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Theorem dmdprd 19703
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 3461 . . . . 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 7170 . . . . . . 7 ((𝑆:𝐼⟶(SubGrp‘𝐺) ∧ 𝐼𝑉) → 𝑆 ∈ V)
43expcom 415 . . . . . 6 (𝐼𝑉 → (𝑆:𝐼⟶(SubGrp‘𝐺) → 𝑆 ∈ V))
54adantr 482 . . . . 5 ((𝐼𝑉 ∧ dom 𝑆 = 𝐼) → (𝑆:𝐼⟶(SubGrp‘𝐺) → 𝑆 ∈ V))
65adantrd 493 . . . 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 3738 . . . . . 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 486 . . . . . . 7 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ 𝑆 ∈ V) → 𝑆 ∈ V)
9 simpr 486 . . . . . . . . . 10 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → = 𝑆)
109dmeqd 5857 . . . . . . . . . . 11 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → dom = dom 𝑆)
11 simplr 767 . . . . . . . . . . 11 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → dom 𝑆 = 𝐼)
1210, 11eqtrd 2777 . . . . . . . . . 10 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → dom = 𝐼)
139, 12feq12d 6651 . . . . . . . . 9 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → (:dom ⟶(SubGrp‘𝐺) ↔ 𝑆:𝐼⟶(SubGrp‘𝐺)))
1412difeq1d 4079 . . . . . . . . . . . 12 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → (dom ∖ {𝑥}) = (𝐼 ∖ {𝑥}))
159fveq1d 6839 . . . . . . . . . . . . 13 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → (𝑥) = (𝑆𝑥))
169fveq1d 6839 . . . . . . . . . . . . . 14 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → (𝑦) = (𝑆𝑦))
1716fveq2d 6841 . . . . . . . . . . . . 13 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → (𝑍‘(𝑦)) = (𝑍‘(𝑆𝑦)))
1815, 17sseq12d 3975 . . . . . . . . . . . 12 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → ((𝑥) ⊆ (𝑍‘(𝑦)) ↔ (𝑆𝑥) ⊆ (𝑍‘(𝑆𝑦))))
1914, 18raleqbidv 3317 . . . . . . . . . . 11 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → (∀𝑦 ∈ (dom ∖ {𝑥})(𝑥) ⊆ (𝑍‘(𝑦)) ↔ ∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆𝑥) ⊆ (𝑍‘(𝑆𝑦))))
209, 14imaeq12d 6010 . . . . . . . . . . . . . . 15 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → ( “ (dom ∖ {𝑥})) = (𝑆 “ (𝐼 ∖ {𝑥})))
2120unieqd 4877 . . . . . . . . . . . . . 14 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → ( “ (dom ∖ {𝑥})) = (𝑆 “ (𝐼 ∖ {𝑥})))
2221fveq2d 6841 . . . . . . . . . . . . 13 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → (𝐾 ( “ (dom ∖ {𝑥}))) = (𝐾 (𝑆 “ (𝐼 ∖ {𝑥}))))
2315, 22ineq12d 4171 . . . . . . . . . . . 12 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → ((𝑥) ∩ (𝐾 ( “ (dom ∖ {𝑥})))) = ((𝑆𝑥) ∩ (𝐾 (𝑆 “ (𝐼 ∖ {𝑥})))))
2423eqeq1d 2739 . . . . . . . . . . 11 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → (((𝑥) ∩ (𝐾 ( “ (dom ∖ {𝑥})))) = { 0 } ↔ ((𝑆𝑥) ∩ (𝐾 (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 }))
2519, 24anbi12d 632 . . . . . . . . . 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 3317 . . . . . . . . 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 632 . . . . . . . 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 3782 . . . . . 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 285 . . . . 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 414 . . . 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 381 . . 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 630 . 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 5104 . . 3 (𝐺dom DProd 𝑆 ↔ ⟨𝐺, 𝑆⟩ ∈ dom DProd )
35 fvex 6850 . . . . . . . . . . 11 (𝑠𝑥) ∈ V
3635rgenw 3066 . . . . . . . . . 10 𝑥 ∈ dom 𝑠(𝑠𝑥) ∈ V
37 ixpexg 8793 . . . . . . . . . 10 (∀𝑥 ∈ dom 𝑠(𝑠𝑥) ∈ V → X𝑥 ∈ dom 𝑠(𝑠𝑥) ∈ V)
3836, 37ax-mp 5 . . . . . . . . 9 X𝑥 ∈ dom 𝑠(𝑠𝑥) ∈ V
3938mptrabex 7169 . . . . . . . 8 (𝑓 ∈ {X𝑥 ∈ dom 𝑠(𝑠𝑥) ∣ finSupp (0g𝑔)} ↦ (𝑔 Σg 𝑓)) ∈ V
4039rnex 7839 . . . . . . 7 ran (𝑓 ∈ {X𝑥 ∈ dom 𝑠(𝑠𝑥) ∣ finSupp (0g𝑔)} ↦ (𝑔 Σg 𝑓)) ∈ V
4140rgen2w 3067 . . . . . 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 19700 . . . . . . 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 7987 . . . . . 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 6675 . . . 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 2829 . . 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 6837 . . . . . . 7 (𝑔 = 𝐺 → (SubGrp‘𝑔) = (SubGrp‘𝐺))
4847feq3d 6650 . . . . . 6 (𝑔 = 𝐺 → (:dom ⟶(SubGrp‘𝑔) ↔ :dom ⟶(SubGrp‘𝐺)))
49 fveq2 6837 . . . . . . . . . . . 12 (𝑔 = 𝐺 → (Cntz‘𝑔) = (Cntz‘𝐺))
50 dmdprd.z . . . . . . . . . . . 12 𝑍 = (Cntz‘𝐺)
5149, 50eqtr4di 2795 . . . . . . . . . . 11 (𝑔 = 𝐺 → (Cntz‘𝑔) = 𝑍)
5251fveq1d 6839 . . . . . . . . . 10 (𝑔 = 𝐺 → ((Cntz‘𝑔)‘(𝑦)) = (𝑍‘(𝑦)))
5352sseq2d 3974 . . . . . . . . 9 (𝑔 = 𝐺 → ((𝑥) ⊆ ((Cntz‘𝑔)‘(𝑦)) ↔ (𝑥) ⊆ (𝑍‘(𝑦))))
5453ralbidv 3172 . . . . . . . 8 (𝑔 = 𝐺 → (∀𝑦 ∈ (dom ∖ {𝑥})(𝑥) ⊆ ((Cntz‘𝑔)‘(𝑦)) ↔ ∀𝑦 ∈ (dom ∖ {𝑥})(𝑥) ⊆ (𝑍‘(𝑦))))
5547fveq2d 6841 . . . . . . . . . . . 12 (𝑔 = 𝐺 → (mrCls‘(SubGrp‘𝑔)) = (mrCls‘(SubGrp‘𝐺)))
56 dmdprd.k . . . . . . . . . . . 12 𝐾 = (mrCls‘(SubGrp‘𝐺))
5755, 56eqtr4di 2795 . . . . . . . . . . 11 (𝑔 = 𝐺 → (mrCls‘(SubGrp‘𝑔)) = 𝐾)
5857fveq1d 6839 . . . . . . . . . 10 (𝑔 = 𝐺 → ((mrCls‘(SubGrp‘𝑔))‘ ( “ (dom ∖ {𝑥}))) = (𝐾 ( “ (dom ∖ {𝑥}))))
5958ineq2d 4170 . . . . . . . . 9 (𝑔 = 𝐺 → ((𝑥) ∩ ((mrCls‘(SubGrp‘𝑔))‘ ( “ (dom ∖ {𝑥})))) = ((𝑥) ∩ (𝐾 ( “ (dom ∖ {𝑥})))))
60 fveq2 6837 . . . . . . . . . . 11 (𝑔 = 𝐺 → (0g𝑔) = (0g𝐺))
61 dmdprd.0 . . . . . . . . . . 11 0 = (0g𝐺)
6260, 61eqtr4di 2795 . . . . . . . . . 10 (𝑔 = 𝐺 → (0g𝑔) = 0 )
6362sneqd 4596 . . . . . . . . 9 (𝑔 = 𝐺 → {(0g𝑔)} = { 0 })
6459, 63eqeq12d 2753 . . . . . . . 8 (𝑔 = 𝐺 → (((𝑥) ∩ ((mrCls‘(SubGrp‘𝑔))‘ ( “ (dom ∖ {𝑥})))) = {(0g𝑔)} ↔ ((𝑥) ∩ (𝐾 ( “ (dom ∖ {𝑥})))) = { 0 }))
6554, 64anbi12d 632 . . . . . . 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 3172 . . . . . 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 632 . . . . 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 2806 . . . 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 5790 . . 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 297 . 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 314 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 397  w3a 1087   = wceq 1541  wcel 2106  {cab 2714  wral 3062  {crab 3405  Vcvv 3443  [wsbc 3737  cdif 3905  cin 3907  wss 3908  {csn 4584  cop 4590   cuni 4863   ciun 4952   class class class wbr 5103  cmpt 5186   × cxp 5628  dom cdm 5630  ran crn 5631  cima 5633  wf 6487  cfv 6491  (class class class)co 7349  Xcixp 8768   finSupp cfsupp 9238  0gc0g 17255   Σg cgsu 17256  mrClscmrc 17397  Grpcgrp 18681  SubGrpcsubg 18853  Cntzccntz 19025   DProd cdprd 19698
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 2708  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7662
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5528  df-xp 5636  df-rel 5637  df-cnv 5638  df-co 5639  df-dm 5640  df-rn 5641  df-res 5642  df-ima 5643  df-iota 6443  df-fun 6493  df-fn 6494  df-f 6495  df-f1 6496  df-fo 6497  df-f1o 6498  df-fv 6499  df-oprab 7353  df-mpo 7354  df-1st 7911  df-2nd 7912  df-ixp 8769  df-dprd 19700
This theorem is referenced by:  dmdprdd  19704  dprdgrp  19710  dprdf  19711  dprdcntz  19713  dprddisj  19714  dprdres  19733  subgdmdprd  19739
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