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Theorem dmdprd 18589
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 3344 . . . . 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 6645 . . . . . . 7 ((𝑆:𝐼⟶(SubGrp‘𝐺) ∧ 𝐼𝑉) → 𝑆 ∈ V)
43expcom 450 . . . . . 6 (𝐼𝑉 → (𝑆:𝐼⟶(SubGrp‘𝐺) → 𝑆 ∈ V))
54adantr 472 . . . . 5 ((𝐼𝑉 ∧ dom 𝑆 = 𝐼) → (𝑆:𝐼⟶(SubGrp‘𝐺) → 𝑆 ∈ V))
65adantrd 485 . . . 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 3569 . . . . . 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 479 . . . . . . 7 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ 𝑆 ∈ V) → 𝑆 ∈ V)
9 simpr 479 . . . . . . . . . 10 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → = 𝑆)
109dmeqd 5473 . . . . . . . . . . 11 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → dom = dom 𝑆)
11 simplr 809 . . . . . . . . . . 11 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → dom 𝑆 = 𝐼)
1210, 11eqtrd 2786 . . . . . . . . . 10 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → dom = 𝐼)
139, 12feq12d 6186 . . . . . . . . 9 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → (:dom ⟶(SubGrp‘𝐺) ↔ 𝑆:𝐼⟶(SubGrp‘𝐺)))
1412difeq1d 3862 . . . . . . . . . . . 12 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → (dom ∖ {𝑥}) = (𝐼 ∖ {𝑥}))
159fveq1d 6346 . . . . . . . . . . . . 13 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → (𝑥) = (𝑆𝑥))
169fveq1d 6346 . . . . . . . . . . . . . 14 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → (𝑦) = (𝑆𝑦))
1716fveq2d 6348 . . . . . . . . . . . . 13 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → (𝑍‘(𝑦)) = (𝑍‘(𝑆𝑦)))
1815, 17sseq12d 3767 . . . . . . . . . . . 12 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → ((𝑥) ⊆ (𝑍‘(𝑦)) ↔ (𝑆𝑥) ⊆ (𝑍‘(𝑆𝑦))))
1914, 18raleqbidv 3283 . . . . . . . . . . 11 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → (∀𝑦 ∈ (dom ∖ {𝑥})(𝑥) ⊆ (𝑍‘(𝑦)) ↔ ∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆𝑥) ⊆ (𝑍‘(𝑆𝑦))))
209, 14imaeq12d 5617 . . . . . . . . . . . . . . 15 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → ( “ (dom ∖ {𝑥})) = (𝑆 “ (𝐼 ∖ {𝑥})))
2120unieqd 4590 . . . . . . . . . . . . . 14 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → ( “ (dom ∖ {𝑥})) = (𝑆 “ (𝐼 ∖ {𝑥})))
2221fveq2d 6348 . . . . . . . . . . . . 13 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → (𝐾 ( “ (dom ∖ {𝑥}))) = (𝐾 (𝑆 “ (𝐼 ∖ {𝑥}))))
2315, 22ineq12d 3950 . . . . . . . . . . . 12 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → ((𝑥) ∩ (𝐾 ( “ (dom ∖ {𝑥})))) = ((𝑆𝑥) ∩ (𝐾 (𝑆 “ (𝐼 ∖ {𝑥})))))
2423eqeq1d 2754 . . . . . . . . . . 11 (((𝐼𝑉 ∧ dom 𝑆 = 𝐼) ∧ = 𝑆) → (((𝑥) ∩ (𝐾 ( “ (dom ∖ {𝑥})))) = { 0 } ↔ ((𝑆𝑥) ∩ (𝐾 (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 }))
2519, 24anbi12d 749 . . . . . . . . . 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 3283 . . . . . . . . 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 749 . . . . . . . 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 753 . . . . . . 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 3605 . . . . . 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 274 . . . . 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 742 . 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 4797 . . 3 (𝐺dom DProd 𝑆 ↔ ⟨𝐺, 𝑆⟩ ∈ dom DProd )
35 fvex 6354 . . . . . . . . . . 11 (𝑠𝑥) ∈ V
3635rgenw 3054 . . . . . . . . . 10 𝑥 ∈ dom 𝑠(𝑠𝑥) ∈ V
37 ixpexg 8090 . . . . . . . . . 10 (∀𝑥 ∈ dom 𝑠(𝑠𝑥) ∈ V → X𝑥 ∈ dom 𝑠(𝑠𝑥) ∈ V)
3836, 37ax-mp 5 . . . . . . . . 9 X𝑥 ∈ dom 𝑠(𝑠𝑥) ∈ V
3938mptrabex 6644 . . . . . . . 8 (𝑓 ∈ {X𝑥 ∈ dom 𝑠(𝑠𝑥) ∣ finSupp (0g𝑔)} ↦ (𝑔 Σg 𝑓)) ∈ V
4039rnex 7257 . . . . . . 7 ran (𝑓 ∈ {X𝑥 ∈ dom 𝑠(𝑠𝑥) ∣ finSupp (0g𝑔)} ↦ (𝑔 Σg 𝑓)) ∈ V
4140rgen2w 3055 . . . . . 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 18586 . . . . . . 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 7396 . . . . . 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 220 . . . . 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 6205 . . . 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 2823 . . 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 6344 . . . . . . 7 (𝑔 = 𝐺 → (SubGrp‘𝑔) = (SubGrp‘𝐺))
4847feq3d 6185 . . . . . 6 (𝑔 = 𝐺 → (:dom ⟶(SubGrp‘𝑔) ↔ :dom ⟶(SubGrp‘𝐺)))
49 fveq2 6344 . . . . . . . . . . . 12 (𝑔 = 𝐺 → (Cntz‘𝑔) = (Cntz‘𝐺))
50 dmdprd.z . . . . . . . . . . . 12 𝑍 = (Cntz‘𝐺)
5149, 50syl6eqr 2804 . . . . . . . . . . 11 (𝑔 = 𝐺 → (Cntz‘𝑔) = 𝑍)
5251fveq1d 6346 . . . . . . . . . 10 (𝑔 = 𝐺 → ((Cntz‘𝑔)‘(𝑦)) = (𝑍‘(𝑦)))
5352sseq2d 3766 . . . . . . . . 9 (𝑔 = 𝐺 → ((𝑥) ⊆ ((Cntz‘𝑔)‘(𝑦)) ↔ (𝑥) ⊆ (𝑍‘(𝑦))))
5453ralbidv 3116 . . . . . . . 8 (𝑔 = 𝐺 → (∀𝑦 ∈ (dom ∖ {𝑥})(𝑥) ⊆ ((Cntz‘𝑔)‘(𝑦)) ↔ ∀𝑦 ∈ (dom ∖ {𝑥})(𝑥) ⊆ (𝑍‘(𝑦))))
5547fveq2d 6348 . . . . . . . . . . . 12 (𝑔 = 𝐺 → (mrCls‘(SubGrp‘𝑔)) = (mrCls‘(SubGrp‘𝐺)))
56 dmdprd.k . . . . . . . . . . . 12 𝐾 = (mrCls‘(SubGrp‘𝐺))
5755, 56syl6eqr 2804 . . . . . . . . . . 11 (𝑔 = 𝐺 → (mrCls‘(SubGrp‘𝑔)) = 𝐾)
5857fveq1d 6346 . . . . . . . . . 10 (𝑔 = 𝐺 → ((mrCls‘(SubGrp‘𝑔))‘ ( “ (dom ∖ {𝑥}))) = (𝐾 ( “ (dom ∖ {𝑥}))))
5958ineq2d 3949 . . . . . . . . 9 (𝑔 = 𝐺 → ((𝑥) ∩ ((mrCls‘(SubGrp‘𝑔))‘ ( “ (dom ∖ {𝑥})))) = ((𝑥) ∩ (𝐾 ( “ (dom ∖ {𝑥})))))
60 fveq2 6344 . . . . . . . . . . 11 (𝑔 = 𝐺 → (0g𝑔) = (0g𝐺))
61 dmdprd.0 . . . . . . . . . . 11 0 = (0g𝐺)
6260, 61syl6eqr 2804 . . . . . . . . . 10 (𝑔 = 𝐺 → (0g𝑔) = 0 )
6362sneqd 4325 . . . . . . . . 9 (𝑔 = 𝐺 → {(0g𝑔)} = { 0 })
6459, 63eqeq12d 2767 . . . . . . . 8 (𝑔 = 𝐺 → (((𝑥) ∩ ((mrCls‘(SubGrp‘𝑔))‘ ( “ (dom ∖ {𝑥})))) = {(0g𝑔)} ↔ ((𝑥) ∩ (𝐾 ( “ (dom ∖ {𝑥})))) = { 0 }))
6554, 64anbi12d 749 . . . . . . 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 3116 . . . . . 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 749 . . . . 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 2871 . . . 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 5408 . . 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 286 . 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 1081 . 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 303 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 196  wa 383  w3a 1072   = wceq 1624  wcel 2131  {cab 2738  wral 3042  {crab 3046  Vcvv 3332  [wsbc 3568  cdif 3704  cin 3706  wss 3707  {csn 4313  cop 4319   cuni 4580   ciun 4664   class class class wbr 4796  cmpt 4873   × cxp 5256  dom cdm 5258  ran crn 5259  cima 5261  wf 6037  cfv 6041  (class class class)co 6805  Xcixp 8066   finSupp cfsupp 8432  0gc0g 16294   Σg cgsu 16295  mrClscmrc 16437  Grpcgrp 17615  SubGrpcsubg 17781  Cntzccntz 17940   DProd cdprd 18584
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-reu 3049  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-oprab 6809  df-mpt2 6810  df-1st 7325  df-2nd 7326  df-ixp 8067  df-dprd 18586
This theorem is referenced by:  dmdprdd  18590  dprdgrp  18596  dprdf  18597  dprdcntz  18599  dprddisj  18600  dprdres  18619  subgdmdprd  18625
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