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Theorem dmdprd 19827
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 3491 . . . . 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 7212 . . . . . . 7 ((𝑆:𝐼⟶(SubGrp‘𝐺) ∧ 𝐼𝑉) → 𝑆 ∈ V)
43expcom 414 . . . . . 6 (𝐼𝑉 → (𝑆:𝐼⟶(SubGrp‘𝐺) → 𝑆 ∈ V))
54adantr 481 . . . . 5 ((𝐼𝑉 ∧ dom 𝑆 = 𝐼) → (𝑆:𝐼⟶(SubGrp‘𝐺) → 𝑆 ∈ V))
65adantrd 492 . . . 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 3774 . . . . . 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 485 . . . . . . 7 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ 𝑆 ∈ V) → 𝑆 ∈ V)
9 simpr 485 . . . . . . . . . 10 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → = 𝑆)
109dmeqd 5897 . . . . . . . . . . 11 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → dom = dom 𝑆)
11 simplr 767 . . . . . . . . . . 11 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → dom 𝑆 = 𝐼)
1210, 11eqtrd 2771 . . . . . . . . . 10 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → dom = 𝐼)
139, 12feq12d 6692 . . . . . . . . 9 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → (:dom ⟶(SubGrp‘𝐺) ↔ 𝑆:𝐼⟶(SubGrp‘𝐺)))
1412difeq1d 4117 . . . . . . . . . . . 12 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → (dom ∖ {𝑥}) = (𝐼 ∖ {𝑥}))
159fveq1d 6880 . . . . . . . . . . . . 13 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → (𝑥) = (𝑆𝑥))
169fveq1d 6880 . . . . . . . . . . . . . 14 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → (𝑦) = (𝑆𝑦))
1716fveq2d 6882 . . . . . . . . . . . . 13 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → (𝑍‘(𝑦)) = (𝑍‘(𝑆𝑦)))
1815, 17sseq12d 4011 . . . . . . . . . . . 12 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → ((𝑥) ⊆ (𝑍‘(𝑦)) ↔ (𝑆𝑥) ⊆ (𝑍‘(𝑆𝑦))))
1914, 18raleqbidv 3341 . . . . . . . . . . 11 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → (∀𝑦 ∈ (dom ∖ {𝑥})(𝑥) ⊆ (𝑍‘(𝑦)) ↔ ∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆𝑥) ⊆ (𝑍‘(𝑆𝑦))))
209, 14imaeq12d 6050 . . . . . . . . . . . . . . 15 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → ( “ (dom ∖ {𝑥})) = (𝑆 “ (𝐼 ∖ {𝑥})))
2120unieqd 4915 . . . . . . . . . . . . . 14 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → ( “ (dom ∖ {𝑥})) = (𝑆 “ (𝐼 ∖ {𝑥})))
2221fveq2d 6882 . . . . . . . . . . . . 13 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → (𝐾 ( “ (dom ∖ {𝑥}))) = (𝐾 (𝑆 “ (𝐼 ∖ {𝑥}))))
2315, 22ineq12d 4209 . . . . . . . . . . . 12 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → ((𝑥) ∩ (𝐾 ( “ (dom ∖ {𝑥})))) = ((𝑆𝑥) ∩ (𝐾 (𝑆 “ (𝐼 ∖ {𝑥})))))
2423eqeq1d 2733 . . . . . . . . . . 11 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → (((𝑥) ∩ (𝐾 ( “ (dom ∖ {𝑥})))) = { 0 } ↔ ((𝑆𝑥) ∩ (𝐾 (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 }))
2519, 24anbi12d 631 . . . . . . . . . 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 3341 . . . . . . . . 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 631 . . . . . . . 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 713 . . . . . . 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 3818 . . . . . 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 284 . . . . 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 413 . . . 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 380 . . 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 629 . 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 5142 . . 3 (𝐺dom DProd 𝑆 ↔ ⟨𝐺, 𝑆⟩ ∈ dom DProd )
35 fvex 6891 . . . . . . . . . . 11 (𝑠𝑥) ∈ V
3635rgenw 3064 . . . . . . . . . 10 𝑥 ∈ dom 𝑠(𝑠𝑥) ∈ V
37 ixpexg 8899 . . . . . . . . . 10 (∀𝑥 ∈ dom 𝑠(𝑠𝑥) ∈ V → X𝑥 ∈ dom 𝑠(𝑠𝑥) ∈ V)
3836, 37ax-mp 5 . . . . . . . . 9 X𝑥 ∈ dom 𝑠(𝑠𝑥) ∈ V
3938mptrabex 7211 . . . . . . . 8 (𝑓 ∈ {X𝑥 ∈ dom 𝑠(𝑠𝑥) ∣ finSupp (0g𝑔)} ↦ (𝑔 Σg 𝑓)) ∈ V
4039rnex 7885 . . . . . . 7 ran (𝑓 ∈ {X𝑥 ∈ dom 𝑠(𝑠𝑥) ∣ finSupp (0g𝑔)} ↦ (𝑔 Σg 𝑓)) ∈ V
4140rgen2w 3065 . . . . . 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 19824 . . . . . . 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 8035 . . . . . 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 229 . . . . 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 6716 . . . 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 2824 . . 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 6878 . . . . . . 7 (𝑔 = 𝐺 → (SubGrp‘𝑔) = (SubGrp‘𝐺))
4847feq3d 6691 . . . . . 6 (𝑔 = 𝐺 → (:dom ⟶(SubGrp‘𝑔) ↔ :dom ⟶(SubGrp‘𝐺)))
49 fveq2 6878 . . . . . . . . . . . 12 (𝑔 = 𝐺 → (Cntz‘𝑔) = (Cntz‘𝐺))
50 dmdprd.z . . . . . . . . . . . 12 𝑍 = (Cntz‘𝐺)
5149, 50eqtr4di 2789 . . . . . . . . . . 11 (𝑔 = 𝐺 → (Cntz‘𝑔) = 𝑍)
5251fveq1d 6880 . . . . . . . . . 10 (𝑔 = 𝐺 → ((Cntz‘𝑔)‘(𝑦)) = (𝑍‘(𝑦)))
5352sseq2d 4010 . . . . . . . . 9 (𝑔 = 𝐺 → ((𝑥) ⊆ ((Cntz‘𝑔)‘(𝑦)) ↔ (𝑥) ⊆ (𝑍‘(𝑦))))
5453ralbidv 3176 . . . . . . . 8 (𝑔 = 𝐺 → (∀𝑦 ∈ (dom ∖ {𝑥})(𝑥) ⊆ ((Cntz‘𝑔)‘(𝑦)) ↔ ∀𝑦 ∈ (dom ∖ {𝑥})(𝑥) ⊆ (𝑍‘(𝑦))))
5547fveq2d 6882 . . . . . . . . . . . 12 (𝑔 = 𝐺 → (mrCls‘(SubGrp‘𝑔)) = (mrCls‘(SubGrp‘𝐺)))
56 dmdprd.k . . . . . . . . . . . 12 𝐾 = (mrCls‘(SubGrp‘𝐺))
5755, 56eqtr4di 2789 . . . . . . . . . . 11 (𝑔 = 𝐺 → (mrCls‘(SubGrp‘𝑔)) = 𝐾)
5857fveq1d 6880 . . . . . . . . . 10 (𝑔 = 𝐺 → ((mrCls‘(SubGrp‘𝑔))‘ ( “ (dom ∖ {𝑥}))) = (𝐾 ( “ (dom ∖ {𝑥}))))
5958ineq2d 4208 . . . . . . . . 9 (𝑔 = 𝐺 → ((𝑥) ∩ ((mrCls‘(SubGrp‘𝑔))‘ ( “ (dom ∖ {𝑥})))) = ((𝑥) ∩ (𝐾 ( “ (dom ∖ {𝑥})))))
60 fveq2 6878 . . . . . . . . . . 11 (𝑔 = 𝐺 → (0g𝑔) = (0g𝐺))
61 dmdprd.0 . . . . . . . . . . 11 0 = (0g𝐺)
6260, 61eqtr4di 2789 . . . . . . . . . 10 (𝑔 = 𝐺 → (0g𝑔) = 0 )
6362sneqd 4634 . . . . . . . . 9 (𝑔 = 𝐺 → {(0g𝑔)} = { 0 })
6459, 63eqeq12d 2747 . . . . . . . 8 (𝑔 = 𝐺 → (((𝑥) ∩ ((mrCls‘(SubGrp‘𝑔))‘ ( “ (dom ∖ {𝑥})))) = {(0g𝑔)} ↔ ((𝑥) ∩ (𝐾 ( “ (dom ∖ {𝑥})))) = { 0 }))
6554, 64anbi12d 631 . . . . . . 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 3176 . . . . . 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 631 . . . . 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 2800 . . . 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 5830 . . 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 296 . 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 1095 . 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 313 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 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  {cab 2708  wral 3060  {crab 3431  Vcvv 3473  [wsbc 3773  cdif 3941  cin 3943  wss 3944  {csn 4622  cop 4628   cuni 4901   ciun 4990   class class class wbr 5141  cmpt 5224   × cxp 5667  dom cdm 5669  ran crn 5670  cima 5672  wf 6528  cfv 6532  (class class class)co 7393  Xcixp 8874   finSupp cfsupp 9344  0gc0g 17367   Σg cgsu 17368  mrClscmrc 17509  Grpcgrp 18794  SubGrpcsubg 18972  Cntzccntz 19145   DProd cdprd 19822
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-oprab 7397  df-mpo 7398  df-1st 7957  df-2nd 7958  df-ixp 8875  df-dprd 19824
This theorem is referenced by:  dmdprdd  19828  dprdgrp  19834  dprdf  19835  dprdcntz  19837  dprddisj  19838  dprdres  19857  subgdmdprd  19863
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