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Theorem prdstopn 21652
 Description: Topology of a structure product. (Contributed by Mario Carneiro, 27-Aug-2015.)
Hypotheses
Ref Expression
prdstopn.y 𝑌 = (𝑆Xs𝑅)
prdstopn.s (𝜑𝑆𝑉)
prdstopn.i (𝜑𝐼𝑊)
prdstopn.r (𝜑𝑅 Fn 𝐼)
prdstopn.o 𝑂 = (TopOpen‘𝑌)
Assertion
Ref Expression
prdstopn (𝜑𝑂 = (∏t‘(TopOpen ∘ 𝑅)))

Proof of Theorem prdstopn
Dummy variables 𝑥 𝑔 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prdstopn.y . . . . . 6 𝑌 = (𝑆Xs𝑅)
2 prdstopn.s . . . . . 6 (𝜑𝑆𝑉)
3 prdstopn.r . . . . . . 7 (𝜑𝑅 Fn 𝐼)
4 prdstopn.i . . . . . . 7 (𝜑𝐼𝑊)
5 fnex 6625 . . . . . . 7 ((𝑅 Fn 𝐼𝐼𝑊) → 𝑅 ∈ V)
63, 4, 5syl2anc 573 . . . . . 6 (𝜑𝑅 ∈ V)
7 eqid 2771 . . . . . 6 (Base‘𝑌) = (Base‘𝑌)
8 eqidd 2772 . . . . . 6 (𝜑 → dom 𝑅 = dom 𝑅)
9 eqid 2771 . . . . . 6 (TopSet‘𝑌) = (TopSet‘𝑌)
101, 2, 6, 7, 8, 9prdstset 16334 . . . . 5 (𝜑 → (TopSet‘𝑌) = (∏t‘(TopOpen ∘ 𝑅)))
11 topnfn 16294 . . . . . . . . . . 11 TopOpen Fn V
12 dffn2 6187 . . . . . . . . . . . 12 (𝑅 Fn 𝐼𝑅:𝐼⟶V)
133, 12sylib 208 . . . . . . . . . . 11 (𝜑𝑅:𝐼⟶V)
14 fnfco 6209 . . . . . . . . . . 11 ((TopOpen Fn V ∧ 𝑅:𝐼⟶V) → (TopOpen ∘ 𝑅) Fn 𝐼)
1511, 13, 14sylancr 575 . . . . . . . . . 10 (𝜑 → (TopOpen ∘ 𝑅) Fn 𝐼)
16 eqid 2771 . . . . . . . . . . 11 {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑦𝐼 (𝑔𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼𝑧)(𝑔𝑦) = ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦𝐼 (𝑔𝑦))} = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑦𝐼 (𝑔𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼𝑧)(𝑔𝑦) = ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦𝐼 (𝑔𝑦))}
1716ptval 21594 . . . . . . . . . 10 ((𝐼𝑊 ∧ (TopOpen ∘ 𝑅) Fn 𝐼) → (∏t‘(TopOpen ∘ 𝑅)) = (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑦𝐼 (𝑔𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼𝑧)(𝑔𝑦) = ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦𝐼 (𝑔𝑦))}))
184, 15, 17syl2anc 573 . . . . . . . . 9 (𝜑 → (∏t‘(TopOpen ∘ 𝑅)) = (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑦𝐼 (𝑔𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼𝑧)(𝑔𝑦) = ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦𝐼 (𝑔𝑦))}))
1918unieqd 4584 . . . . . . . 8 (𝜑 (∏t‘(TopOpen ∘ 𝑅)) = (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑦𝐼 (𝑔𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼𝑧)(𝑔𝑦) = ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦𝐼 (𝑔𝑦))}))
20 simpl2 1229 . . . . . . . . . . . . . . . 16 (((𝑔 Fn 𝐼 ∧ ∀𝑦𝐼 (𝑔𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼𝑧)(𝑔𝑦) = ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦𝐼 (𝑔𝑦)) → ∀𝑦𝐼 (𝑔𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦))
21 fvco2 6415 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑅 Fn 𝐼𝑦𝐼) → ((TopOpen ∘ 𝑅)‘𝑦) = (TopOpen‘(𝑅𝑦)))
223, 21sylan 569 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑦𝐼) → ((TopOpen ∘ 𝑅)‘𝑦) = (TopOpen‘(𝑅𝑦)))
23 eqid 2771 . . . . . . . . . . . . . . . . . . . . . . 23 (Base‘(𝑅𝑦)) = (Base‘(𝑅𝑦))
24 eqid 2771 . . . . . . . . . . . . . . . . . . . . . . 23 (TopSet‘(𝑅𝑦)) = (TopSet‘(𝑅𝑦))
2523, 24topnval 16303 . . . . . . . . . . . . . . . . . . . . . 22 ((TopSet‘(𝑅𝑦)) ↾t (Base‘(𝑅𝑦))) = (TopOpen‘(𝑅𝑦))
26 restsspw 16300 . . . . . . . . . . . . . . . . . . . . . 22 ((TopSet‘(𝑅𝑦)) ↾t (Base‘(𝑅𝑦))) ⊆ 𝒫 (Base‘(𝑅𝑦))
2725, 26eqsstr3i 3785 . . . . . . . . . . . . . . . . . . . . 21 (TopOpen‘(𝑅𝑦)) ⊆ 𝒫 (Base‘(𝑅𝑦))
2822, 27syl6eqss 3804 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑦𝐼) → ((TopOpen ∘ 𝑅)‘𝑦) ⊆ 𝒫 (Base‘(𝑅𝑦)))
2928sseld 3751 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑦𝐼) → ((𝑔𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) → (𝑔𝑦) ∈ 𝒫 (Base‘(𝑅𝑦))))
30 fvex 6342 . . . . . . . . . . . . . . . . . . . 20 (𝑔𝑦) ∈ V
3130elpw 4303 . . . . . . . . . . . . . . . . . . 19 ((𝑔𝑦) ∈ 𝒫 (Base‘(𝑅𝑦)) ↔ (𝑔𝑦) ⊆ (Base‘(𝑅𝑦)))
3229, 31syl6ib 241 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑦𝐼) → ((𝑔𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) → (𝑔𝑦) ⊆ (Base‘(𝑅𝑦))))
3332ralimdva 3111 . . . . . . . . . . . . . . . . 17 (𝜑 → (∀𝑦𝐼 (𝑔𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) → ∀𝑦𝐼 (𝑔𝑦) ⊆ (Base‘(𝑅𝑦))))
3433imp 393 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ∀𝑦𝐼 (𝑔𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦)) → ∀𝑦𝐼 (𝑔𝑦) ⊆ (Base‘(𝑅𝑦)))
3520, 34sylan2 580 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑔 Fn 𝐼 ∧ ∀𝑦𝐼 (𝑔𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼𝑧)(𝑔𝑦) = ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦𝐼 (𝑔𝑦))) → ∀𝑦𝐼 (𝑔𝑦) ⊆ (Base‘(𝑅𝑦)))
36 ss2ixp 8075 . . . . . . . . . . . . . . 15 (∀𝑦𝐼 (𝑔𝑦) ⊆ (Base‘(𝑅𝑦)) → X𝑦𝐼 (𝑔𝑦) ⊆ X𝑦𝐼 (Base‘(𝑅𝑦)))
3735, 36syl 17 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑔 Fn 𝐼 ∧ ∀𝑦𝐼 (𝑔𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼𝑧)(𝑔𝑦) = ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦𝐼 (𝑔𝑦))) → X𝑦𝐼 (𝑔𝑦) ⊆ X𝑦𝐼 (Base‘(𝑅𝑦)))
38 simprr 756 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑔 Fn 𝐼 ∧ ∀𝑦𝐼 (𝑔𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼𝑧)(𝑔𝑦) = ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦𝐼 (𝑔𝑦))) → 𝑥 = X𝑦𝐼 (𝑔𝑦))
391, 7, 2, 4, 3prdsbas2 16337 . . . . . . . . . . . . . . 15 (𝜑 → (Base‘𝑌) = X𝑦𝐼 (Base‘(𝑅𝑦)))
4039adantr 466 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑔 Fn 𝐼 ∧ ∀𝑦𝐼 (𝑔𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼𝑧)(𝑔𝑦) = ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦𝐼 (𝑔𝑦))) → (Base‘𝑌) = X𝑦𝐼 (Base‘(𝑅𝑦)))
4137, 38, 403sstr4d 3797 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑔 Fn 𝐼 ∧ ∀𝑦𝐼 (𝑔𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼𝑧)(𝑔𝑦) = ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦𝐼 (𝑔𝑦))) → 𝑥 ⊆ (Base‘𝑌))
4241ex 397 . . . . . . . . . . . 12 (𝜑 → (((𝑔 Fn 𝐼 ∧ ∀𝑦𝐼 (𝑔𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼𝑧)(𝑔𝑦) = ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦𝐼 (𝑔𝑦)) → 𝑥 ⊆ (Base‘𝑌)))
4342exlimdv 2013 . . . . . . . . . . 11 (𝜑 → (∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑦𝐼 (𝑔𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼𝑧)(𝑔𝑦) = ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦𝐼 (𝑔𝑦)) → 𝑥 ⊆ (Base‘𝑌)))
44 selpw 4304 . . . . . . . . . . 11 (𝑥 ∈ 𝒫 (Base‘𝑌) ↔ 𝑥 ⊆ (Base‘𝑌))
4543, 44syl6ibr 242 . . . . . . . . . 10 (𝜑 → (∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑦𝐼 (𝑔𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼𝑧)(𝑔𝑦) = ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦𝐼 (𝑔𝑦)) → 𝑥 ∈ 𝒫 (Base‘𝑌)))
4645abssdv 3825 . . . . . . . . 9 (𝜑 → {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑦𝐼 (𝑔𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼𝑧)(𝑔𝑦) = ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦𝐼 (𝑔𝑦))} ⊆ 𝒫 (Base‘𝑌))
47 fvex 6342 . . . . . . . . . . 11 (Base‘𝑌) ∈ V
4847pwex 4981 . . . . . . . . . 10 𝒫 (Base‘𝑌) ∈ V
4948ssex 4936 . . . . . . . . 9 ({𝑥 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑦𝐼 (𝑔𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼𝑧)(𝑔𝑦) = ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦𝐼 (𝑔𝑦))} ⊆ 𝒫 (Base‘𝑌) → {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑦𝐼 (𝑔𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼𝑧)(𝑔𝑦) = ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦𝐼 (𝑔𝑦))} ∈ V)
50 unitg 20992 . . . . . . . . 9 ({𝑥 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑦𝐼 (𝑔𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼𝑧)(𝑔𝑦) = ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦𝐼 (𝑔𝑦))} ∈ V → (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑦𝐼 (𝑔𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼𝑧)(𝑔𝑦) = ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦𝐼 (𝑔𝑦))}) = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑦𝐼 (𝑔𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼𝑧)(𝑔𝑦) = ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦𝐼 (𝑔𝑦))})
5146, 49, 503syl 18 . . . . . . . 8 (𝜑 (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑦𝐼 (𝑔𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼𝑧)(𝑔𝑦) = ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦𝐼 (𝑔𝑦))}) = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑦𝐼 (𝑔𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼𝑧)(𝑔𝑦) = ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦𝐼 (𝑔𝑦))})
5219, 51eqtrd 2805 . . . . . . 7 (𝜑 (∏t‘(TopOpen ∘ 𝑅)) = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑦𝐼 (𝑔𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼𝑧)(𝑔𝑦) = ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦𝐼 (𝑔𝑦))})
53 sspwuni 4745 . . . . . . . 8 ({𝑥 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑦𝐼 (𝑔𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼𝑧)(𝑔𝑦) = ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦𝐼 (𝑔𝑦))} ⊆ 𝒫 (Base‘𝑌) ↔ {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑦𝐼 (𝑔𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼𝑧)(𝑔𝑦) = ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦𝐼 (𝑔𝑦))} ⊆ (Base‘𝑌))
5446, 53sylib 208 . . . . . . 7 (𝜑 {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑦𝐼 (𝑔𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼𝑧)(𝑔𝑦) = ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦𝐼 (𝑔𝑦))} ⊆ (Base‘𝑌))
5552, 54eqsstrd 3788 . . . . . 6 (𝜑 (∏t‘(TopOpen ∘ 𝑅)) ⊆ (Base‘𝑌))
56 sspwuni 4745 . . . . . 6 ((∏t‘(TopOpen ∘ 𝑅)) ⊆ 𝒫 (Base‘𝑌) ↔ (∏t‘(TopOpen ∘ 𝑅)) ⊆ (Base‘𝑌))
5755, 56sylibr 224 . . . . 5 (𝜑 → (∏t‘(TopOpen ∘ 𝑅)) ⊆ 𝒫 (Base‘𝑌))
5810, 57eqsstrd 3788 . . . 4 (𝜑 → (TopSet‘𝑌) ⊆ 𝒫 (Base‘𝑌))
597, 9topnid 16304 . . . 4 ((TopSet‘𝑌) ⊆ 𝒫 (Base‘𝑌) → (TopSet‘𝑌) = (TopOpen‘𝑌))
6058, 59syl 17 . . 3 (𝜑 → (TopSet‘𝑌) = (TopOpen‘𝑌))
61 prdstopn.o . . 3 𝑂 = (TopOpen‘𝑌)
6260, 61syl6eqr 2823 . 2 (𝜑 → (TopSet‘𝑌) = 𝑂)
6362, 10eqtr3d 2807 1 (𝜑𝑂 = (∏t‘(TopOpen ∘ 𝑅)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 382   ∧ w3a 1071   = wceq 1631  ∃wex 1852   ∈ wcel 2145  {cab 2757  ∀wral 3061  ∃wrex 3062  Vcvv 3351   ∖ cdif 3720   ⊆ wss 3723  𝒫 cpw 4297  ∪ cuni 4574  dom cdm 5249   ∘ ccom 5253   Fn wfn 6026  ⟶wf 6027  ‘cfv 6031  (class class class)co 6793  Xcixp 8062  Fincfn 8109  Basecbs 16064  TopSetcts 16155   ↾t crest 16289  TopOpenctopn 16290  topGenctg 16306  ∏tcpt 16307  Xscprds 16314 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-oadd 7717  df-er 7896  df-map 8011  df-ixp 8063  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-sup 8504  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-nn 11223  df-2 11281  df-3 11282  df-4 11283  df-5 11284  df-6 11285  df-7 11286  df-8 11287  df-9 11288  df-n0 11495  df-z 11580  df-dec 11696  df-uz 11889  df-fz 12534  df-struct 16066  df-ndx 16067  df-slot 16068  df-base 16070  df-plusg 16162  df-mulr 16163  df-sca 16165  df-vsca 16166  df-ip 16167  df-tset 16168  df-ple 16169  df-ds 16172  df-hom 16174  df-cco 16175  df-rest 16291  df-topn 16292  df-topgen 16312  df-pt 16313  df-prds 16316 This theorem is referenced by:  xpstopnlem2  21835  prdstmdd  22147  prdstgpd  22148  prdsxmslem2  22554
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