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Theorem dmdprd 19203
Description: The domain of definition of the internal direct product, which states that 𝑆 is a family of subgroups that mutually commute and have trivial intersections. (Contributed by Mario Carneiro, 25-Apr-2016.) (Proof shortened by AV, 11-Jul-2019.)
Hypotheses
Ref Expression
dmdprd.z 𝑍 = (Cntz‘𝐺)
dmdprd.0 0 = (0g𝐺)
dmdprd.k 𝐾 = (mrCls‘(SubGrp‘𝐺))
Assertion
Ref Expression
dmdprd ((𝐼𝑉 ∧ dom 𝑆 = 𝐼) → (𝐺dom DProd 𝑆 ↔ (𝐺 ∈ Grp ∧ 𝑆:𝐼⟶(SubGrp‘𝐺) ∧ ∀𝑥𝐼 (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆𝑥) ⊆ (𝑍‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ (𝐾 (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 }))))
Distinct variable groups:   𝑥,𝑦,𝐺   𝑥,𝐼,𝑦   𝑥,𝑆,𝑦   𝑥,𝑉,𝑦
Allowed substitution hints:   𝐾(𝑥,𝑦)   0 (𝑥,𝑦)   𝑍(𝑥,𝑦)

Proof of Theorem dmdprd
Dummy variables 𝑔 𝑓 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3429 . . . . 5 (𝑆 ∈ { ∣ (:dom ⟶(SubGrp‘𝐺) ∧ ∀𝑥 ∈ dom (∀𝑦 ∈ (dom ∖ {𝑥})(𝑥) ⊆ (𝑍‘(𝑦)) ∧ ((𝑥) ∩ (𝐾 ( “ (dom ∖ {𝑥})))) = { 0 }))} → 𝑆 ∈ V)
21a1i 11 . . . 4 ((𝐼𝑉 ∧ dom 𝑆 = 𝐼) → (𝑆 ∈ { ∣ (:dom ⟶(SubGrp‘𝐺) ∧ ∀𝑥 ∈ dom (∀𝑦 ∈ (dom ∖ {𝑥})(𝑥) ⊆ (𝑍‘(𝑦)) ∧ ((𝑥) ∩ (𝐾 ( “ (dom ∖ {𝑥})))) = { 0 }))} → 𝑆 ∈ V))
3 fex 6987 . . . . . . 7 ((𝑆:𝐼⟶(SubGrp‘𝐺) ∧ 𝐼𝑉) → 𝑆 ∈ V)
43expcom 417 . . . . . 6 (𝐼𝑉 → (𝑆:𝐼⟶(SubGrp‘𝐺) → 𝑆 ∈ V))
54adantr 484 . . . . 5 ((𝐼𝑉 ∧ dom 𝑆 = 𝐼) → (𝑆:𝐼⟶(SubGrp‘𝐺) → 𝑆 ∈ V))
65adantrd 495 . . . 4 ((𝐼𝑉 ∧ dom 𝑆 = 𝐼) → ((𝑆:𝐼⟶(SubGrp‘𝐺) ∧ ∀𝑥𝐼 (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆𝑥) ⊆ (𝑍‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ (𝐾 (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 })) → 𝑆 ∈ V))
7 df-sbc 3700 . . . . . 6 ([𝑆 / ](:dom ⟶(SubGrp‘𝐺) ∧ ∀𝑥 ∈ dom (∀𝑦 ∈ (dom ∖ {𝑥})(𝑥) ⊆ (𝑍‘(𝑦)) ∧ ((𝑥) ∩ (𝐾 ( “ (dom ∖ {𝑥})))) = { 0 })) ↔ 𝑆 ∈ { ∣ (:dom ⟶(SubGrp‘𝐺) ∧ ∀𝑥 ∈ dom (∀𝑦 ∈ (dom ∖ {𝑥})(𝑥) ⊆ (𝑍‘(𝑦)) ∧ ((𝑥) ∩ (𝐾 ( “ (dom ∖ {𝑥})))) = { 0 }))})
8 simpr 488 . . . . . . 7 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ 𝑆 ∈ V) → 𝑆 ∈ V)
9 simpr 488 . . . . . . . . . 10 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → = 𝑆)
109dmeqd 5752 . . . . . . . . . . 11 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → dom = dom 𝑆)
11 simplr 768 . . . . . . . . . . 11 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → dom 𝑆 = 𝐼)
1210, 11eqtrd 2794 . . . . . . . . . 10 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → dom = 𝐼)
139, 12feq12d 6492 . . . . . . . . 9 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → (:dom ⟶(SubGrp‘𝐺) ↔ 𝑆:𝐼⟶(SubGrp‘𝐺)))
1412difeq1d 4030 . . . . . . . . . . . 12 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → (dom ∖ {𝑥}) = (𝐼 ∖ {𝑥}))
159fveq1d 6666 . . . . . . . . . . . . 13 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → (𝑥) = (𝑆𝑥))
169fveq1d 6666 . . . . . . . . . . . . . 14 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → (𝑦) = (𝑆𝑦))
1716fveq2d 6668 . . . . . . . . . . . . 13 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → (𝑍‘(𝑦)) = (𝑍‘(𝑆𝑦)))
1815, 17sseq12d 3928 . . . . . . . . . . . 12 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → ((𝑥) ⊆ (𝑍‘(𝑦)) ↔ (𝑆𝑥) ⊆ (𝑍‘(𝑆𝑦))))
1914, 18raleqbidv 3320 . . . . . . . . . . 11 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → (∀𝑦 ∈ (dom ∖ {𝑥})(𝑥) ⊆ (𝑍‘(𝑦)) ↔ ∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆𝑥) ⊆ (𝑍‘(𝑆𝑦))))
209, 14imaeq12d 5908 . . . . . . . . . . . . . . 15 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → ( “ (dom ∖ {𝑥})) = (𝑆 “ (𝐼 ∖ {𝑥})))
2120unieqd 4816 . . . . . . . . . . . . . 14 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → ( “ (dom ∖ {𝑥})) = (𝑆 “ (𝐼 ∖ {𝑥})))
2221fveq2d 6668 . . . . . . . . . . . . 13 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → (𝐾 ( “ (dom ∖ {𝑥}))) = (𝐾 (𝑆 “ (𝐼 ∖ {𝑥}))))
2315, 22ineq12d 4121 . . . . . . . . . . . 12 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → ((𝑥) ∩ (𝐾 ( “ (dom ∖ {𝑥})))) = ((𝑆𝑥) ∩ (𝐾 (𝑆 “ (𝐼 ∖ {𝑥})))))
2423eqeq1d 2761 . . . . . . . . . . 11 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → (((𝑥) ∩ (𝐾 ( “ (dom ∖ {𝑥})))) = { 0 } ↔ ((𝑆𝑥) ∩ (𝐾 (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 }))
2519, 24anbi12d 633 . . . . . . . . . 10 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → ((∀𝑦 ∈ (dom ∖ {𝑥})(𝑥) ⊆ (𝑍‘(𝑦)) ∧ ((𝑥) ∩ (𝐾 ( “ (dom ∖ {𝑥})))) = { 0 }) ↔ (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆𝑥) ⊆ (𝑍‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ (𝐾 (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 })))
2612, 25raleqbidv 3320 . . . . . . . . 9 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → (∀𝑥 ∈ dom (∀𝑦 ∈ (dom ∖ {𝑥})(𝑥) ⊆ (𝑍‘(𝑦)) ∧ ((𝑥) ∩ (𝐾 ( “ (dom ∖ {𝑥})))) = { 0 }) ↔ ∀𝑥𝐼 (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆𝑥) ⊆ (𝑍‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ (𝐾 (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 })))
2713, 26anbi12d 633 . . . . . . . 8 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → ((:dom ⟶(SubGrp‘𝐺) ∧ ∀𝑥 ∈ dom (∀𝑦 ∈ (dom ∖ {𝑥})(𝑥) ⊆ (𝑍‘(𝑦)) ∧ ((𝑥) ∩ (𝐾 ( “ (dom ∖ {𝑥})))) = { 0 })) ↔ (𝑆:𝐼⟶(SubGrp‘𝐺) ∧ ∀𝑥𝐼 (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆𝑥) ⊆ (𝑍‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ (𝐾 (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 }))))
2827adantlr 714 . . . . . . 7 ((((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ 𝑆 ∈ V) ∧ = 𝑆) → ((:dom ⟶(SubGrp‘𝐺) ∧ ∀𝑥 ∈ dom (∀𝑦 ∈ (dom ∖ {𝑥})(𝑥) ⊆ (𝑍‘(𝑦)) ∧ ((𝑥) ∩ (𝐾 ( “ (dom ∖ {𝑥})))) = { 0 })) ↔ (𝑆:𝐼⟶(SubGrp‘𝐺) ∧ ∀𝑥𝐼 (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆𝑥) ⊆ (𝑍‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ (𝐾 (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 }))))
298, 28sbcied 3742 . . . . . 6 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ 𝑆 ∈ V) → ([𝑆 / ](:dom ⟶(SubGrp‘𝐺) ∧ ∀𝑥 ∈ dom (∀𝑦 ∈ (dom ∖ {𝑥})(𝑥) ⊆ (𝑍‘(𝑦)) ∧ ((𝑥) ∩ (𝐾 ( “ (dom ∖ {𝑥})))) = { 0 })) ↔ (𝑆:𝐼⟶(SubGrp‘𝐺) ∧ ∀𝑥𝐼 (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆𝑥) ⊆ (𝑍‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ (𝐾 (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 }))))
307, 29bitr3id 288 . . . . 5 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ 𝑆 ∈ V) → (𝑆 ∈ { ∣ (:dom ⟶(SubGrp‘𝐺) ∧ ∀𝑥 ∈ dom (∀𝑦 ∈ (dom ∖ {𝑥})(𝑥) ⊆ (𝑍‘(𝑦)) ∧ ((𝑥) ∩ (𝐾 ( “ (dom ∖ {𝑥})))) = { 0 }))} ↔ (𝑆:𝐼⟶(SubGrp‘𝐺) ∧ ∀𝑥𝐼 (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆𝑥) ⊆ (𝑍‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ (𝐾 (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 }))))
3130ex 416 . . . 4 ((𝐼𝑉 ∧ dom 𝑆 = 𝐼) → (𝑆 ∈ V → (𝑆 ∈ { ∣ (:dom ⟶(SubGrp‘𝐺) ∧ ∀𝑥 ∈ dom (∀𝑦 ∈ (dom ∖ {𝑥})(𝑥) ⊆ (𝑍‘(𝑦)) ∧ ((𝑥) ∩ (𝐾 ( “ (dom ∖ {𝑥})))) = { 0 }))} ↔ (𝑆:𝐼⟶(SubGrp‘𝐺) ∧ ∀𝑥𝐼 (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆𝑥) ⊆ (𝑍‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ (𝐾 (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 })))))
322, 6, 31pm5.21ndd 384 . . 3 ((𝐼𝑉 ∧ dom 𝑆 = 𝐼) → (𝑆 ∈ { ∣ (:dom ⟶(SubGrp‘𝐺) ∧ ∀𝑥 ∈ dom (∀𝑦 ∈ (dom ∖ {𝑥})(𝑥) ⊆ (𝑍‘(𝑦)) ∧ ((𝑥) ∩ (𝐾 ( “ (dom ∖ {𝑥})))) = { 0 }))} ↔ (𝑆:𝐼⟶(SubGrp‘𝐺) ∧ ∀𝑥𝐼 (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆𝑥) ⊆ (𝑍‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ (𝐾 (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 }))))
3332anbi2d 631 . 2 ((𝐼𝑉 ∧ dom 𝑆 = 𝐼) → ((𝐺 ∈ Grp ∧ 𝑆 ∈ { ∣ (:dom ⟶(SubGrp‘𝐺) ∧ ∀𝑥 ∈ dom (∀𝑦 ∈ (dom ∖ {𝑥})(𝑥) ⊆ (𝑍‘(𝑦)) ∧ ((𝑥) ∩ (𝐾 ( “ (dom ∖ {𝑥})))) = { 0 }))}) ↔ (𝐺 ∈ Grp ∧ (𝑆:𝐼⟶(SubGrp‘𝐺) ∧ ∀𝑥𝐼 (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆𝑥) ⊆ (𝑍‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ (𝐾 (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 })))))
34 df-br 5038 . . 3 (𝐺dom DProd 𝑆 ↔ ⟨𝐺, 𝑆⟩ ∈ dom DProd )
35 fvex 6677 . . . . . . . . . . 11 (𝑠𝑥) ∈ V
3635rgenw 3083 . . . . . . . . . 10 𝑥 ∈ dom 𝑠(𝑠𝑥) ∈ V
37 ixpexg 8518 . . . . . . . . . 10 (∀𝑥 ∈ dom 𝑠(𝑠𝑥) ∈ V → X𝑥 ∈ dom 𝑠(𝑠𝑥) ∈ V)
3836, 37ax-mp 5 . . . . . . . . 9 X𝑥 ∈ dom 𝑠(𝑠𝑥) ∈ V
3938mptrabex 6986 . . . . . . . 8 (𝑓 ∈ {X𝑥 ∈ dom 𝑠(𝑠𝑥) ∣ finSupp (0g𝑔)} ↦ (𝑔 Σg 𝑓)) ∈ V
4039rnex 7629 . . . . . . 7 ran (𝑓 ∈ {X𝑥 ∈ dom 𝑠(𝑠𝑥) ∣ finSupp (0g𝑔)} ↦ (𝑔 Σg 𝑓)) ∈ V
4140rgen2w 3084 . . . . . 6 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)) ∈ V
42 df-dprd 19200 . . . . . . 7 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)))
4342fmpox 7776 . . . . . 6 (∀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)) ∈ V ↔ DProd : 𝑔 ∈ Grp ({𝑔} × { ∣ (:dom ⟶(SubGrp‘𝑔) ∧ ∀𝑥 ∈ dom (∀𝑦 ∈ (dom ∖ {𝑥})(𝑥) ⊆ ((Cntz‘𝑔)‘(𝑦)) ∧ ((𝑥) ∩ ((mrCls‘(SubGrp‘𝑔))‘ ( “ (dom ∖ {𝑥})))) = {(0g𝑔)}))})⟶V)
4441, 43mpbi 233 . . . . 5 DProd : 𝑔 ∈ Grp ({𝑔} × { ∣ (:dom ⟶(SubGrp‘𝑔) ∧ ∀𝑥 ∈ dom (∀𝑦 ∈ (dom ∖ {𝑥})(𝑥) ⊆ ((Cntz‘𝑔)‘(𝑦)) ∧ ((𝑥) ∩ ((mrCls‘(SubGrp‘𝑔))‘ ( “ (dom ∖ {𝑥})))) = {(0g𝑔)}))})⟶V
4544fdmi 6515 . . . 4 dom DProd = 𝑔 ∈ Grp ({𝑔} × { ∣ (:dom ⟶(SubGrp‘𝑔) ∧ ∀𝑥 ∈ dom (∀𝑦 ∈ (dom ∖ {𝑥})(𝑥) ⊆ ((Cntz‘𝑔)‘(𝑦)) ∧ ((𝑥) ∩ ((mrCls‘(SubGrp‘𝑔))‘ ( “ (dom ∖ {𝑥})))) = {(0g𝑔)}))})
4645eleq2i 2844 . . 3 (⟨𝐺, 𝑆⟩ ∈ dom DProd ↔ ⟨𝐺, 𝑆⟩ ∈ 𝑔 ∈ Grp ({𝑔} × { ∣ (:dom ⟶(SubGrp‘𝑔) ∧ ∀𝑥 ∈ dom (∀𝑦 ∈ (dom ∖ {𝑥})(𝑥) ⊆ ((Cntz‘𝑔)‘(𝑦)) ∧ ((𝑥) ∩ ((mrCls‘(SubGrp‘𝑔))‘ ( “ (dom ∖ {𝑥})))) = {(0g𝑔)}))}))
47 fveq2 6664 . . . . . . 7 (𝑔 = 𝐺 → (SubGrp‘𝑔) = (SubGrp‘𝐺))
4847feq3d 6491 . . . . . 6 (𝑔 = 𝐺 → (:dom ⟶(SubGrp‘𝑔) ↔ :dom ⟶(SubGrp‘𝐺)))
49 fveq2 6664 . . . . . . . . . . . 12 (𝑔 = 𝐺 → (Cntz‘𝑔) = (Cntz‘𝐺))
50 dmdprd.z . . . . . . . . . . . 12 𝑍 = (Cntz‘𝐺)
5149, 50eqtr4di 2812 . . . . . . . . . . 11 (𝑔 = 𝐺 → (Cntz‘𝑔) = 𝑍)
5251fveq1d 6666 . . . . . . . . . 10 (𝑔 = 𝐺 → ((Cntz‘𝑔)‘(𝑦)) = (𝑍‘(𝑦)))
5352sseq2d 3927 . . . . . . . . 9 (𝑔 = 𝐺 → ((𝑥) ⊆ ((Cntz‘𝑔)‘(𝑦)) ↔ (𝑥) ⊆ (𝑍‘(𝑦))))
5453ralbidv 3127 . . . . . . . 8 (𝑔 = 𝐺 → (∀𝑦 ∈ (dom ∖ {𝑥})(𝑥) ⊆ ((Cntz‘𝑔)‘(𝑦)) ↔ ∀𝑦 ∈ (dom ∖ {𝑥})(𝑥) ⊆ (𝑍‘(𝑦))))
5547fveq2d 6668 . . . . . . . . . . . 12 (𝑔 = 𝐺 → (mrCls‘(SubGrp‘𝑔)) = (mrCls‘(SubGrp‘𝐺)))
56 dmdprd.k . . . . . . . . . . . 12 𝐾 = (mrCls‘(SubGrp‘𝐺))
5755, 56eqtr4di 2812 . . . . . . . . . . 11 (𝑔 = 𝐺 → (mrCls‘(SubGrp‘𝑔)) = 𝐾)
5857fveq1d 6666 . . . . . . . . . 10 (𝑔 = 𝐺 → ((mrCls‘(SubGrp‘𝑔))‘ ( “ (dom ∖ {𝑥}))) = (𝐾 ( “ (dom ∖ {𝑥}))))
5958ineq2d 4120 . . . . . . . . 9 (𝑔 = 𝐺 → ((𝑥) ∩ ((mrCls‘(SubGrp‘𝑔))‘ ( “ (dom ∖ {𝑥})))) = ((𝑥) ∩ (𝐾 ( “ (dom ∖ {𝑥})))))
60 fveq2 6664 . . . . . . . . . . 11 (𝑔 = 𝐺 → (0g𝑔) = (0g𝐺))
61 dmdprd.0 . . . . . . . . . . 11 0 = (0g𝐺)
6260, 61eqtr4di 2812 . . . . . . . . . 10 (𝑔 = 𝐺 → (0g𝑔) = 0 )
6362sneqd 4538 . . . . . . . . 9 (𝑔 = 𝐺 → {(0g𝑔)} = { 0 })
6459, 63eqeq12d 2775 . . . . . . . 8 (𝑔 = 𝐺 → (((𝑥) ∩ ((mrCls‘(SubGrp‘𝑔))‘ ( “ (dom ∖ {𝑥})))) = {(0g𝑔)} ↔ ((𝑥) ∩ (𝐾 ( “ (dom ∖ {𝑥})))) = { 0 }))
6554, 64anbi12d 633 . . . . . . 7 (𝑔 = 𝐺 → ((∀𝑦 ∈ (dom ∖ {𝑥})(𝑥) ⊆ ((Cntz‘𝑔)‘(𝑦)) ∧ ((𝑥) ∩ ((mrCls‘(SubGrp‘𝑔))‘ ( “ (dom ∖ {𝑥})))) = {(0g𝑔)}) ↔ (∀𝑦 ∈ (dom ∖ {𝑥})(𝑥) ⊆ (𝑍‘(𝑦)) ∧ ((𝑥) ∩ (𝐾 ( “ (dom ∖ {𝑥})))) = { 0 })))
6665ralbidv 3127 . . . . . 6 (𝑔 = 𝐺 → (∀𝑥 ∈ dom (∀𝑦 ∈ (dom ∖ {𝑥})(𝑥) ⊆ ((Cntz‘𝑔)‘(𝑦)) ∧ ((𝑥) ∩ ((mrCls‘(SubGrp‘𝑔))‘ ( “ (dom ∖ {𝑥})))) = {(0g𝑔)}) ↔ ∀𝑥 ∈ dom (∀𝑦 ∈ (dom ∖ {𝑥})(𝑥) ⊆ (𝑍‘(𝑦)) ∧ ((𝑥) ∩ (𝐾 ( “ (dom ∖ {𝑥})))) = { 0 })))
6748, 66anbi12d 633 . . . . 5 (𝑔 = 𝐺 → ((:dom ⟶(SubGrp‘𝑔) ∧ ∀𝑥 ∈ dom (∀𝑦 ∈ (dom ∖ {𝑥})(𝑥) ⊆ ((Cntz‘𝑔)‘(𝑦)) ∧ ((𝑥) ∩ ((mrCls‘(SubGrp‘𝑔))‘ ( “ (dom ∖ {𝑥})))) = {(0g𝑔)})) ↔ (:dom ⟶(SubGrp‘𝐺) ∧ ∀𝑥 ∈ dom (∀𝑦 ∈ (dom ∖ {𝑥})(𝑥) ⊆ (𝑍‘(𝑦)) ∧ ((𝑥) ∩ (𝐾 ( “ (dom ∖ {𝑥})))) = { 0 }))))
6867abbidv 2823 . . . 4 (𝑔 = 𝐺 → { ∣ (:dom ⟶(SubGrp‘𝑔) ∧ ∀𝑥 ∈ dom (∀𝑦 ∈ (dom ∖ {𝑥})(𝑥) ⊆ ((Cntz‘𝑔)‘(𝑦)) ∧ ((𝑥) ∩ ((mrCls‘(SubGrp‘𝑔))‘ ( “ (dom ∖ {𝑥})))) = {(0g𝑔)}))} = { ∣ (:dom ⟶(SubGrp‘𝐺) ∧ ∀𝑥 ∈ dom (∀𝑦 ∈ (dom ∖ {𝑥})(𝑥) ⊆ (𝑍‘(𝑦)) ∧ ((𝑥) ∩ (𝐾 ( “ (dom ∖ {𝑥})))) = { 0 }))})
6968opeliunxp2 5685 . . 3 (⟨𝐺, 𝑆⟩ ∈ 𝑔 ∈ Grp ({𝑔} × { ∣ (:dom ⟶(SubGrp‘𝑔) ∧ ∀𝑥 ∈ dom (∀𝑦 ∈ (dom ∖ {𝑥})(𝑥) ⊆ ((Cntz‘𝑔)‘(𝑦)) ∧ ((𝑥) ∩ ((mrCls‘(SubGrp‘𝑔))‘ ( “ (dom ∖ {𝑥})))) = {(0g𝑔)}))}) ↔ (𝐺 ∈ Grp ∧ 𝑆 ∈ { ∣ (:dom ⟶(SubGrp‘𝐺) ∧ ∀𝑥 ∈ dom (∀𝑦 ∈ (dom ∖ {𝑥})(𝑥) ⊆ (𝑍‘(𝑦)) ∧ ((𝑥) ∩ (𝐾 ( “ (dom ∖ {𝑥})))) = { 0 }))}))
7034, 46, 693bitri 300 . 2 (𝐺dom DProd 𝑆 ↔ (𝐺 ∈ Grp ∧ 𝑆 ∈ { ∣ (:dom ⟶(SubGrp‘𝐺) ∧ ∀𝑥 ∈ dom (∀𝑦 ∈ (dom ∖ {𝑥})(𝑥) ⊆ (𝑍‘(𝑦)) ∧ ((𝑥) ∩ (𝐾 ( “ (dom ∖ {𝑥})))) = { 0 }))}))
71 3anass 1093 . 2 ((𝐺 ∈ Grp ∧ 𝑆:𝐼⟶(SubGrp‘𝐺) ∧ ∀𝑥𝐼 (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆𝑥) ⊆ (𝑍‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ (𝐾 (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 })) ↔ (𝐺 ∈ Grp ∧ (𝑆:𝐼⟶(SubGrp‘𝐺) ∧ ∀𝑥𝐼 (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆𝑥) ⊆ (𝑍‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ (𝐾 (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 }))))
7233, 70, 713bitr4g 317 1 ((𝐼𝑉 ∧ dom 𝑆 = 𝐼) → (𝐺dom DProd 𝑆 ↔ (𝐺 ∈ Grp ∧ 𝑆:𝐼⟶(SubGrp‘𝐺) ∧ ∀𝑥𝐼 (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆𝑥) ⊆ (𝑍‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ (𝐾 (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 }))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1085   = wceq 1539  wcel 2112  {cab 2736  wral 3071  {crab 3075  Vcvv 3410  [wsbc 3699  cdif 3858  cin 3860  wss 3861  {csn 4526  cop 4532   cuni 4802   ciun 4887   class class class wbr 5037  cmpt 5117   × cxp 5527  dom cdm 5529  ran crn 5530  cima 5532  wf 6337  cfv 6341  (class class class)co 7157  Xcixp 8493   finSupp cfsupp 8880  0gc0g 16786   Σg cgsu 16787  mrClscmrc 16927  Grpcgrp 18184  SubGrpcsubg 18355  Cntzccntz 18527   DProd cdprd 19198
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5161  ax-sep 5174  ax-nul 5181  ax-pow 5239  ax-pr 5303  ax-un 7466
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-reu 3078  df-rab 3080  df-v 3412  df-sbc 3700  df-csb 3809  df-dif 3864  df-un 3866  df-in 3868  df-ss 3878  df-nul 4229  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4803  df-iun 4889  df-br 5038  df-opab 5100  df-mpt 5118  df-id 5435  df-xp 5535  df-rel 5536  df-cnv 5537  df-co 5538  df-dm 5539  df-rn 5540  df-res 5541  df-ima 5542  df-iota 6300  df-fun 6343  df-fn 6344  df-f 6345  df-f1 6346  df-fo 6347  df-f1o 6348  df-fv 6349  df-oprab 7161  df-mpo 7162  df-1st 7700  df-2nd 7701  df-ixp 8494  df-dprd 19200
This theorem is referenced by:  dmdprdd  19204  dprdgrp  19210  dprdf  19211  dprdcntz  19213  dprddisj  19214  dprdres  19233  subgdmdprd  19239
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