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Theorem prdstmdd 22735
Description: The product of a family of topological monoids is a topological monoid. (Contributed by Mario Carneiro, 22-Sep-2015.)
Hypotheses
Ref Expression
prdstmdd.y 𝑌 = (𝑆Xs𝑅)
prdstmdd.i (𝜑𝐼𝑊)
prdstmdd.s (𝜑𝑆𝑉)
prdstmdd.r (𝜑𝑅:𝐼⟶TopMnd)
Assertion
Ref Expression
prdstmdd (𝜑𝑌 ∈ TopMnd)

Proof of Theorem prdstmdd
Dummy variables 𝑓 𝑔 𝑘 𝑥 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prdstmdd.y . . 3 𝑌 = (𝑆Xs𝑅)
2 prdstmdd.i . . 3 (𝜑𝐼𝑊)
3 prdstmdd.s . . 3 (𝜑𝑆𝑉)
4 prdstmdd.r . . . 4 (𝜑𝑅:𝐼⟶TopMnd)
5 tmdmnd 22686 . . . . 5 (𝑥 ∈ TopMnd → 𝑥 ∈ Mnd)
65ssriv 3974 . . . 4 TopMnd ⊆ Mnd
7 fss 6530 . . . 4 ((𝑅:𝐼⟶TopMnd ∧ TopMnd ⊆ Mnd) → 𝑅:𝐼⟶Mnd)
84, 6, 7sylancl 588 . . 3 (𝜑𝑅:𝐼⟶Mnd)
91, 2, 3, 8prdsmndd 17947 . 2 (𝜑𝑌 ∈ Mnd)
10 tmdtps 22687 . . . . 5 (𝑥 ∈ TopMnd → 𝑥 ∈ TopSp)
1110ssriv 3974 . . . 4 TopMnd ⊆ TopSp
12 fss 6530 . . . 4 ((𝑅:𝐼⟶TopMnd ∧ TopMnd ⊆ TopSp) → 𝑅:𝐼⟶TopSp)
134, 11, 12sylancl 588 . . 3 (𝜑𝑅:𝐼⟶TopSp)
141, 3, 2, 13prdstps 22240 . 2 (𝜑𝑌 ∈ TopSp)
15 eqid 2824 . . . . . . 7 (Base‘𝑌) = (Base‘𝑌)
1633ad2ant1 1129 . . . . . . 7 ((𝜑𝑓 ∈ (Base‘𝑌) ∧ 𝑔 ∈ (Base‘𝑌)) → 𝑆𝑉)
1723ad2ant1 1129 . . . . . . 7 ((𝜑𝑓 ∈ (Base‘𝑌) ∧ 𝑔 ∈ (Base‘𝑌)) → 𝐼𝑊)
184ffnd 6518 . . . . . . . 8 (𝜑𝑅 Fn 𝐼)
19183ad2ant1 1129 . . . . . . 7 ((𝜑𝑓 ∈ (Base‘𝑌) ∧ 𝑔 ∈ (Base‘𝑌)) → 𝑅 Fn 𝐼)
20 simp2 1133 . . . . . . 7 ((𝜑𝑓 ∈ (Base‘𝑌) ∧ 𝑔 ∈ (Base‘𝑌)) → 𝑓 ∈ (Base‘𝑌))
21 simp3 1134 . . . . . . 7 ((𝜑𝑓 ∈ (Base‘𝑌) ∧ 𝑔 ∈ (Base‘𝑌)) → 𝑔 ∈ (Base‘𝑌))
22 eqid 2824 . . . . . . 7 (+g𝑌) = (+g𝑌)
231, 15, 16, 17, 19, 20, 21, 22prdsplusgval 16749 . . . . . 6 ((𝜑𝑓 ∈ (Base‘𝑌) ∧ 𝑔 ∈ (Base‘𝑌)) → (𝑓(+g𝑌)𝑔) = (𝑘𝐼 ↦ ((𝑓𝑘)(+g‘(𝑅𝑘))(𝑔𝑘))))
2423mpoeq3dva 7234 . . . . 5 (𝜑 → (𝑓 ∈ (Base‘𝑌), 𝑔 ∈ (Base‘𝑌) ↦ (𝑓(+g𝑌)𝑔)) = (𝑓 ∈ (Base‘𝑌), 𝑔 ∈ (Base‘𝑌) ↦ (𝑘𝐼 ↦ ((𝑓𝑘)(+g‘(𝑅𝑘))(𝑔𝑘)))))
25 eqid 2824 . . . . . 6 (+𝑓𝑌) = (+𝑓𝑌)
2615, 22, 25plusffval 17861 . . . . 5 (+𝑓𝑌) = (𝑓 ∈ (Base‘𝑌), 𝑔 ∈ (Base‘𝑌) ↦ (𝑓(+g𝑌)𝑔))
27 vex 3500 . . . . . . . . . 10 𝑓 ∈ V
28 vex 3500 . . . . . . . . . 10 𝑔 ∈ V
2927, 28op1std 7702 . . . . . . . . 9 (𝑧 = ⟨𝑓, 𝑔⟩ → (1st𝑧) = 𝑓)
3029fveq1d 6675 . . . . . . . 8 (𝑧 = ⟨𝑓, 𝑔⟩ → ((1st𝑧)‘𝑘) = (𝑓𝑘))
3127, 28op2ndd 7703 . . . . . . . . 9 (𝑧 = ⟨𝑓, 𝑔⟩ → (2nd𝑧) = 𝑔)
3231fveq1d 6675 . . . . . . . 8 (𝑧 = ⟨𝑓, 𝑔⟩ → ((2nd𝑧)‘𝑘) = (𝑔𝑘))
3330, 32oveq12d 7177 . . . . . . 7 (𝑧 = ⟨𝑓, 𝑔⟩ → (((1st𝑧)‘𝑘)(+g‘(𝑅𝑘))((2nd𝑧)‘𝑘)) = ((𝑓𝑘)(+g‘(𝑅𝑘))(𝑔𝑘)))
3433mpteq2dv 5165 . . . . . 6 (𝑧 = ⟨𝑓, 𝑔⟩ → (𝑘𝐼 ↦ (((1st𝑧)‘𝑘)(+g‘(𝑅𝑘))((2nd𝑧)‘𝑘))) = (𝑘𝐼 ↦ ((𝑓𝑘)(+g‘(𝑅𝑘))(𝑔𝑘))))
3534mpompt 7269 . . . . 5 (𝑧 ∈ ((Base‘𝑌) × (Base‘𝑌)) ↦ (𝑘𝐼 ↦ (((1st𝑧)‘𝑘)(+g‘(𝑅𝑘))((2nd𝑧)‘𝑘)))) = (𝑓 ∈ (Base‘𝑌), 𝑔 ∈ (Base‘𝑌) ↦ (𝑘𝐼 ↦ ((𝑓𝑘)(+g‘(𝑅𝑘))(𝑔𝑘))))
3624, 26, 353eqtr4g 2884 . . . 4 (𝜑 → (+𝑓𝑌) = (𝑧 ∈ ((Base‘𝑌) × (Base‘𝑌)) ↦ (𝑘𝐼 ↦ (((1st𝑧)‘𝑘)(+g‘(𝑅𝑘))((2nd𝑧)‘𝑘)))))
37 eqid 2824 . . . . 5 (∏t‘(TopOpen ∘ 𝑅)) = (∏t‘(TopOpen ∘ 𝑅))
38 eqid 2824 . . . . . . . 8 (TopOpen‘𝑌) = (TopOpen‘𝑌)
3915, 38istps 21545 . . . . . . 7 (𝑌 ∈ TopSp ↔ (TopOpen‘𝑌) ∈ (TopOn‘(Base‘𝑌)))
4014, 39sylib 220 . . . . . 6 (𝜑 → (TopOpen‘𝑌) ∈ (TopOn‘(Base‘𝑌)))
41 txtopon 22202 . . . . . 6 (((TopOpen‘𝑌) ∈ (TopOn‘(Base‘𝑌)) ∧ (TopOpen‘𝑌) ∈ (TopOn‘(Base‘𝑌))) → ((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) ∈ (TopOn‘((Base‘𝑌) × (Base‘𝑌))))
4240, 40, 41syl2anc 586 . . . . 5 (𝜑 → ((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) ∈ (TopOn‘((Base‘𝑌) × (Base‘𝑌))))
43 topnfn 16702 . . . . . . . 8 TopOpen Fn V
44 ssv 3994 . . . . . . . 8 TopSp ⊆ V
45 fnssres 6473 . . . . . . . 8 ((TopOpen Fn V ∧ TopSp ⊆ V) → (TopOpen ↾ TopSp) Fn TopSp)
4643, 44, 45mp2an 690 . . . . . . 7 (TopOpen ↾ TopSp) Fn TopSp
47 fvres 6692 . . . . . . . . 9 (𝑥 ∈ TopSp → ((TopOpen ↾ TopSp)‘𝑥) = (TopOpen‘𝑥))
48 eqid 2824 . . . . . . . . . 10 (TopOpen‘𝑥) = (TopOpen‘𝑥)
4948tpstop 21548 . . . . . . . . 9 (𝑥 ∈ TopSp → (TopOpen‘𝑥) ∈ Top)
5047, 49eqeltrd 2916 . . . . . . . 8 (𝑥 ∈ TopSp → ((TopOpen ↾ TopSp)‘𝑥) ∈ Top)
5150rgen 3151 . . . . . . 7 𝑥 ∈ TopSp ((TopOpen ↾ TopSp)‘𝑥) ∈ Top
52 ffnfv 6885 . . . . . . 7 ((TopOpen ↾ TopSp):TopSp⟶Top ↔ ((TopOpen ↾ TopSp) Fn TopSp ∧ ∀𝑥 ∈ TopSp ((TopOpen ↾ TopSp)‘𝑥) ∈ Top))
5346, 51, 52mpbir2an 709 . . . . . 6 (TopOpen ↾ TopSp):TopSp⟶Top
54 fco2 6536 . . . . . 6 (((TopOpen ↾ TopSp):TopSp⟶Top ∧ 𝑅:𝐼⟶TopSp) → (TopOpen ∘ 𝑅):𝐼⟶Top)
5553, 13, 54sylancr 589 . . . . 5 (𝜑 → (TopOpen ∘ 𝑅):𝐼⟶Top)
5633mpompt 7269 . . . . . 6 (𝑧 ∈ ((Base‘𝑌) × (Base‘𝑌)) ↦ (((1st𝑧)‘𝑘)(+g‘(𝑅𝑘))((2nd𝑧)‘𝑘))) = (𝑓 ∈ (Base‘𝑌), 𝑔 ∈ (Base‘𝑌) ↦ ((𝑓𝑘)(+g‘(𝑅𝑘))(𝑔𝑘)))
57 eqid 2824 . . . . . . . 8 (TopOpen‘(𝑅𝑘)) = (TopOpen‘(𝑅𝑘))
58 eqid 2824 . . . . . . . 8 (+g‘(𝑅𝑘)) = (+g‘(𝑅𝑘))
594ffvelrnda 6854 . . . . . . . 8 ((𝜑𝑘𝐼) → (𝑅𝑘) ∈ TopMnd)
6040adantr 483 . . . . . . . 8 ((𝜑𝑘𝐼) → (TopOpen‘𝑌) ∈ (TopOn‘(Base‘𝑌)))
6160, 60cnmpt1st 22279 . . . . . . . . 9 ((𝜑𝑘𝐼) → (𝑓 ∈ (Base‘𝑌), 𝑔 ∈ (Base‘𝑌) ↦ 𝑓) ∈ (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn (TopOpen‘𝑌)))
621, 3, 2, 18, 38prdstopn 22239 . . . . . . . . . . . . . . 15 (𝜑 → (TopOpen‘𝑌) = (∏t‘(TopOpen ∘ 𝑅)))
6362adantr 483 . . . . . . . . . . . . . 14 ((𝜑𝑘𝐼) → (TopOpen‘𝑌) = (∏t‘(TopOpen ∘ 𝑅)))
6463, 60eqeltrrd 2917 . . . . . . . . . . . . 13 ((𝜑𝑘𝐼) → (∏t‘(TopOpen ∘ 𝑅)) ∈ (TopOn‘(Base‘𝑌)))
65 toponuni 21525 . . . . . . . . . . . . 13 ((∏t‘(TopOpen ∘ 𝑅)) ∈ (TopOn‘(Base‘𝑌)) → (Base‘𝑌) = (∏t‘(TopOpen ∘ 𝑅)))
6664, 65syl 17 . . . . . . . . . . . 12 ((𝜑𝑘𝐼) → (Base‘𝑌) = (∏t‘(TopOpen ∘ 𝑅)))
6766mpteq1d 5158 . . . . . . . . . . 11 ((𝜑𝑘𝐼) → (𝑥 ∈ (Base‘𝑌) ↦ (𝑥𝑘)) = (𝑥 (∏t‘(TopOpen ∘ 𝑅)) ↦ (𝑥𝑘)))
682adantr 483 . . . . . . . . . . . 12 ((𝜑𝑘𝐼) → 𝐼𝑊)
6955adantr 483 . . . . . . . . . . . 12 ((𝜑𝑘𝐼) → (TopOpen ∘ 𝑅):𝐼⟶Top)
70 simpr 487 . . . . . . . . . . . 12 ((𝜑𝑘𝐼) → 𝑘𝐼)
71 eqid 2824 . . . . . . . . . . . . 13 (∏t‘(TopOpen ∘ 𝑅)) = (∏t‘(TopOpen ∘ 𝑅))
7271, 37ptpjcn 22222 . . . . . . . . . . . 12 ((𝐼𝑊 ∧ (TopOpen ∘ 𝑅):𝐼⟶Top ∧ 𝑘𝐼) → (𝑥 (∏t‘(TopOpen ∘ 𝑅)) ↦ (𝑥𝑘)) ∈ ((∏t‘(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)‘𝑘)))
7368, 69, 70, 72syl3anc 1367 . . . . . . . . . . 11 ((𝜑𝑘𝐼) → (𝑥 (∏t‘(TopOpen ∘ 𝑅)) ↦ (𝑥𝑘)) ∈ ((∏t‘(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)‘𝑘)))
7467, 73eqeltrd 2916 . . . . . . . . . 10 ((𝜑𝑘𝐼) → (𝑥 ∈ (Base‘𝑌) ↦ (𝑥𝑘)) ∈ ((∏t‘(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)‘𝑘)))
7563eqcomd 2830 . . . . . . . . . . 11 ((𝜑𝑘𝐼) → (∏t‘(TopOpen ∘ 𝑅)) = (TopOpen‘𝑌))
76 fvco3 6763 . . . . . . . . . . . 12 ((𝑅:𝐼⟶TopMnd ∧ 𝑘𝐼) → ((TopOpen ∘ 𝑅)‘𝑘) = (TopOpen‘(𝑅𝑘)))
774, 76sylan 582 . . . . . . . . . . 11 ((𝜑𝑘𝐼) → ((TopOpen ∘ 𝑅)‘𝑘) = (TopOpen‘(𝑅𝑘)))
7875, 77oveq12d 7177 . . . . . . . . . 10 ((𝜑𝑘𝐼) → ((∏t‘(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)‘𝑘)) = ((TopOpen‘𝑌) Cn (TopOpen‘(𝑅𝑘))))
7974, 78eleqtrd 2918 . . . . . . . . 9 ((𝜑𝑘𝐼) → (𝑥 ∈ (Base‘𝑌) ↦ (𝑥𝑘)) ∈ ((TopOpen‘𝑌) Cn (TopOpen‘(𝑅𝑘))))
80 fveq1 6672 . . . . . . . . 9 (𝑥 = 𝑓 → (𝑥𝑘) = (𝑓𝑘))
8160, 60, 61, 60, 79, 80cnmpt21 22282 . . . . . . . 8 ((𝜑𝑘𝐼) → (𝑓 ∈ (Base‘𝑌), 𝑔 ∈ (Base‘𝑌) ↦ (𝑓𝑘)) ∈ (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn (TopOpen‘(𝑅𝑘))))
8260, 60cnmpt2nd 22280 . . . . . . . . 9 ((𝜑𝑘𝐼) → (𝑓 ∈ (Base‘𝑌), 𝑔 ∈ (Base‘𝑌) ↦ 𝑔) ∈ (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn (TopOpen‘𝑌)))
83 fveq1 6672 . . . . . . . . 9 (𝑥 = 𝑔 → (𝑥𝑘) = (𝑔𝑘))
8460, 60, 82, 60, 79, 83cnmpt21 22282 . . . . . . . 8 ((𝜑𝑘𝐼) → (𝑓 ∈ (Base‘𝑌), 𝑔 ∈ (Base‘𝑌) ↦ (𝑔𝑘)) ∈ (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn (TopOpen‘(𝑅𝑘))))
8557, 58, 59, 60, 60, 81, 84cnmpt2plusg 22699 . . . . . . 7 ((𝜑𝑘𝐼) → (𝑓 ∈ (Base‘𝑌), 𝑔 ∈ (Base‘𝑌) ↦ ((𝑓𝑘)(+g‘(𝑅𝑘))(𝑔𝑘))) ∈ (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn (TopOpen‘(𝑅𝑘))))
8677oveq2d 7175 . . . . . . 7 ((𝜑𝑘𝐼) → (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn ((TopOpen ∘ 𝑅)‘𝑘)) = (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn (TopOpen‘(𝑅𝑘))))
8785, 86eleqtrrd 2919 . . . . . 6 ((𝜑𝑘𝐼) → (𝑓 ∈ (Base‘𝑌), 𝑔 ∈ (Base‘𝑌) ↦ ((𝑓𝑘)(+g‘(𝑅𝑘))(𝑔𝑘))) ∈ (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn ((TopOpen ∘ 𝑅)‘𝑘)))
8856, 87eqeltrid 2920 . . . . 5 ((𝜑𝑘𝐼) → (𝑧 ∈ ((Base‘𝑌) × (Base‘𝑌)) ↦ (((1st𝑧)‘𝑘)(+g‘(𝑅𝑘))((2nd𝑧)‘𝑘))) ∈ (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn ((TopOpen ∘ 𝑅)‘𝑘)))
8937, 42, 2, 55, 88ptcn 22238 . . . 4 (𝜑 → (𝑧 ∈ ((Base‘𝑌) × (Base‘𝑌)) ↦ (𝑘𝐼 ↦ (((1st𝑧)‘𝑘)(+g‘(𝑅𝑘))((2nd𝑧)‘𝑘)))) ∈ (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn (∏t‘(TopOpen ∘ 𝑅))))
9036, 89eqeltrd 2916 . . 3 (𝜑 → (+𝑓𝑌) ∈ (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn (∏t‘(TopOpen ∘ 𝑅))))
9162oveq2d 7175 . . 3 (𝜑 → (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn (TopOpen‘𝑌)) = (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn (∏t‘(TopOpen ∘ 𝑅))))
9290, 91eleqtrrd 2919 . 2 (𝜑 → (+𝑓𝑌) ∈ (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn (TopOpen‘𝑌)))
9325, 38istmd 22685 . 2 (𝑌 ∈ TopMnd ↔ (𝑌 ∈ Mnd ∧ 𝑌 ∈ TopSp ∧ (+𝑓𝑌) ∈ (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn (TopOpen‘𝑌))))
949, 14, 92, 93syl3anbrc 1339 1 (𝜑𝑌 ∈ TopMnd)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  w3a 1083   = wceq 1536  wcel 2113  wral 3141  Vcvv 3497  wss 3939  cop 4576   cuni 4841  cmpt 5149   × cxp 5556  cres 5560  ccom 5562   Fn wfn 6353  wf 6354  cfv 6358  (class class class)co 7159  cmpo 7161  1st c1st 7690  2nd c2nd 7691  Basecbs 16486  +gcplusg 16568  TopOpenctopn 16698  tcpt 16715  Xscprds 16722  +𝑓cplusf 17852  Mndcmnd 17914  Topctop 21504  TopOnctopon 21521  TopSpctps 21543   Cn ccn 21835   ×t ctx 22171  TopMndctmd 22681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-rep 5193  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464  ax-cnex 10596  ax-resscn 10597  ax-1cn 10598  ax-icn 10599  ax-addcl 10600  ax-addrcl 10601  ax-mulcl 10602  ax-mulrcl 10603  ax-mulcom 10604  ax-addass 10605  ax-mulass 10606  ax-distr 10607  ax-i2m1 10608  ax-1ne0 10609  ax-1rid 10610  ax-rnegex 10611  ax-rrecex 10612  ax-cnre 10613  ax-pre-lttri 10614  ax-pre-lttrn 10615  ax-pre-ltadd 10616  ax-pre-mulgt0 10617
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-nel 3127  df-ral 3146  df-rex 3147  df-reu 3148  df-rmo 3149  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-tp 4575  df-op 4577  df-uni 4842  df-int 4880  df-iun 4924  df-iin 4925  df-br 5070  df-opab 5132  df-mpt 5150  df-tr 5176  df-id 5463  df-eprel 5468  df-po 5477  df-so 5478  df-fr 5517  df-we 5519  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-pred 6151  df-ord 6197  df-on 6198  df-lim 6199  df-suc 6200  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-riota 7117  df-ov 7162  df-oprab 7163  df-mpo 7164  df-om 7584  df-1st 7692  df-2nd 7693  df-wrecs 7950  df-recs 8011  df-rdg 8049  df-1o 8105  df-oadd 8109  df-er 8292  df-map 8411  df-ixp 8465  df-en 8513  df-dom 8514  df-sdom 8515  df-fin 8516  df-fi 8878  df-sup 8909  df-pnf 10680  df-mnf 10681  df-xr 10682  df-ltxr 10683  df-le 10684  df-sub 10875  df-neg 10876  df-nn 11642  df-2 11703  df-3 11704  df-4 11705  df-5 11706  df-6 11707  df-7 11708  df-8 11709  df-9 11710  df-n0 11901  df-z 11985  df-dec 12102  df-uz 12247  df-fz 12896  df-struct 16488  df-ndx 16489  df-slot 16490  df-base 16492  df-plusg 16581  df-mulr 16582  df-sca 16584  df-vsca 16585  df-ip 16586  df-tset 16587  df-ple 16588  df-ds 16590  df-hom 16592  df-cco 16593  df-rest 16699  df-topn 16700  df-0g 16718  df-topgen 16720  df-pt 16721  df-prds 16724  df-plusf 17854  df-mgm 17855  df-sgrp 17904  df-mnd 17915  df-top 21505  df-topon 21522  df-topsp 21544  df-bases 21557  df-cn 21838  df-cnp 21839  df-tx 22173  df-tmd 22683
This theorem is referenced by:  prdstgpd  22736
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