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Theorem dmdprd 18220
 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 3185 . . . . 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 6394 . . . . . . 7 ((𝑆:𝐼⟶(SubGrp‘𝐺) ∧ 𝐼𝑉) → 𝑆 ∈ V)
43expcom 450 . . . . . 6 (𝐼𝑉 → (𝑆:𝐼⟶(SubGrp‘𝐺) → 𝑆 ∈ V))
54adantr 480 . . . . 5 ((𝐼𝑉 ∧ dom 𝑆 = 𝐼) → (𝑆:𝐼⟶(SubGrp‘𝐺) → 𝑆 ∈ V))
65adantrd 483 . . . 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 3403 . . . . . 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 476 . . . . . . 7 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ 𝑆 ∈ V) → 𝑆 ∈ V)
9 simpr 476 . . . . . . . . . 10 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → = 𝑆)
109dmeqd 5248 . . . . . . . . . . 11 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → dom = dom 𝑆)
11 simplr 788 . . . . . . . . . . 11 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → dom 𝑆 = 𝐼)
1210, 11eqtrd 2644 . . . . . . . . . 10 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → dom = 𝐼)
139, 12feq12d 5946 . . . . . . . . 9 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → (:dom ⟶(SubGrp‘𝐺) ↔ 𝑆:𝐼⟶(SubGrp‘𝐺)))
1412difeq1d 3689 . . . . . . . . . . . 12 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → (dom ∖ {𝑥}) = (𝐼 ∖ {𝑥}))
159fveq1d 6105 . . . . . . . . . . . . 13 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → (𝑥) = (𝑆𝑥))
169fveq1d 6105 . . . . . . . . . . . . . 14 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → (𝑦) = (𝑆𝑦))
1716fveq2d 6107 . . . . . . . . . . . . 13 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → (𝑍‘(𝑦)) = (𝑍‘(𝑆𝑦)))
1815, 17sseq12d 3597 . . . . . . . . . . . 12 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → ((𝑥) ⊆ (𝑍‘(𝑦)) ↔ (𝑆𝑥) ⊆ (𝑍‘(𝑆𝑦))))
1914, 18raleqbidv 3129 . . . . . . . . . . 11 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → (∀𝑦 ∈ (dom ∖ {𝑥})(𝑥) ⊆ (𝑍‘(𝑦)) ↔ ∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆𝑥) ⊆ (𝑍‘(𝑆𝑦))))
209, 14imaeq12d 5386 . . . . . . . . . . . . . . 15 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → ( “ (dom ∖ {𝑥})) = (𝑆 “ (𝐼 ∖ {𝑥})))
2120unieqd 4382 . . . . . . . . . . . . . 14 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → ( “ (dom ∖ {𝑥})) = (𝑆 “ (𝐼 ∖ {𝑥})))
2221fveq2d 6107 . . . . . . . . . . . . 13 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → (𝐾 ( “ (dom ∖ {𝑥}))) = (𝐾 (𝑆 “ (𝐼 ∖ {𝑥}))))
2315, 22ineq12d 3777 . . . . . . . . . . . 12 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → ((𝑥) ∩ (𝐾 ( “ (dom ∖ {𝑥})))) = ((𝑆𝑥) ∩ (𝐾 (𝑆 “ (𝐼 ∖ {𝑥})))))
2423eqeq1d 2612 . . . . . . . . . . 11 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → (((𝑥) ∩ (𝐾 ( “ (dom ∖ {𝑥})))) = { 0 } ↔ ((𝑆𝑥) ∩ (𝐾 (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 }))
2519, 24anbi12d 743 . . . . . . . . . 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 3129 . . . . . . . . 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 743 . . . . . . . 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 747 . . . . . . 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 3439 . . . . . 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, 29syl5bbr 273 . . . . 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 449 . . . 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 368 . . 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 736 . 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 4584 . . 3 (𝐺dom DProd 𝑆 ↔ ⟨𝐺, 𝑆⟩ ∈ dom DProd )
35 fvex 6113 . . . . . . . . . . 11 (𝑠𝑥) ∈ V
3635rgenw 2908 . . . . . . . . . 10 𝑥 ∈ dom 𝑠(𝑠𝑥) ∈ V
37 ixpexg 7818 . . . . . . . . . 10 (∀𝑥 ∈ dom 𝑠(𝑠𝑥) ∈ V → X𝑥 ∈ dom 𝑠(𝑠𝑥) ∈ V)
3836, 37ax-mp 5 . . . . . . . . 9 X𝑥 ∈ dom 𝑠(𝑠𝑥) ∈ V
3938mptrabex 6392 . . . . . . . 8 (𝑓 ∈ {X𝑥 ∈ dom 𝑠(𝑠𝑥) ∣ finSupp (0g𝑔)} ↦ (𝑔 Σg 𝑓)) ∈ V
4039rnex 6992 . . . . . . 7 ran (𝑓 ∈ {X𝑥 ∈ dom 𝑠(𝑠𝑥) ∣ finSupp (0g𝑔)} ↦ (𝑔 Σg 𝑓)) ∈ V
4140rgen2w 2909 . . . . . 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 18217 . . . . . . 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)))
4342fmpt2x 7125 . . . . . 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 219 . . . . 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 5965 . . . 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 2680 . . 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 6103 . . . . . . 7 (𝑔 = 𝐺 → (SubGrp‘𝑔) = (SubGrp‘𝐺))
4847feq3d 5945 . . . . . 6 (𝑔 = 𝐺 → (:dom ⟶(SubGrp‘𝑔) ↔ :dom ⟶(SubGrp‘𝐺)))
49 fveq2 6103 . . . . . . . . . . . 12 (𝑔 = 𝐺 → (Cntz‘𝑔) = (Cntz‘𝐺))
50 dmdprd.z . . . . . . . . . . . 12 𝑍 = (Cntz‘𝐺)
5149, 50syl6eqr 2662 . . . . . . . . . . 11 (𝑔 = 𝐺 → (Cntz‘𝑔) = 𝑍)
5251fveq1d 6105 . . . . . . . . . 10 (𝑔 = 𝐺 → ((Cntz‘𝑔)‘(𝑦)) = (𝑍‘(𝑦)))
5352sseq2d 3596 . . . . . . . . 9 (𝑔 = 𝐺 → ((𝑥) ⊆ ((Cntz‘𝑔)‘(𝑦)) ↔ (𝑥) ⊆ (𝑍‘(𝑦))))
5453ralbidv 2969 . . . . . . . 8 (𝑔 = 𝐺 → (∀𝑦 ∈ (dom ∖ {𝑥})(𝑥) ⊆ ((Cntz‘𝑔)‘(𝑦)) ↔ ∀𝑦 ∈ (dom ∖ {𝑥})(𝑥) ⊆ (𝑍‘(𝑦))))
5547fveq2d 6107 . . . . . . . . . . . 12 (𝑔 = 𝐺 → (mrCls‘(SubGrp‘𝑔)) = (mrCls‘(SubGrp‘𝐺)))
56 dmdprd.k . . . . . . . . . . . 12 𝐾 = (mrCls‘(SubGrp‘𝐺))
5755, 56syl6eqr 2662 . . . . . . . . . . 11 (𝑔 = 𝐺 → (mrCls‘(SubGrp‘𝑔)) = 𝐾)
5857fveq1d 6105 . . . . . . . . . 10 (𝑔 = 𝐺 → ((mrCls‘(SubGrp‘𝑔))‘ ( “ (dom ∖ {𝑥}))) = (𝐾 ( “ (dom ∖ {𝑥}))))
5958ineq2d 3776 . . . . . . . . 9 (𝑔 = 𝐺 → ((𝑥) ∩ ((mrCls‘(SubGrp‘𝑔))‘ ( “ (dom ∖ {𝑥})))) = ((𝑥) ∩ (𝐾 ( “ (dom ∖ {𝑥})))))
60 fveq2 6103 . . . . . . . . . . 11 (𝑔 = 𝐺 → (0g𝑔) = (0g𝐺))
61 dmdprd.0 . . . . . . . . . . 11 0 = (0g𝐺)
6260, 61syl6eqr 2662 . . . . . . . . . 10 (𝑔 = 𝐺 → (0g𝑔) = 0 )
6362sneqd 4137 . . . . . . . . 9 (𝑔 = 𝐺 → {(0g𝑔)} = { 0 })
6459, 63eqeq12d 2625 . . . . . . . 8 (𝑔 = 𝐺 → (((𝑥) ∩ ((mrCls‘(SubGrp‘𝑔))‘ ( “ (dom ∖ {𝑥})))) = {(0g𝑔)} ↔ ((𝑥) ∩ (𝐾 ( “ (dom ∖ {𝑥})))) = { 0 }))
6554, 64anbi12d 743 . . . . . . 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 2969 . . . . . 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 743 . . . . 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 2728 . . . 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 5182 . . 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 285 . 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 1035 . 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 302 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 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  {cab 2596  ∀wral 2896  {crab 2900  Vcvv 3173  [wsbc 3402   ∖ cdif 3537   ∩ cin 3539   ⊆ wss 3540  {csn 4125  ⟨cop 4131  ∪ cuni 4372  ∪ ciun 4455   class class class wbr 4583   ↦ cmpt 4643   × cxp 5036  dom cdm 5038  ran crn 5039   “ cima 5041  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  Xcixp 7794   finSupp cfsupp 8158  0gc0g 15923   Σg cgsu 15924  mrClscmrc 16066  Grpcgrp 17245  SubGrpcsubg 17411  Cntzccntz 17571   DProd cdprd 18215 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-ixp 7795  df-dprd 18217 This theorem is referenced by:  dmdprdd  18221  dprdgrp  18227  dprdf  18228  dprdcntz  18230  dprddisj  18231  dprdres  18250  subgdmdprd  18256
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