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Theorem prdstmdd 21737
 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 21689 . . . . 5 (𝑥 ∈ TopMnd → 𝑥 ∈ Mnd)
65ssriv 3572 . . . 4 TopMnd ⊆ Mnd
7 fss 5969 . . . 4 ((𝑅:𝐼⟶TopMnd ∧ TopMnd ⊆ Mnd) → 𝑅:𝐼⟶Mnd)
84, 6, 7sylancl 693 . . 3 (𝜑𝑅:𝐼⟶Mnd)
91, 2, 3, 8prdsmndd 17146 . 2 (𝜑𝑌 ∈ Mnd)
10 tmdtps 21690 . . . . 5 (𝑥 ∈ TopMnd → 𝑥 ∈ TopSp)
1110ssriv 3572 . . . 4 TopMnd ⊆ TopSp
12 fss 5969 . . . 4 ((𝑅:𝐼⟶TopMnd ∧ TopMnd ⊆ TopSp) → 𝑅:𝐼⟶TopSp)
134, 11, 12sylancl 693 . . 3 (𝜑𝑅:𝐼⟶TopSp)
141, 3, 2, 13prdstps 21242 . 2 (𝜑𝑌 ∈ TopSp)
15 eqid 2610 . . . . . . 7 (Base‘𝑌) = (Base‘𝑌)
1633ad2ant1 1075 . . . . . . 7 ((𝜑𝑓 ∈ (Base‘𝑌) ∧ 𝑔 ∈ (Base‘𝑌)) → 𝑆𝑉)
1723ad2ant1 1075 . . . . . . 7 ((𝜑𝑓 ∈ (Base‘𝑌) ∧ 𝑔 ∈ (Base‘𝑌)) → 𝐼𝑊)
18 ffn 5958 . . . . . . . . 9 (𝑅:𝐼⟶TopMnd → 𝑅 Fn 𝐼)
194, 18syl 17 . . . . . . . 8 (𝜑𝑅 Fn 𝐼)
20193ad2ant1 1075 . . . . . . 7 ((𝜑𝑓 ∈ (Base‘𝑌) ∧ 𝑔 ∈ (Base‘𝑌)) → 𝑅 Fn 𝐼)
21 simp2 1055 . . . . . . 7 ((𝜑𝑓 ∈ (Base‘𝑌) ∧ 𝑔 ∈ (Base‘𝑌)) → 𝑓 ∈ (Base‘𝑌))
22 simp3 1056 . . . . . . 7 ((𝜑𝑓 ∈ (Base‘𝑌) ∧ 𝑔 ∈ (Base‘𝑌)) → 𝑔 ∈ (Base‘𝑌))
23 eqid 2610 . . . . . . 7 (+g𝑌) = (+g𝑌)
241, 15, 16, 17, 20, 21, 22, 23prdsplusgval 15956 . . . . . 6 ((𝜑𝑓 ∈ (Base‘𝑌) ∧ 𝑔 ∈ (Base‘𝑌)) → (𝑓(+g𝑌)𝑔) = (𝑘𝐼 ↦ ((𝑓𝑘)(+g‘(𝑅𝑘))(𝑔𝑘))))
2524mpt2eq3dva 6617 . . . . 5 (𝜑 → (𝑓 ∈ (Base‘𝑌), 𝑔 ∈ (Base‘𝑌) ↦ (𝑓(+g𝑌)𝑔)) = (𝑓 ∈ (Base‘𝑌), 𝑔 ∈ (Base‘𝑌) ↦ (𝑘𝐼 ↦ ((𝑓𝑘)(+g‘(𝑅𝑘))(𝑔𝑘)))))
26 eqid 2610 . . . . . 6 (+𝑓𝑌) = (+𝑓𝑌)
2715, 23, 26plusffval 17070 . . . . 5 (+𝑓𝑌) = (𝑓 ∈ (Base‘𝑌), 𝑔 ∈ (Base‘𝑌) ↦ (𝑓(+g𝑌)𝑔))
28 vex 3176 . . . . . . . . . 10 𝑓 ∈ V
29 vex 3176 . . . . . . . . . 10 𝑔 ∈ V
3028, 29op1std 7069 . . . . . . . . 9 (𝑧 = ⟨𝑓, 𝑔⟩ → (1st𝑧) = 𝑓)
3130fveq1d 6105 . . . . . . . 8 (𝑧 = ⟨𝑓, 𝑔⟩ → ((1st𝑧)‘𝑘) = (𝑓𝑘))
3228, 29op2ndd 7070 . . . . . . . . 9 (𝑧 = ⟨𝑓, 𝑔⟩ → (2nd𝑧) = 𝑔)
3332fveq1d 6105 . . . . . . . 8 (𝑧 = ⟨𝑓, 𝑔⟩ → ((2nd𝑧)‘𝑘) = (𝑔𝑘))
3431, 33oveq12d 6567 . . . . . . 7 (𝑧 = ⟨𝑓, 𝑔⟩ → (((1st𝑧)‘𝑘)(+g‘(𝑅𝑘))((2nd𝑧)‘𝑘)) = ((𝑓𝑘)(+g‘(𝑅𝑘))(𝑔𝑘)))
3534mpteq2dv 4673 . . . . . 6 (𝑧 = ⟨𝑓, 𝑔⟩ → (𝑘𝐼 ↦ (((1st𝑧)‘𝑘)(+g‘(𝑅𝑘))((2nd𝑧)‘𝑘))) = (𝑘𝐼 ↦ ((𝑓𝑘)(+g‘(𝑅𝑘))(𝑔𝑘))))
3635mpt2mpt 6650 . . . . 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 2669 . . . 4 (𝜑 → (+𝑓𝑌) = (𝑧 ∈ ((Base‘𝑌) × (Base‘𝑌)) ↦ (𝑘𝐼 ↦ (((1st𝑧)‘𝑘)(+g‘(𝑅𝑘))((2nd𝑧)‘𝑘)))))
38 eqid 2610 . . . . 5 (∏t‘(TopOpen ∘ 𝑅)) = (∏t‘(TopOpen ∘ 𝑅))
39 eqid 2610 . . . . . . . 8 (TopOpen‘𝑌) = (TopOpen‘𝑌)
4015, 39istps 20551 . . . . . . 7 (𝑌 ∈ TopSp ↔ (TopOpen‘𝑌) ∈ (TopOn‘(Base‘𝑌)))
4114, 40sylib 207 . . . . . 6 (𝜑 → (TopOpen‘𝑌) ∈ (TopOn‘(Base‘𝑌)))
42 txtopon 21204 . . . . . 6 (((TopOpen‘𝑌) ∈ (TopOn‘(Base‘𝑌)) ∧ (TopOpen‘𝑌) ∈ (TopOn‘(Base‘𝑌))) → ((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) ∈ (TopOn‘((Base‘𝑌) × (Base‘𝑌))))
4341, 41, 42syl2anc 691 . . . . 5 (𝜑 → ((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) ∈ (TopOn‘((Base‘𝑌) × (Base‘𝑌))))
44 topnfn 15909 . . . . . . . 8 TopOpen Fn V
45 ssv 3588 . . . . . . . 8 TopSp ⊆ V
46 fnssres 5918 . . . . . . . 8 ((TopOpen Fn V ∧ TopSp ⊆ V) → (TopOpen ↾ TopSp) Fn TopSp)
4744, 45, 46mp2an 704 . . . . . . 7 (TopOpen ↾ TopSp) Fn TopSp
48 fvres 6117 . . . . . . . . 9 (𝑥 ∈ TopSp → ((TopOpen ↾ TopSp)‘𝑥) = (TopOpen‘𝑥))
49 eqid 2610 . . . . . . . . . 10 (TopOpen‘𝑥) = (TopOpen‘𝑥)
5049tpstop 20554 . . . . . . . . 9 (𝑥 ∈ TopSp → (TopOpen‘𝑥) ∈ Top)
5148, 50eqeltrd 2688 . . . . . . . 8 (𝑥 ∈ TopSp → ((TopOpen ↾ TopSp)‘𝑥) ∈ Top)
5251rgen 2906 . . . . . . 7 𝑥 ∈ TopSp ((TopOpen ↾ TopSp)‘𝑥) ∈ Top
53 ffnfv 6295 . . . . . . 7 ((TopOpen ↾ TopSp):TopSp⟶Top ↔ ((TopOpen ↾ TopSp) Fn TopSp ∧ ∀𝑥 ∈ TopSp ((TopOpen ↾ TopSp)‘𝑥) ∈ Top))
5447, 52, 53mpbir2an 957 . . . . . 6 (TopOpen ↾ TopSp):TopSp⟶Top
55 fco2 5972 . . . . . 6 (((TopOpen ↾ TopSp):TopSp⟶Top ∧ 𝑅:𝐼⟶TopSp) → (TopOpen ∘ 𝑅):𝐼⟶Top)
5654, 13, 55sylancr 694 . . . . 5 (𝜑 → (TopOpen ∘ 𝑅):𝐼⟶Top)
5734mpt2mpt 6650 . . . . . 6 (𝑧 ∈ ((Base‘𝑌) × (Base‘𝑌)) ↦ (((1st𝑧)‘𝑘)(+g‘(𝑅𝑘))((2nd𝑧)‘𝑘))) = (𝑓 ∈ (Base‘𝑌), 𝑔 ∈ (Base‘𝑌) ↦ ((𝑓𝑘)(+g‘(𝑅𝑘))(𝑔𝑘)))
58 eqid 2610 . . . . . . . 8 (TopOpen‘(𝑅𝑘)) = (TopOpen‘(𝑅𝑘))
59 eqid 2610 . . . . . . . 8 (+g‘(𝑅𝑘)) = (+g‘(𝑅𝑘))
604ffvelrnda 6267 . . . . . . . 8 ((𝜑𝑘𝐼) → (𝑅𝑘) ∈ TopMnd)
6141adantr 480 . . . . . . . 8 ((𝜑𝑘𝐼) → (TopOpen‘𝑌) ∈ (TopOn‘(Base‘𝑌)))
6261, 61cnmpt1st 21281 . . . . . . . . 9 ((𝜑𝑘𝐼) → (𝑓 ∈ (Base‘𝑌), 𝑔 ∈ (Base‘𝑌) ↦ 𝑓) ∈ (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn (TopOpen‘𝑌)))
631, 3, 2, 19, 39prdstopn 21241 . . . . . . . . . . . . . . 15 (𝜑 → (TopOpen‘𝑌) = (∏t‘(TopOpen ∘ 𝑅)))
6463adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑘𝐼) → (TopOpen‘𝑌) = (∏t‘(TopOpen ∘ 𝑅)))
6564, 61eqeltrrd 2689 . . . . . . . . . . . . 13 ((𝜑𝑘𝐼) → (∏t‘(TopOpen ∘ 𝑅)) ∈ (TopOn‘(Base‘𝑌)))
66 toponuni 20542 . . . . . . . . . . . . 13 ((∏t‘(TopOpen ∘ 𝑅)) ∈ (TopOn‘(Base‘𝑌)) → (Base‘𝑌) = (∏t‘(TopOpen ∘ 𝑅)))
6765, 66syl 17 . . . . . . . . . . . 12 ((𝜑𝑘𝐼) → (Base‘𝑌) = (∏t‘(TopOpen ∘ 𝑅)))
6867mpteq1d 4666 . . . . . . . . . . 11 ((𝜑𝑘𝐼) → (𝑥 ∈ (Base‘𝑌) ↦ (𝑥𝑘)) = (𝑥 (∏t‘(TopOpen ∘ 𝑅)) ↦ (𝑥𝑘)))
692adantr 480 . . . . . . . . . . . 12 ((𝜑𝑘𝐼) → 𝐼𝑊)
7056adantr 480 . . . . . . . . . . . 12 ((𝜑𝑘𝐼) → (TopOpen ∘ 𝑅):𝐼⟶Top)
71 simpr 476 . . . . . . . . . . . 12 ((𝜑𝑘𝐼) → 𝑘𝐼)
72 eqid 2610 . . . . . . . . . . . . 13 (∏t‘(TopOpen ∘ 𝑅)) = (∏t‘(TopOpen ∘ 𝑅))
7372, 38ptpjcn 21224 . . . . . . . . . . . 12 ((𝐼𝑊 ∧ (TopOpen ∘ 𝑅):𝐼⟶Top ∧ 𝑘𝐼) → (𝑥 (∏t‘(TopOpen ∘ 𝑅)) ↦ (𝑥𝑘)) ∈ ((∏t‘(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)‘𝑘)))
7469, 70, 71, 73syl3anc 1318 . . . . . . . . . . 11 ((𝜑𝑘𝐼) → (𝑥 (∏t‘(TopOpen ∘ 𝑅)) ↦ (𝑥𝑘)) ∈ ((∏t‘(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)‘𝑘)))
7568, 74eqeltrd 2688 . . . . . . . . . 10 ((𝜑𝑘𝐼) → (𝑥 ∈ (Base‘𝑌) ↦ (𝑥𝑘)) ∈ ((∏t‘(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)‘𝑘)))
7664eqcomd 2616 . . . . . . . . . . 11 ((𝜑𝑘𝐼) → (∏t‘(TopOpen ∘ 𝑅)) = (TopOpen‘𝑌))
77 fvco3 6185 . . . . . . . . . . . 12 ((𝑅:𝐼⟶TopMnd ∧ 𝑘𝐼) → ((TopOpen ∘ 𝑅)‘𝑘) = (TopOpen‘(𝑅𝑘)))
784, 77sylan 487 . . . . . . . . . . 11 ((𝜑𝑘𝐼) → ((TopOpen ∘ 𝑅)‘𝑘) = (TopOpen‘(𝑅𝑘)))
7976, 78oveq12d 6567 . . . . . . . . . 10 ((𝜑𝑘𝐼) → ((∏t‘(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)‘𝑘)) = ((TopOpen‘𝑌) Cn (TopOpen‘(𝑅𝑘))))
8075, 79eleqtrd 2690 . . . . . . . . 9 ((𝜑𝑘𝐼) → (𝑥 ∈ (Base‘𝑌) ↦ (𝑥𝑘)) ∈ ((TopOpen‘𝑌) Cn (TopOpen‘(𝑅𝑘))))
81 fveq1 6102 . . . . . . . . 9 (𝑥 = 𝑓 → (𝑥𝑘) = (𝑓𝑘))
8261, 61, 62, 61, 80, 81cnmpt21 21284 . . . . . . . 8 ((𝜑𝑘𝐼) → (𝑓 ∈ (Base‘𝑌), 𝑔 ∈ (Base‘𝑌) ↦ (𝑓𝑘)) ∈ (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn (TopOpen‘(𝑅𝑘))))
8361, 61cnmpt2nd 21282 . . . . . . . . 9 ((𝜑𝑘𝐼) → (𝑓 ∈ (Base‘𝑌), 𝑔 ∈ (Base‘𝑌) ↦ 𝑔) ∈ (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn (TopOpen‘𝑌)))
84 fveq1 6102 . . . . . . . . 9 (𝑥 = 𝑔 → (𝑥𝑘) = (𝑔𝑘))
8561, 61, 83, 61, 80, 84cnmpt21 21284 . . . . . . . 8 ((𝜑𝑘𝐼) → (𝑓 ∈ (Base‘𝑌), 𝑔 ∈ (Base‘𝑌) ↦ (𝑔𝑘)) ∈ (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn (TopOpen‘(𝑅𝑘))))
8658, 59, 60, 61, 61, 82, 85cnmpt2plusg 21702 . . . . . . 7 ((𝜑𝑘𝐼) → (𝑓 ∈ (Base‘𝑌), 𝑔 ∈ (Base‘𝑌) ↦ ((𝑓𝑘)(+g‘(𝑅𝑘))(𝑔𝑘))) ∈ (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn (TopOpen‘(𝑅𝑘))))
8778oveq2d 6565 . . . . . . 7 ((𝜑𝑘𝐼) → (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn ((TopOpen ∘ 𝑅)‘𝑘)) = (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn (TopOpen‘(𝑅𝑘))))
8886, 87eleqtrrd 2691 . . . . . 6 ((𝜑𝑘𝐼) → (𝑓 ∈ (Base‘𝑌), 𝑔 ∈ (Base‘𝑌) ↦ ((𝑓𝑘)(+g‘(𝑅𝑘))(𝑔𝑘))) ∈ (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn ((TopOpen ∘ 𝑅)‘𝑘)))
8957, 88syl5eqel 2692 . . . . 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 21240 . . . 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 2688 . . 3 (𝜑 → (+𝑓𝑌) ∈ (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn (∏t‘(TopOpen ∘ 𝑅))))
9263oveq2d 6565 . . 3 (𝜑 → (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn (TopOpen‘𝑌)) = (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn (∏t‘(TopOpen ∘ 𝑅))))
9391, 92eleqtrrd 2691 . 2 (𝜑 → (+𝑓𝑌) ∈ (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn (TopOpen‘𝑌)))
9426, 39istmd 21688 . 2 (𝑌 ∈ TopMnd ↔ (𝑌 ∈ Mnd ∧ 𝑌 ∈ TopSp ∧ (+𝑓𝑌) ∈ (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn (TopOpen‘𝑌))))
959, 14, 93, 94syl3anbrc 1239 1 (𝜑𝑌 ∈ TopMnd)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   ⊆ wss 3540  ⟨cop 4131  ∪ cuni 4372   ↦ cmpt 4643   × cxp 5036   ↾ cres 5040   ∘ ccom 5042   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  1st c1st 7057  2nd c2nd 7058  Basecbs 15695  +gcplusg 15768  TopOpenctopn 15905  ∏tcpt 15922  Xscprds 15929  +𝑓cplusf 17062  Mndcmnd 17117  Topctop 20517  TopOnctopon 20518  TopSpctps 20519   Cn ccn 20838   ×t ctx 21173  TopMndctmd 21684 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  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  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-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  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-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  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-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  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-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fi 8200  df-sup 8231  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-fz 12198  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-plusg 15781  df-mulr 15782  df-sca 15784  df-vsca 15785  df-ip 15786  df-tset 15787  df-ple 15788  df-ds 15791  df-hom 15793  df-cco 15794  df-rest 15906  df-topn 15907  df-0g 15925  df-topgen 15927  df-pt 15928  df-prds 15931  df-plusf 17064  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-top 20521  df-bases 20522  df-topon 20523  df-topsp 20524  df-cn 20841  df-cnp 20842  df-tx 21175  df-tmd 21686 This theorem is referenced by:  prdstgpd  21738
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