0g𝑔)}))} ↦ ran (𝑓 ∈ {X𝑥 ∈ dom 𝑠(𝑠𝑥) ∣ finSupp (0g𝑔)} ↦ (𝑔 Σg 𝑓)))
Distinct variable group:   𝑔,,𝑓,𝑠,𝑥,𝑦

Detailed syntax breakdown of Definition df-dprd
StepHypRef Expression
1 cdprd 18313 . 2 class DProd
2 vg . . 3 setvar 𝑔
3 vs . . 3 setvar 𝑠
4 cgrp 17343 . . 3 class Grp
5 vh . . . . . . . 8 setvar
65cv 1479 . . . . . . 7 class
76cdm 5074 . . . . . 6 class dom
82cv 1479 . . . . . . 7 class 𝑔
9 csubg 17509 . . . . . . 7 class SubGrp
108, 9cfv 5847 . . . . . 6 class (SubGrp‘𝑔)
117, 10, 6wf 5843 . . . . 5 wff :dom ⟶(SubGrp‘𝑔)
12 vx . . . . . . . . . . 11 setvar 𝑥
1312cv 1479 . . . . . . . . . 10 class 𝑥
1413, 6cfv 5847 . . . . . . . . 9 class (𝑥)
15 vy . . . . . . . . . . . 12 setvar 𝑦
1615cv 1479 . . . . . . . . . . 11 class 𝑦
1716, 6cfv 5847 . . . . . . . . . 10 class (𝑦)
18 ccntz 17669 . . . . . . . . . . 11 class Cntz
198, 18cfv 5847 . . . . . . . . . 10 class (Cntz‘𝑔)
2017, 19cfv 5847 . . . . . . . . 9 class ((Cntz‘𝑔)‘(𝑦))
2114, 20wss 3555 . . . . . . . 8 wff (𝑥) ⊆ ((Cntz‘𝑔)‘(𝑦))
2213csn 4148 . . . . . . . . 9 class {𝑥}
237, 22cdif 3552 . . . . . . . 8 class (dom ∖ {𝑥})
2421, 15, 23wral 2907 . . . . . . 7 wff 𝑦 ∈ (dom ∖ {𝑥})(𝑥) ⊆ ((Cntz‘𝑔)‘(𝑦))
256, 23cima 5077 . . . . . . . . . . 11 class ( “ (dom ∖ {𝑥}))
2625cuni 4402 . . . . . . . . . 10 class ( “ (dom ∖ {𝑥}))
27 cmrc 16164 . . . . . . . . . . 11 class mrCls
2810, 27cfv 5847 . . . . . . . . . 10 class (mrCls‘(SubGrp‘𝑔))
2926, 28cfv 5847 . . . . . . . . 9 class ((mrCls‘(SubGrp‘𝑔))‘ ( “ (dom ∖ {𝑥})))
3014, 29cin 3554 . . . . . . . 8 class ((𝑥) ∩ ((mrCls‘(SubGrp‘𝑔))‘ ( “ (dom ∖ {𝑥}))))
31 c0g 16021 . . . . . . . . . 10 class 0g
328, 31cfv 5847 . . . . . . . . 9 class (0g𝑔)
3332csn 4148 . . . . . . . 8 class {(0g𝑔)}
3430, 33wceq 1480 . . . . . . 7 wff ((𝑥) ∩ ((mrCls‘(SubGrp‘𝑔))‘ ( “ (dom ∖ {𝑥})))) = {(0g𝑔)}
3524, 34wa 384 . . . . . 6 wff (∀𝑦 ∈ (dom ∖ {𝑥})(𝑥) ⊆ ((Cntz‘𝑔)‘(𝑦)) ∧ ((𝑥) ∩ ((mrCls‘(SubGrp‘𝑔))‘ ( “ (dom ∖ {𝑥})))) = {(0g𝑔)})
3635, 12, 7wral 2907 . . . . 5 wff 𝑥 ∈ dom (∀𝑦 ∈ (dom ∖ {𝑥})(𝑥) ⊆ ((Cntz‘𝑔)‘(𝑦)) ∧ ((𝑥) ∩ ((mrCls‘(SubGrp‘𝑔))‘ ( “ (dom ∖ {𝑥})))) = {(0g𝑔)})
3711, 36wa 384 . . . 4 wff (:dom ⟶(SubGrp‘𝑔) ∧ ∀𝑥 ∈ dom (∀𝑦 ∈ (dom ∖ {𝑥})(𝑥) ⊆ ((Cntz‘𝑔)‘(𝑦)) ∧ ((𝑥) ∩ ((mrCls‘(SubGrp‘𝑔))‘ ( “ (dom ∖ {𝑥})))) = {(0g𝑔)}))
3837, 5cab 2607 . . 3 class { ∣ (:dom ⟶(SubGrp‘𝑔) ∧ ∀𝑥 ∈ dom (∀𝑦 ∈ (dom ∖ {𝑥})(𝑥) ⊆ ((Cntz‘𝑔)‘(𝑦)) ∧ ((𝑥) ∩ ((mrCls‘(SubGrp‘𝑔))‘ ( “ (dom ∖ {𝑥})))) = {(0g𝑔)}))}
39 vf . . . . 5 setvar 𝑓
40 cfsupp 8219 . . . . . . 7 class finSupp
416, 32, 40wbr 4613 . . . . . 6 wff finSupp (0g𝑔)
423cv 1479 . . . . . . . 8 class 𝑠
4342cdm 5074 . . . . . . 7 class dom 𝑠
4413, 42cfv 5847 . . . . . . 7 class (𝑠𝑥)
4512, 43, 44cixp 7852 . . . . . 6 class X𝑥 ∈ dom 𝑠(𝑠𝑥)
4641, 5, 45crab 2911 . . . . 5 class {X𝑥 ∈ dom 𝑠(𝑠𝑥) ∣ finSupp (0g𝑔)}
4739cv 1479 . . . . . 6 class 𝑓
48 cgsu 16022 . . . . . 6 class Σg
498, 47, 48co 6604 . . . . 5 class (𝑔 Σg 𝑓)
5039, 46, 49cmpt 4673 . . . 4 class (𝑓 ∈ {X𝑥 ∈ dom 𝑠(𝑠𝑥) ∣ finSupp (0g𝑔)} ↦ (𝑔 Σg 𝑓))
5150crn 5075 . . 3 class ran (𝑓 ∈ {X𝑥 ∈ dom 𝑠(𝑠𝑥) ∣ finSupp (0g𝑔)} ↦ (𝑔 Σg 𝑓))
522, 3, 4, 38, 51cmpt2 6606 . 2 class (𝑔 ∈ Grp, 𝑠 ∈ { ∣ (:dom ⟶(SubGrp‘𝑔) ∧ ∀𝑥 ∈ dom (∀𝑦 ∈ (dom ∖ {𝑥})(𝑥) ⊆ ((Cntz‘𝑔)‘(𝑦)) ∧ ((𝑥) ∩ ((mrCls‘(SubGrp‘𝑔))‘ ( “ (dom ∖ {𝑥})))) = {(0g𝑔)}))} ↦ ran (𝑓 ∈ {X𝑥 ∈ dom 𝑠(𝑠𝑥) ∣ finSupp (0g𝑔)} ↦ (𝑔 Σg 𝑓)))
531, 52wceq 1480 1 wff DProd = (𝑔 ∈ Grp, 𝑠 ∈ { ∣ (:dom ⟶(SubGrp‘𝑔) ∧ ∀𝑥 ∈ dom (∀𝑦 ∈ (dom ∖ {𝑥})(𝑥) ⊆ ((Cntz‘𝑔)‘(𝑦)) ∧ ((𝑥) ∩ ((mrCls‘(SubGrp‘𝑔))‘ ( “ (dom ∖ {𝑥})))) = {(0g𝑔)}))} ↦ ran (𝑓 ∈ {X𝑥 ∈ dom 𝑠(𝑠𝑥) ∣ finSupp (0g𝑔)} ↦ (𝑔 Σg 𝑓)))
Colors of variables: wff setvar class
This definition is referenced by:  reldmdprd  18317  dmdprd  18318  dprdval  18323
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