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Theorem prdstmdd 21837
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 21789 . . . . 5 (𝑥 ∈ TopMnd → 𝑥 ∈ Mnd)
65ssriv 3587 . . . 4 TopMnd ⊆ Mnd
7 fss 6013 . . . 4 ((𝑅:𝐼⟶TopMnd ∧ TopMnd ⊆ Mnd) → 𝑅:𝐼⟶Mnd)
84, 6, 7sylancl 693 . . 3 (𝜑𝑅:𝐼⟶Mnd)
91, 2, 3, 8prdsmndd 17244 . 2 (𝜑𝑌 ∈ Mnd)
10 tmdtps 21790 . . . . 5 (𝑥 ∈ TopMnd → 𝑥 ∈ TopSp)
1110ssriv 3587 . . . 4 TopMnd ⊆ TopSp
12 fss 6013 . . . 4 ((𝑅:𝐼⟶TopMnd ∧ TopMnd ⊆ TopSp) → 𝑅:𝐼⟶TopSp)
134, 11, 12sylancl 693 . . 3 (𝜑𝑅:𝐼⟶TopSp)
141, 3, 2, 13prdstps 21342 . 2 (𝜑𝑌 ∈ TopSp)
15 eqid 2621 . . . . . . 7 (Base‘𝑌) = (Base‘𝑌)
1633ad2ant1 1080 . . . . . . 7 ((𝜑𝑓 ∈ (Base‘𝑌) ∧ 𝑔 ∈ (Base‘𝑌)) → 𝑆𝑉)
1723ad2ant1 1080 . . . . . . 7 ((𝜑𝑓 ∈ (Base‘𝑌) ∧ 𝑔 ∈ (Base‘𝑌)) → 𝐼𝑊)
18 ffn 6002 . . . . . . . . 9 (𝑅:𝐼⟶TopMnd → 𝑅 Fn 𝐼)
194, 18syl 17 . . . . . . . 8 (𝜑𝑅 Fn 𝐼)
20193ad2ant1 1080 . . . . . . 7 ((𝜑𝑓 ∈ (Base‘𝑌) ∧ 𝑔 ∈ (Base‘𝑌)) → 𝑅 Fn 𝐼)
21 simp2 1060 . . . . . . 7 ((𝜑𝑓 ∈ (Base‘𝑌) ∧ 𝑔 ∈ (Base‘𝑌)) → 𝑓 ∈ (Base‘𝑌))
22 simp3 1061 . . . . . . 7 ((𝜑𝑓 ∈ (Base‘𝑌) ∧ 𝑔 ∈ (Base‘𝑌)) → 𝑔 ∈ (Base‘𝑌))
23 eqid 2621 . . . . . . 7 (+g𝑌) = (+g𝑌)
241, 15, 16, 17, 20, 21, 22, 23prdsplusgval 16054 . . . . . 6 ((𝜑𝑓 ∈ (Base‘𝑌) ∧ 𝑔 ∈ (Base‘𝑌)) → (𝑓(+g𝑌)𝑔) = (𝑘𝐼 ↦ ((𝑓𝑘)(+g‘(𝑅𝑘))(𝑔𝑘))))
2524mpt2eq3dva 6672 . . . . 5 (𝜑 → (𝑓 ∈ (Base‘𝑌), 𝑔 ∈ (Base‘𝑌) ↦ (𝑓(+g𝑌)𝑔)) = (𝑓 ∈ (Base‘𝑌), 𝑔 ∈ (Base‘𝑌) ↦ (𝑘𝐼 ↦ ((𝑓𝑘)(+g‘(𝑅𝑘))(𝑔𝑘)))))
26 eqid 2621 . . . . . 6 (+𝑓𝑌) = (+𝑓𝑌)
2715, 23, 26plusffval 17168 . . . . 5 (+𝑓𝑌) = (𝑓 ∈ (Base‘𝑌), 𝑔 ∈ (Base‘𝑌) ↦ (𝑓(+g𝑌)𝑔))
28 vex 3189 . . . . . . . . . 10 𝑓 ∈ V
29 vex 3189 . . . . . . . . . 10 𝑔 ∈ V
3028, 29op1std 7123 . . . . . . . . 9 (𝑧 = ⟨𝑓, 𝑔⟩ → (1st𝑧) = 𝑓)
3130fveq1d 6150 . . . . . . . 8 (𝑧 = ⟨𝑓, 𝑔⟩ → ((1st𝑧)‘𝑘) = (𝑓𝑘))
3228, 29op2ndd 7124 . . . . . . . . 9 (𝑧 = ⟨𝑓, 𝑔⟩ → (2nd𝑧) = 𝑔)
3332fveq1d 6150 . . . . . . . 8 (𝑧 = ⟨𝑓, 𝑔⟩ → ((2nd𝑧)‘𝑘) = (𝑔𝑘))
3431, 33oveq12d 6622 . . . . . . 7 (𝑧 = ⟨𝑓, 𝑔⟩ → (((1st𝑧)‘𝑘)(+g‘(𝑅𝑘))((2nd𝑧)‘𝑘)) = ((𝑓𝑘)(+g‘(𝑅𝑘))(𝑔𝑘)))
3534mpteq2dv 4705 . . . . . 6 (𝑧 = ⟨𝑓, 𝑔⟩ → (𝑘𝐼 ↦ (((1st𝑧)‘𝑘)(+g‘(𝑅𝑘))((2nd𝑧)‘𝑘))) = (𝑘𝐼 ↦ ((𝑓𝑘)(+g‘(𝑅𝑘))(𝑔𝑘))))
3635mpt2mpt 6705 . . . . 5 (𝑧 ∈ ((Base‘𝑌) × (Base‘𝑌)) ↦ (𝑘𝐼 ↦ (((1st𝑧)‘𝑘)(+g‘(𝑅𝑘))((2nd𝑧)‘𝑘)))) = (𝑓 ∈ (Base‘𝑌), 𝑔 ∈ (Base‘𝑌) ↦ (𝑘𝐼 ↦ ((𝑓𝑘)(+g‘(𝑅𝑘))(𝑔𝑘))))
3725, 27, 363eqtr4g 2680 . . . 4 (𝜑 → (+𝑓𝑌) = (𝑧 ∈ ((Base‘𝑌) × (Base‘𝑌)) ↦ (𝑘𝐼 ↦ (((1st𝑧)‘𝑘)(+g‘(𝑅𝑘))((2nd𝑧)‘𝑘)))))
38 eqid 2621 . . . . 5 (∏t‘(TopOpen ∘ 𝑅)) = (∏t‘(TopOpen ∘ 𝑅))
39 eqid 2621 . . . . . . . 8 (TopOpen‘𝑌) = (TopOpen‘𝑌)
4015, 39istps 20651 . . . . . . 7 (𝑌 ∈ TopSp ↔ (TopOpen‘𝑌) ∈ (TopOn‘(Base‘𝑌)))
4114, 40sylib 208 . . . . . 6 (𝜑 → (TopOpen‘𝑌) ∈ (TopOn‘(Base‘𝑌)))
42 txtopon 21304 . . . . . 6 (((TopOpen‘𝑌) ∈ (TopOn‘(Base‘𝑌)) ∧ (TopOpen‘𝑌) ∈ (TopOn‘(Base‘𝑌))) → ((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) ∈ (TopOn‘((Base‘𝑌) × (Base‘𝑌))))
4341, 41, 42syl2anc 692 . . . . 5 (𝜑 → ((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) ∈ (TopOn‘((Base‘𝑌) × (Base‘𝑌))))
44 topnfn 16007 . . . . . . . 8 TopOpen Fn V
45 ssv 3604 . . . . . . . 8 TopSp ⊆ V
46 fnssres 5962 . . . . . . . 8 ((TopOpen Fn V ∧ TopSp ⊆ V) → (TopOpen ↾ TopSp) Fn TopSp)
4744, 45, 46mp2an 707 . . . . . . 7 (TopOpen ↾ TopSp) Fn TopSp
48 fvres 6164 . . . . . . . . 9 (𝑥 ∈ TopSp → ((TopOpen ↾ TopSp)‘𝑥) = (TopOpen‘𝑥))
49 eqid 2621 . . . . . . . . . 10 (TopOpen‘𝑥) = (TopOpen‘𝑥)
5049tpstop 20654 . . . . . . . . 9 (𝑥 ∈ TopSp → (TopOpen‘𝑥) ∈ Top)
5148, 50eqeltrd 2698 . . . . . . . 8 (𝑥 ∈ TopSp → ((TopOpen ↾ TopSp)‘𝑥) ∈ Top)
5251rgen 2917 . . . . . . 7 𝑥 ∈ TopSp ((TopOpen ↾ TopSp)‘𝑥) ∈ Top
53 ffnfv 6343 . . . . . . 7 ((TopOpen ↾ TopSp):TopSp⟶Top ↔ ((TopOpen ↾ TopSp) Fn TopSp ∧ ∀𝑥 ∈ TopSp ((TopOpen ↾ TopSp)‘𝑥) ∈ Top))
5447, 52, 53mpbir2an 954 . . . . . 6 (TopOpen ↾ TopSp):TopSp⟶Top
55 fco2 6016 . . . . . 6 (((TopOpen ↾ TopSp):TopSp⟶Top ∧ 𝑅:𝐼⟶TopSp) → (TopOpen ∘ 𝑅):𝐼⟶Top)
5654, 13, 55sylancr 694 . . . . 5 (𝜑 → (TopOpen ∘ 𝑅):𝐼⟶Top)
5734mpt2mpt 6705 . . . . . 6 (𝑧 ∈ ((Base‘𝑌) × (Base‘𝑌)) ↦ (((1st𝑧)‘𝑘)(+g‘(𝑅𝑘))((2nd𝑧)‘𝑘))) = (𝑓 ∈ (Base‘𝑌), 𝑔 ∈ (Base‘𝑌) ↦ ((𝑓𝑘)(+g‘(𝑅𝑘))(𝑔𝑘)))
58 eqid 2621 . . . . . . . 8 (TopOpen‘(𝑅𝑘)) = (TopOpen‘(𝑅𝑘))
59 eqid 2621 . . . . . . . 8 (+g‘(𝑅𝑘)) = (+g‘(𝑅𝑘))
604ffvelrnda 6315 . . . . . . . 8 ((𝜑𝑘𝐼) → (𝑅𝑘) ∈ TopMnd)
6141adantr 481 . . . . . . . 8 ((𝜑𝑘𝐼) → (TopOpen‘𝑌) ∈ (TopOn‘(Base‘𝑌)))
6261, 61cnmpt1st 21381 . . . . . . . . 9 ((𝜑𝑘𝐼) → (𝑓 ∈ (Base‘𝑌), 𝑔 ∈ (Base‘𝑌) ↦ 𝑓) ∈ (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn (TopOpen‘𝑌)))
631, 3, 2, 19, 39prdstopn 21341 . . . . . . . . . . . . . . 15 (𝜑 → (TopOpen‘𝑌) = (∏t‘(TopOpen ∘ 𝑅)))
6463adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑘𝐼) → (TopOpen‘𝑌) = (∏t‘(TopOpen ∘ 𝑅)))
6564, 61eqeltrrd 2699 . . . . . . . . . . . . 13 ((𝜑𝑘𝐼) → (∏t‘(TopOpen ∘ 𝑅)) ∈ (TopOn‘(Base‘𝑌)))
66 toponuni 20642 . . . . . . . . . . . . 13 ((∏t‘(TopOpen ∘ 𝑅)) ∈ (TopOn‘(Base‘𝑌)) → (Base‘𝑌) = (∏t‘(TopOpen ∘ 𝑅)))
6765, 66syl 17 . . . . . . . . . . . 12 ((𝜑𝑘𝐼) → (Base‘𝑌) = (∏t‘(TopOpen ∘ 𝑅)))
6867mpteq1d 4698 . . . . . . . . . . 11 ((𝜑𝑘𝐼) → (𝑥 ∈ (Base‘𝑌) ↦ (𝑥𝑘)) = (𝑥 (∏t‘(TopOpen ∘ 𝑅)) ↦ (𝑥𝑘)))
692adantr 481 . . . . . . . . . . . 12 ((𝜑𝑘𝐼) → 𝐼𝑊)
7056adantr 481 . . . . . . . . . . . 12 ((𝜑𝑘𝐼) → (TopOpen ∘ 𝑅):𝐼⟶Top)
71 simpr 477 . . . . . . . . . . . 12 ((𝜑𝑘𝐼) → 𝑘𝐼)
72 eqid 2621 . . . . . . . . . . . . 13 (∏t‘(TopOpen ∘ 𝑅)) = (∏t‘(TopOpen ∘ 𝑅))
7372, 38ptpjcn 21324 . . . . . . . . . . . 12 ((𝐼𝑊 ∧ (TopOpen ∘ 𝑅):𝐼⟶Top ∧ 𝑘𝐼) → (𝑥 (∏t‘(TopOpen ∘ 𝑅)) ↦ (𝑥𝑘)) ∈ ((∏t‘(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)‘𝑘)))
7469, 70, 71, 73syl3anc 1323 . . . . . . . . . . 11 ((𝜑𝑘𝐼) → (𝑥 (∏t‘(TopOpen ∘ 𝑅)) ↦ (𝑥𝑘)) ∈ ((∏t‘(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)‘𝑘)))
7568, 74eqeltrd 2698 . . . . . . . . . 10 ((𝜑𝑘𝐼) → (𝑥 ∈ (Base‘𝑌) ↦ (𝑥𝑘)) ∈ ((∏t‘(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)‘𝑘)))
7664eqcomd 2627 . . . . . . . . . . 11 ((𝜑𝑘𝐼) → (∏t‘(TopOpen ∘ 𝑅)) = (TopOpen‘𝑌))
77 fvco3 6232 . . . . . . . . . . . 12 ((𝑅:𝐼⟶TopMnd ∧ 𝑘𝐼) → ((TopOpen ∘ 𝑅)‘𝑘) = (TopOpen‘(𝑅𝑘)))
784, 77sylan 488 . . . . . . . . . . 11 ((𝜑𝑘𝐼) → ((TopOpen ∘ 𝑅)‘𝑘) = (TopOpen‘(𝑅𝑘)))
7976, 78oveq12d 6622 . . . . . . . . . 10 ((𝜑𝑘𝐼) → ((∏t‘(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)‘𝑘)) = ((TopOpen‘𝑌) Cn (TopOpen‘(𝑅𝑘))))
8075, 79eleqtrd 2700 . . . . . . . . 9 ((𝜑𝑘𝐼) → (𝑥 ∈ (Base‘𝑌) ↦ (𝑥𝑘)) ∈ ((TopOpen‘𝑌) Cn (TopOpen‘(𝑅𝑘))))
81 fveq1 6147 . . . . . . . . 9 (𝑥 = 𝑓 → (𝑥𝑘) = (𝑓𝑘))
8261, 61, 62, 61, 80, 81cnmpt21 21384 . . . . . . . 8 ((𝜑𝑘𝐼) → (𝑓 ∈ (Base‘𝑌), 𝑔 ∈ (Base‘𝑌) ↦ (𝑓𝑘)) ∈ (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn (TopOpen‘(𝑅𝑘))))
8361, 61cnmpt2nd 21382 . . . . . . . . 9 ((𝜑𝑘𝐼) → (𝑓 ∈ (Base‘𝑌), 𝑔 ∈ (Base‘𝑌) ↦ 𝑔) ∈ (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn (TopOpen‘𝑌)))
84 fveq1 6147 . . . . . . . . 9 (𝑥 = 𝑔 → (𝑥𝑘) = (𝑔𝑘))
8561, 61, 83, 61, 80, 84cnmpt21 21384 . . . . . . . 8 ((𝜑𝑘𝐼) → (𝑓 ∈ (Base‘𝑌), 𝑔 ∈ (Base‘𝑌) ↦ (𝑔𝑘)) ∈ (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn (TopOpen‘(𝑅𝑘))))
8658, 59, 60, 61, 61, 82, 85cnmpt2plusg 21802 . . . . . . 7 ((𝜑𝑘𝐼) → (𝑓 ∈ (Base‘𝑌), 𝑔 ∈ (Base‘𝑌) ↦ ((𝑓𝑘)(+g‘(𝑅𝑘))(𝑔𝑘))) ∈ (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn (TopOpen‘(𝑅𝑘))))
8778oveq2d 6620 . . . . . . 7 ((𝜑𝑘𝐼) → (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn ((TopOpen ∘ 𝑅)‘𝑘)) = (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn (TopOpen‘(𝑅𝑘))))
8886, 87eleqtrrd 2701 . . . . . 6 ((𝜑𝑘𝐼) → (𝑓 ∈ (Base‘𝑌), 𝑔 ∈ (Base‘𝑌) ↦ ((𝑓𝑘)(+g‘(𝑅𝑘))(𝑔𝑘))) ∈ (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn ((TopOpen ∘ 𝑅)‘𝑘)))
8957, 88syl5eqel 2702 . . . . 5 ((𝜑𝑘𝐼) → (𝑧 ∈ ((Base‘𝑌) × (Base‘𝑌)) ↦ (((1st𝑧)‘𝑘)(+g‘(𝑅𝑘))((2nd𝑧)‘𝑘))) ∈ (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn ((TopOpen ∘ 𝑅)‘𝑘)))
9038, 43, 2, 56, 89ptcn 21340 . . . 4 (𝜑 → (𝑧 ∈ ((Base‘𝑌) × (Base‘𝑌)) ↦ (𝑘𝐼 ↦ (((1st𝑧)‘𝑘)(+g‘(𝑅𝑘))((2nd𝑧)‘𝑘)))) ∈ (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn (∏t‘(TopOpen ∘ 𝑅))))
9137, 90eqeltrd 2698 . . 3 (𝜑 → (+𝑓𝑌) ∈ (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn (∏t‘(TopOpen ∘ 𝑅))))
9263oveq2d 6620 . . 3 (𝜑 → (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn (TopOpen‘𝑌)) = (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn (∏t‘(TopOpen ∘ 𝑅))))
9391, 92eleqtrrd 2701 . 2 (𝜑 → (+𝑓𝑌) ∈ (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn (TopOpen‘𝑌)))
9426, 39istmd 21788 . 2 (𝑌 ∈ TopMnd ↔ (𝑌 ∈ Mnd ∧ 𝑌 ∈ TopSp ∧ (+𝑓𝑌) ∈ (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn (TopOpen‘𝑌))))
959, 14, 93, 94syl3anbrc 1244 1 (𝜑𝑌 ∈ TopMnd)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  Vcvv 3186  wss 3555  cop 4154   cuni 4402  cmpt 4673   × cxp 5072  cres 5076  ccom 5078   Fn wfn 5842  wf 5843  cfv 5847  (class class class)co 6604  cmpt2 6606  1st c1st 7111  2nd c2nd 7112  Basecbs 15781  +gcplusg 15862  TopOpenctopn 16003  tcpt 16020  Xscprds 16027  +𝑓cplusf 17160  Mndcmnd 17215  Topctop 20617  TopOnctopon 20618  TopSpctps 20619   Cn ccn 20938   ×t ctx 21273  TopMndctmd 21784
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-iin 4488  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-map 7804  df-ixp 7853  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-fi 8261  df-sup 8292  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-2 11023  df-3 11024  df-4 11025  df-5 11026  df-6 11027  df-7 11028  df-8 11029  df-9 11030  df-n0 11237  df-z 11322  df-dec 11438  df-uz 11632  df-fz 12269  df-struct 15783  df-ndx 15784  df-slot 15785  df-base 15786  df-plusg 15875  df-mulr 15876  df-sca 15878  df-vsca 15879  df-ip 15880  df-tset 15881  df-ple 15882  df-ds 15885  df-hom 15887  df-cco 15888  df-rest 16004  df-topn 16005  df-0g 16023  df-topgen 16025  df-pt 16026  df-prds 16029  df-plusf 17162  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-top 20621  df-bases 20622  df-topon 20623  df-topsp 20624  df-cn 20941  df-cnp 20942  df-tx 21275  df-tmd 21786
This theorem is referenced by:  prdstgpd  21838
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