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Definition df-dprd 19037
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 19035 . 2 class DProd
2 vg . . 3 setvar 𝑔
3 vs . . 3 setvar 𝑠
4 cgrp 18033 . . 3 class Grp
5 vh . . . . . . . 8 setvar
65cv 1529 . . . . . . 7 class
76cdm 5554 . . . . . 6 class dom
82cv 1529 . . . . . . 7 class 𝑔
9 csubg 18203 . . . . . . 7 class SubGrp
108, 9cfv 6352 . . . . . 6 class (SubGrp‘𝑔)
117, 10, 6wf 6348 . . . . 5 wff :dom ⟶(SubGrp‘𝑔)
12 vx . . . . . . . . . . 11 setvar 𝑥
1312cv 1529 . . . . . . . . . 10 class 𝑥
1413, 6cfv 6352 . . . . . . . . 9 class (𝑥)
15 vy . . . . . . . . . . . 12 setvar 𝑦
1615cv 1529 . . . . . . . . . . 11 class 𝑦
1716, 6cfv 6352 . . . . . . . . . 10 class (𝑦)
18 ccntz 18375 . . . . . . . . . . 11 class Cntz
198, 18cfv 6352 . . . . . . . . . 10 class (Cntz‘𝑔)
2017, 19cfv 6352 . . . . . . . . 9 class ((Cntz‘𝑔)‘(𝑦))
2114, 20wss 3940 . . . . . . . 8 wff (𝑥) ⊆ ((Cntz‘𝑔)‘(𝑦))
2213csn 4564 . . . . . . . . 9 class {𝑥}
237, 22cdif 3937 . . . . . . . 8 class (dom ∖ {𝑥})
2421, 15, 23wral 3143 . . . . . . 7 wff 𝑦 ∈ (dom ∖ {𝑥})(𝑥) ⊆ ((Cntz‘𝑔)‘(𝑦))
256, 23cima 5557 . . . . . . . . . . 11 class ( “ (dom ∖ {𝑥}))
2625cuni 4837 . . . . . . . . . 10 class ( “ (dom ∖ {𝑥}))
27 cmrc 16844 . . . . . . . . . . 11 class mrCls
2810, 27cfv 6352 . . . . . . . . . 10 class (mrCls‘(SubGrp‘𝑔))
2926, 28cfv 6352 . . . . . . . . 9 class ((mrCls‘(SubGrp‘𝑔))‘ ( “ (dom ∖ {𝑥})))
3014, 29cin 3939 . . . . . . . 8 class ((𝑥) ∩ ((mrCls‘(SubGrp‘𝑔))‘ ( “ (dom ∖ {𝑥}))))
31 c0g 16703 . . . . . . . . . 10 class 0g
328, 31cfv 6352 . . . . . . . . 9 class (0g𝑔)
3332csn 4564 . . . . . . . 8 class {(0g𝑔)}
3430, 33wceq 1530 . . . . . . 7 wff ((𝑥) ∩ ((mrCls‘(SubGrp‘𝑔))‘ ( “ (dom ∖ {𝑥})))) = {(0g𝑔)}
3524, 34wa 396 . . . . . 6 wff (∀𝑦 ∈ (dom ∖ {𝑥})(𝑥) ⊆ ((Cntz‘𝑔)‘(𝑦)) ∧ ((𝑥) ∩ ((mrCls‘(SubGrp‘𝑔))‘ ( “ (dom ∖ {𝑥})))) = {(0g𝑔)})
3635, 12, 7wral 3143 . . . . 5 wff 𝑥 ∈ dom (∀𝑦 ∈ (dom ∖ {𝑥})(𝑥) ⊆ ((Cntz‘𝑔)‘(𝑦)) ∧ ((𝑥) ∩ ((mrCls‘(SubGrp‘𝑔))‘ ( “ (dom ∖ {𝑥})))) = {(0g𝑔)})
3711, 36wa 396 . . . 4 wff (:dom ⟶(SubGrp‘𝑔) ∧ ∀𝑥 ∈ dom (∀𝑦 ∈ (dom ∖ {𝑥})(𝑥) ⊆ ((Cntz‘𝑔)‘(𝑦)) ∧ ((𝑥) ∩ ((mrCls‘(SubGrp‘𝑔))‘ ( “ (dom ∖ {𝑥})))) = {(0g𝑔)}))
3837, 5cab 2804 . . 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 8822 . . . . . . 7 class finSupp
416, 32, 40wbr 5063 . . . . . 6 wff finSupp (0g𝑔)
423cv 1529 . . . . . . . 8 class 𝑠
4342cdm 5554 . . . . . . 7 class dom 𝑠
4413, 42cfv 6352 . . . . . . 7 class (𝑠𝑥)
4512, 43, 44cixp 8450 . . . . . 6 class X𝑥 ∈ dom 𝑠(𝑠𝑥)
4641, 5, 45crab 3147 . . . . 5 class {X𝑥 ∈ dom 𝑠(𝑠𝑥) ∣ finSupp (0g𝑔)}
4739cv 1529 . . . . . 6 class 𝑓
48 cgsu 16704 . . . . . 6 class Σg
498, 47, 48co 7148 . . . . 5 class (𝑔 Σg 𝑓)
5039, 46, 49cmpt 5143 . . . 4 class (𝑓 ∈ {X𝑥 ∈ dom 𝑠(𝑠𝑥) ∣ finSupp (0g𝑔)} ↦ (𝑔 Σg 𝑓))
5150crn 5555 . . 3 class ran (𝑓 ∈ {X𝑥 ∈ dom 𝑠(𝑠𝑥) ∣ finSupp (0g𝑔)} ↦ (𝑔 Σg 𝑓))
522, 3, 4, 38, 51cmpo 7150 . 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 1530 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  19039  dmdprd  19040  dprdval  19045
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