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Theorem prdstmdd 24132
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 24083 . . . . 5 (𝑥 ∈ TopMnd → 𝑥 ∈ Mnd)
65ssriv 3987 . . . 4 TopMnd ⊆ Mnd
7 fss 6752 . . . 4 ((𝑅:𝐼⟶TopMnd ∧ TopMnd ⊆ Mnd) → 𝑅:𝐼⟶Mnd)
84, 6, 7sylancl 586 . . 3 (𝜑𝑅:𝐼⟶Mnd)
91, 2, 3, 8prdsmndd 18783 . 2 (𝜑𝑌 ∈ Mnd)
10 tmdtps 24084 . . . . 5 (𝑥 ∈ TopMnd → 𝑥 ∈ TopSp)
1110ssriv 3987 . . . 4 TopMnd ⊆ TopSp
12 fss 6752 . . . 4 ((𝑅:𝐼⟶TopMnd ∧ TopMnd ⊆ TopSp) → 𝑅:𝐼⟶TopSp)
134, 11, 12sylancl 586 . . 3 (𝜑𝑅:𝐼⟶TopSp)
141, 3, 2, 13prdstps 23637 . 2 (𝜑𝑌 ∈ TopSp)
15 eqid 2737 . . . . . . 7 (Base‘𝑌) = (Base‘𝑌)
1633ad2ant1 1134 . . . . . . 7 ((𝜑𝑓 ∈ (Base‘𝑌) ∧ 𝑔 ∈ (Base‘𝑌)) → 𝑆𝑉)
1723ad2ant1 1134 . . . . . . 7 ((𝜑𝑓 ∈ (Base‘𝑌) ∧ 𝑔 ∈ (Base‘𝑌)) → 𝐼𝑊)
184ffnd 6737 . . . . . . . 8 (𝜑𝑅 Fn 𝐼)
19183ad2ant1 1134 . . . . . . 7 ((𝜑𝑓 ∈ (Base‘𝑌) ∧ 𝑔 ∈ (Base‘𝑌)) → 𝑅 Fn 𝐼)
20 simp2 1138 . . . . . . 7 ((𝜑𝑓 ∈ (Base‘𝑌) ∧ 𝑔 ∈ (Base‘𝑌)) → 𝑓 ∈ (Base‘𝑌))
21 simp3 1139 . . . . . . 7 ((𝜑𝑓 ∈ (Base‘𝑌) ∧ 𝑔 ∈ (Base‘𝑌)) → 𝑔 ∈ (Base‘𝑌))
22 eqid 2737 . . . . . . 7 (+g𝑌) = (+g𝑌)
231, 15, 16, 17, 19, 20, 21, 22prdsplusgval 17518 . . . . . 6 ((𝜑𝑓 ∈ (Base‘𝑌) ∧ 𝑔 ∈ (Base‘𝑌)) → (𝑓(+g𝑌)𝑔) = (𝑘𝐼 ↦ ((𝑓𝑘)(+g‘(𝑅𝑘))(𝑔𝑘))))
2423mpoeq3dva 7510 . . . . 5 (𝜑 → (𝑓 ∈ (Base‘𝑌), 𝑔 ∈ (Base‘𝑌) ↦ (𝑓(+g𝑌)𝑔)) = (𝑓 ∈ (Base‘𝑌), 𝑔 ∈ (Base‘𝑌) ↦ (𝑘𝐼 ↦ ((𝑓𝑘)(+g‘(𝑅𝑘))(𝑔𝑘)))))
25 eqid 2737 . . . . . 6 (+𝑓𝑌) = (+𝑓𝑌)
2615, 22, 25plusffval 18659 . . . . 5 (+𝑓𝑌) = (𝑓 ∈ (Base‘𝑌), 𝑔 ∈ (Base‘𝑌) ↦ (𝑓(+g𝑌)𝑔))
27 vex 3484 . . . . . . . . . 10 𝑓 ∈ V
28 vex 3484 . . . . . . . . . 10 𝑔 ∈ V
2927, 28op1std 8024 . . . . . . . . 9 (𝑧 = ⟨𝑓, 𝑔⟩ → (1st𝑧) = 𝑓)
3029fveq1d 6908 . . . . . . . 8 (𝑧 = ⟨𝑓, 𝑔⟩ → ((1st𝑧)‘𝑘) = (𝑓𝑘))
3127, 28op2ndd 8025 . . . . . . . . 9 (𝑧 = ⟨𝑓, 𝑔⟩ → (2nd𝑧) = 𝑔)
3231fveq1d 6908 . . . . . . . 8 (𝑧 = ⟨𝑓, 𝑔⟩ → ((2nd𝑧)‘𝑘) = (𝑔𝑘))
3330, 32oveq12d 7449 . . . . . . 7 (𝑧 = ⟨𝑓, 𝑔⟩ → (((1st𝑧)‘𝑘)(+g‘(𝑅𝑘))((2nd𝑧)‘𝑘)) = ((𝑓𝑘)(+g‘(𝑅𝑘))(𝑔𝑘)))
3433mpteq2dv 5244 . . . . . 6 (𝑧 = ⟨𝑓, 𝑔⟩ → (𝑘𝐼 ↦ (((1st𝑧)‘𝑘)(+g‘(𝑅𝑘))((2nd𝑧)‘𝑘))) = (𝑘𝐼 ↦ ((𝑓𝑘)(+g‘(𝑅𝑘))(𝑔𝑘))))
3534mpompt 7547 . . . . 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 2802 . . . 4 (𝜑 → (+𝑓𝑌) = (𝑧 ∈ ((Base‘𝑌) × (Base‘𝑌)) ↦ (𝑘𝐼 ↦ (((1st𝑧)‘𝑘)(+g‘(𝑅𝑘))((2nd𝑧)‘𝑘)))))
37 eqid 2737 . . . . 5 (∏t‘(TopOpen ∘ 𝑅)) = (∏t‘(TopOpen ∘ 𝑅))
38 eqid 2737 . . . . . . . 8 (TopOpen‘𝑌) = (TopOpen‘𝑌)
3915, 38istps 22940 . . . . . . 7 (𝑌 ∈ TopSp ↔ (TopOpen‘𝑌) ∈ (TopOn‘(Base‘𝑌)))
4014, 39sylib 218 . . . . . 6 (𝜑 → (TopOpen‘𝑌) ∈ (TopOn‘(Base‘𝑌)))
41 txtopon 23599 . . . . . 6 (((TopOpen‘𝑌) ∈ (TopOn‘(Base‘𝑌)) ∧ (TopOpen‘𝑌) ∈ (TopOn‘(Base‘𝑌))) → ((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) ∈ (TopOn‘((Base‘𝑌) × (Base‘𝑌))))
4240, 40, 41syl2anc 584 . . . . 5 (𝜑 → ((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) ∈ (TopOn‘((Base‘𝑌) × (Base‘𝑌))))
43 topnfn 17470 . . . . . . . 8 TopOpen Fn V
44 ssv 4008 . . . . . . . 8 TopSp ⊆ V
45 fnssres 6691 . . . . . . . 8 ((TopOpen Fn V ∧ TopSp ⊆ V) → (TopOpen ↾ TopSp) Fn TopSp)
4643, 44, 45mp2an 692 . . . . . . 7 (TopOpen ↾ TopSp) Fn TopSp
47 fvres 6925 . . . . . . . . 9 (𝑥 ∈ TopSp → ((TopOpen ↾ TopSp)‘𝑥) = (TopOpen‘𝑥))
48 eqid 2737 . . . . . . . . . 10 (TopOpen‘𝑥) = (TopOpen‘𝑥)
4948tpstop 22943 . . . . . . . . 9 (𝑥 ∈ TopSp → (TopOpen‘𝑥) ∈ Top)
5047, 49eqeltrd 2841 . . . . . . . 8 (𝑥 ∈ TopSp → ((TopOpen ↾ TopSp)‘𝑥) ∈ Top)
5150rgen 3063 . . . . . . 7 𝑥 ∈ TopSp ((TopOpen ↾ TopSp)‘𝑥) ∈ Top
52 ffnfv 7139 . . . . . . 7 ((TopOpen ↾ TopSp):TopSp⟶Top ↔ ((TopOpen ↾ TopSp) Fn TopSp ∧ ∀𝑥 ∈ TopSp ((TopOpen ↾ TopSp)‘𝑥) ∈ Top))
5346, 51, 52mpbir2an 711 . . . . . 6 (TopOpen ↾ TopSp):TopSp⟶Top
54 fco2 6762 . . . . . 6 (((TopOpen ↾ TopSp):TopSp⟶Top ∧ 𝑅:𝐼⟶TopSp) → (TopOpen ∘ 𝑅):𝐼⟶Top)
5553, 13, 54sylancr 587 . . . . 5 (𝜑 → (TopOpen ∘ 𝑅):𝐼⟶Top)
5633mpompt 7547 . . . . . 6 (𝑧 ∈ ((Base‘𝑌) × (Base‘𝑌)) ↦ (((1st𝑧)‘𝑘)(+g‘(𝑅𝑘))((2nd𝑧)‘𝑘))) = (𝑓 ∈ (Base‘𝑌), 𝑔 ∈ (Base‘𝑌) ↦ ((𝑓𝑘)(+g‘(𝑅𝑘))(𝑔𝑘)))
57 eqid 2737 . . . . . . . 8 (TopOpen‘(𝑅𝑘)) = (TopOpen‘(𝑅𝑘))
58 eqid 2737 . . . . . . . 8 (+g‘(𝑅𝑘)) = (+g‘(𝑅𝑘))
594ffvelcdmda 7104 . . . . . . . 8 ((𝜑𝑘𝐼) → (𝑅𝑘) ∈ TopMnd)
6040adantr 480 . . . . . . . 8 ((𝜑𝑘𝐼) → (TopOpen‘𝑌) ∈ (TopOn‘(Base‘𝑌)))
6160, 60cnmpt1st 23676 . . . . . . . . 9 ((𝜑𝑘𝐼) → (𝑓 ∈ (Base‘𝑌), 𝑔 ∈ (Base‘𝑌) ↦ 𝑓) ∈ (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn (TopOpen‘𝑌)))
621, 3, 2, 18, 38prdstopn 23636 . . . . . . . . . . . . . . 15 (𝜑 → (TopOpen‘𝑌) = (∏t‘(TopOpen ∘ 𝑅)))
6362adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑘𝐼) → (TopOpen‘𝑌) = (∏t‘(TopOpen ∘ 𝑅)))
6463, 60eqeltrrd 2842 . . . . . . . . . . . . 13 ((𝜑𝑘𝐼) → (∏t‘(TopOpen ∘ 𝑅)) ∈ (TopOn‘(Base‘𝑌)))
65 toponuni 22920 . . . . . . . . . . . . 13 ((∏t‘(TopOpen ∘ 𝑅)) ∈ (TopOn‘(Base‘𝑌)) → (Base‘𝑌) = (∏t‘(TopOpen ∘ 𝑅)))
6664, 65syl 17 . . . . . . . . . . . 12 ((𝜑𝑘𝐼) → (Base‘𝑌) = (∏t‘(TopOpen ∘ 𝑅)))
6766mpteq1d 5237 . . . . . . . . . . 11 ((𝜑𝑘𝐼) → (𝑥 ∈ (Base‘𝑌) ↦ (𝑥𝑘)) = (𝑥 (∏t‘(TopOpen ∘ 𝑅)) ↦ (𝑥𝑘)))
682adantr 480 . . . . . . . . . . . 12 ((𝜑𝑘𝐼) → 𝐼𝑊)
6955adantr 480 . . . . . . . . . . . 12 ((𝜑𝑘𝐼) → (TopOpen ∘ 𝑅):𝐼⟶Top)
70 simpr 484 . . . . . . . . . . . 12 ((𝜑𝑘𝐼) → 𝑘𝐼)
71 eqid 2737 . . . . . . . . . . . . 13 (∏t‘(TopOpen ∘ 𝑅)) = (∏t‘(TopOpen ∘ 𝑅))
7271, 37ptpjcn 23619 . . . . . . . . . . . 12 ((𝐼𝑊 ∧ (TopOpen ∘ 𝑅):𝐼⟶Top ∧ 𝑘𝐼) → (𝑥 (∏t‘(TopOpen ∘ 𝑅)) ↦ (𝑥𝑘)) ∈ ((∏t‘(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)‘𝑘)))
7368, 69, 70, 72syl3anc 1373 . . . . . . . . . . 11 ((𝜑𝑘𝐼) → (𝑥 (∏t‘(TopOpen ∘ 𝑅)) ↦ (𝑥𝑘)) ∈ ((∏t‘(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)‘𝑘)))
7467, 73eqeltrd 2841 . . . . . . . . . 10 ((𝜑𝑘𝐼) → (𝑥 ∈ (Base‘𝑌) ↦ (𝑥𝑘)) ∈ ((∏t‘(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)‘𝑘)))
7563eqcomd 2743 . . . . . . . . . . 11 ((𝜑𝑘𝐼) → (∏t‘(TopOpen ∘ 𝑅)) = (TopOpen‘𝑌))
76 fvco3 7008 . . . . . . . . . . . 12 ((𝑅:𝐼⟶TopMnd ∧ 𝑘𝐼) → ((TopOpen ∘ 𝑅)‘𝑘) = (TopOpen‘(𝑅𝑘)))
774, 76sylan 580 . . . . . . . . . . 11 ((𝜑𝑘𝐼) → ((TopOpen ∘ 𝑅)‘𝑘) = (TopOpen‘(𝑅𝑘)))
7875, 77oveq12d 7449 . . . . . . . . . 10 ((𝜑𝑘𝐼) → ((∏t‘(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)‘𝑘)) = ((TopOpen‘𝑌) Cn (TopOpen‘(𝑅𝑘))))
7974, 78eleqtrd 2843 . . . . . . . . 9 ((𝜑𝑘𝐼) → (𝑥 ∈ (Base‘𝑌) ↦ (𝑥𝑘)) ∈ ((TopOpen‘𝑌) Cn (TopOpen‘(𝑅𝑘))))
80 fveq1 6905 . . . . . . . . 9 (𝑥 = 𝑓 → (𝑥𝑘) = (𝑓𝑘))
8160, 60, 61, 60, 79, 80cnmpt21 23679 . . . . . . . 8 ((𝜑𝑘𝐼) → (𝑓 ∈ (Base‘𝑌), 𝑔 ∈ (Base‘𝑌) ↦ (𝑓𝑘)) ∈ (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn (TopOpen‘(𝑅𝑘))))
8260, 60cnmpt2nd 23677 . . . . . . . . 9 ((𝜑𝑘𝐼) → (𝑓 ∈ (Base‘𝑌), 𝑔 ∈ (Base‘𝑌) ↦ 𝑔) ∈ (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn (TopOpen‘𝑌)))
83 fveq1 6905 . . . . . . . . 9 (𝑥 = 𝑔 → (𝑥𝑘) = (𝑔𝑘))
8460, 60, 82, 60, 79, 83cnmpt21 23679 . . . . . . . 8 ((𝜑𝑘𝐼) → (𝑓 ∈ (Base‘𝑌), 𝑔 ∈ (Base‘𝑌) ↦ (𝑔𝑘)) ∈ (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn (TopOpen‘(𝑅𝑘))))
8557, 58, 59, 60, 60, 81, 84cnmpt2plusg 24096 . . . . . . 7 ((𝜑𝑘𝐼) → (𝑓 ∈ (Base‘𝑌), 𝑔 ∈ (Base‘𝑌) ↦ ((𝑓𝑘)(+g‘(𝑅𝑘))(𝑔𝑘))) ∈ (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn (TopOpen‘(𝑅𝑘))))
8677oveq2d 7447 . . . . . . 7 ((𝜑𝑘𝐼) → (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn ((TopOpen ∘ 𝑅)‘𝑘)) = (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn (TopOpen‘(𝑅𝑘))))
8785, 86eleqtrrd 2844 . . . . . 6 ((𝜑𝑘𝐼) → (𝑓 ∈ (Base‘𝑌), 𝑔 ∈ (Base‘𝑌) ↦ ((𝑓𝑘)(+g‘(𝑅𝑘))(𝑔𝑘))) ∈ (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn ((TopOpen ∘ 𝑅)‘𝑘)))
8856, 87eqeltrid 2845 . . . . 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 23635 . . . 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 2841 . . 3 (𝜑 → (+𝑓𝑌) ∈ (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn (∏t‘(TopOpen ∘ 𝑅))))
9162oveq2d 7447 . . 3 (𝜑 → (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn (TopOpen‘𝑌)) = (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn (∏t‘(TopOpen ∘ 𝑅))))
9290, 91eleqtrrd 2844 . 2 (𝜑 → (+𝑓𝑌) ∈ (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn (TopOpen‘𝑌)))
9325, 38istmd 24082 . 2 (𝑌 ∈ TopMnd ↔ (𝑌 ∈ Mnd ∧ 𝑌 ∈ TopSp ∧ (+𝑓𝑌) ∈ (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn (TopOpen‘𝑌))))
949, 14, 92, 93syl3anbrc 1344 1 (𝜑𝑌 ∈ TopMnd)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  Vcvv 3480  wss 3951  cop 4632   cuni 4907  cmpt 5225   × cxp 5683  cres 5687  ccom 5689   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431  cmpo 7433  1st c1st 8012  2nd c2nd 8013  Basecbs 17247  +gcplusg 17297  TopOpenctopn 17466  tcpt 17483  Xscprds 17490  +𝑓cplusf 18650  Mndcmnd 18747  Topctop 22899  TopOnctopon 22916  TopSpctps 22938   Cn ccn 23232   ×t ctx 23568  TopMndctmd 24078
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 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-map 8868  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fi 9451  df-sup 9482  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-fz 13548  df-struct 17184  df-slot 17219  df-ndx 17231  df-base 17248  df-plusg 17310  df-mulr 17311  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-hom 17321  df-cco 17322  df-rest 17467  df-topn 17468  df-0g 17486  df-topgen 17488  df-pt 17489  df-prds 17492  df-plusf 18652  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-top 22900  df-topon 22917  df-topsp 22939  df-bases 22953  df-cn 23235  df-cnp 23236  df-tx 23570  df-tmd 24080
This theorem is referenced by:  prdstgpd  24133
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