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Theorem prdstmdd 24148
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 24099 . . . . 5 (𝑥 ∈ TopMnd → 𝑥 ∈ Mnd)
65ssriv 3999 . . . 4 TopMnd ⊆ Mnd
7 fss 6753 . . . 4 ((𝑅:𝐼⟶TopMnd ∧ TopMnd ⊆ Mnd) → 𝑅:𝐼⟶Mnd)
84, 6, 7sylancl 586 . . 3 (𝜑𝑅:𝐼⟶Mnd)
91, 2, 3, 8prdsmndd 18796 . 2 (𝜑𝑌 ∈ Mnd)
10 tmdtps 24100 . . . . 5 (𝑥 ∈ TopMnd → 𝑥 ∈ TopSp)
1110ssriv 3999 . . . 4 TopMnd ⊆ TopSp
12 fss 6753 . . . 4 ((𝑅:𝐼⟶TopMnd ∧ TopMnd ⊆ TopSp) → 𝑅:𝐼⟶TopSp)
134, 11, 12sylancl 586 . . 3 (𝜑𝑅:𝐼⟶TopSp)
141, 3, 2, 13prdstps 23653 . 2 (𝜑𝑌 ∈ TopSp)
15 eqid 2735 . . . . . . 7 (Base‘𝑌) = (Base‘𝑌)
1633ad2ant1 1132 . . . . . . 7 ((𝜑𝑓 ∈ (Base‘𝑌) ∧ 𝑔 ∈ (Base‘𝑌)) → 𝑆𝑉)
1723ad2ant1 1132 . . . . . . 7 ((𝜑𝑓 ∈ (Base‘𝑌) ∧ 𝑔 ∈ (Base‘𝑌)) → 𝐼𝑊)
184ffnd 6738 . . . . . . . 8 (𝜑𝑅 Fn 𝐼)
19183ad2ant1 1132 . . . . . . 7 ((𝜑𝑓 ∈ (Base‘𝑌) ∧ 𝑔 ∈ (Base‘𝑌)) → 𝑅 Fn 𝐼)
20 simp2 1136 . . . . . . 7 ((𝜑𝑓 ∈ (Base‘𝑌) ∧ 𝑔 ∈ (Base‘𝑌)) → 𝑓 ∈ (Base‘𝑌))
21 simp3 1137 . . . . . . 7 ((𝜑𝑓 ∈ (Base‘𝑌) ∧ 𝑔 ∈ (Base‘𝑌)) → 𝑔 ∈ (Base‘𝑌))
22 eqid 2735 . . . . . . 7 (+g𝑌) = (+g𝑌)
231, 15, 16, 17, 19, 20, 21, 22prdsplusgval 17520 . . . . . 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 2735 . . . . . 6 (+𝑓𝑌) = (+𝑓𝑌)
2615, 22, 25plusffval 18672 . . . . 5 (+𝑓𝑌) = (𝑓 ∈ (Base‘𝑌), 𝑔 ∈ (Base‘𝑌) ↦ (𝑓(+g𝑌)𝑔))
27 vex 3482 . . . . . . . . . 10 𝑓 ∈ V
28 vex 3482 . . . . . . . . . 10 𝑔 ∈ V
2927, 28op1std 8023 . . . . . . . . 9 (𝑧 = ⟨𝑓, 𝑔⟩ → (1st𝑧) = 𝑓)
3029fveq1d 6909 . . . . . . . 8 (𝑧 = ⟨𝑓, 𝑔⟩ → ((1st𝑧)‘𝑘) = (𝑓𝑘))
3127, 28op2ndd 8024 . . . . . . . . 9 (𝑧 = ⟨𝑓, 𝑔⟩ → (2nd𝑧) = 𝑔)
3231fveq1d 6909 . . . . . . . 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 5250 . . . . . 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 2800 . . . 4 (𝜑 → (+𝑓𝑌) = (𝑧 ∈ ((Base‘𝑌) × (Base‘𝑌)) ↦ (𝑘𝐼 ↦ (((1st𝑧)‘𝑘)(+g‘(𝑅𝑘))((2nd𝑧)‘𝑘)))))
37 eqid 2735 . . . . 5 (∏t‘(TopOpen ∘ 𝑅)) = (∏t‘(TopOpen ∘ 𝑅))
38 eqid 2735 . . . . . . . 8 (TopOpen‘𝑌) = (TopOpen‘𝑌)
3915, 38istps 22956 . . . . . . 7 (𝑌 ∈ TopSp ↔ (TopOpen‘𝑌) ∈ (TopOn‘(Base‘𝑌)))
4014, 39sylib 218 . . . . . 6 (𝜑 → (TopOpen‘𝑌) ∈ (TopOn‘(Base‘𝑌)))
41 txtopon 23615 . . . . . 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 17472 . . . . . . . 8 TopOpen Fn V
44 ssv 4020 . . . . . . . 8 TopSp ⊆ V
45 fnssres 6692 . . . . . . . 8 ((TopOpen Fn V ∧ TopSp ⊆ V) → (TopOpen ↾ TopSp) Fn TopSp)
4643, 44, 45mp2an 692 . . . . . . 7 (TopOpen ↾ TopSp) Fn TopSp
47 fvres 6926 . . . . . . . . 9 (𝑥 ∈ TopSp → ((TopOpen ↾ TopSp)‘𝑥) = (TopOpen‘𝑥))
48 eqid 2735 . . . . . . . . . 10 (TopOpen‘𝑥) = (TopOpen‘𝑥)
4948tpstop 22959 . . . . . . . . 9 (𝑥 ∈ TopSp → (TopOpen‘𝑥) ∈ Top)
5047, 49eqeltrd 2839 . . . . . . . 8 (𝑥 ∈ TopSp → ((TopOpen ↾ TopSp)‘𝑥) ∈ Top)
5150rgen 3061 . . . . . . 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 6763 . . . . . 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 2735 . . . . . . . 8 (TopOpen‘(𝑅𝑘)) = (TopOpen‘(𝑅𝑘))
58 eqid 2735 . . . . . . . 8 (+g‘(𝑅𝑘)) = (+g‘(𝑅𝑘))
594ffvelcdmda 7104 . . . . . . . 8 ((𝜑𝑘𝐼) → (𝑅𝑘) ∈ TopMnd)
6040adantr 480 . . . . . . . 8 ((𝜑𝑘𝐼) → (TopOpen‘𝑌) ∈ (TopOn‘(Base‘𝑌)))
6160, 60cnmpt1st 23692 . . . . . . . . 9 ((𝜑𝑘𝐼) → (𝑓 ∈ (Base‘𝑌), 𝑔 ∈ (Base‘𝑌) ↦ 𝑓) ∈ (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn (TopOpen‘𝑌)))
621, 3, 2, 18, 38prdstopn 23652 . . . . . . . . . . . . . . 15 (𝜑 → (TopOpen‘𝑌) = (∏t‘(TopOpen ∘ 𝑅)))
6362adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑘𝐼) → (TopOpen‘𝑌) = (∏t‘(TopOpen ∘ 𝑅)))
6463, 60eqeltrrd 2840 . . . . . . . . . . . . 13 ((𝜑𝑘𝐼) → (∏t‘(TopOpen ∘ 𝑅)) ∈ (TopOn‘(Base‘𝑌)))
65 toponuni 22936 . . . . . . . . . . . . 13 ((∏t‘(TopOpen ∘ 𝑅)) ∈ (TopOn‘(Base‘𝑌)) → (Base‘𝑌) = (∏t‘(TopOpen ∘ 𝑅)))
6664, 65syl 17 . . . . . . . . . . . 12 ((𝜑𝑘𝐼) → (Base‘𝑌) = (∏t‘(TopOpen ∘ 𝑅)))
6766mpteq1d 5243 . . . . . . . . . . 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 2735 . . . . . . . . . . . . 13 (∏t‘(TopOpen ∘ 𝑅)) = (∏t‘(TopOpen ∘ 𝑅))
7271, 37ptpjcn 23635 . . . . . . . . . . . 12 ((𝐼𝑊 ∧ (TopOpen ∘ 𝑅):𝐼⟶Top ∧ 𝑘𝐼) → (𝑥 (∏t‘(TopOpen ∘ 𝑅)) ↦ (𝑥𝑘)) ∈ ((∏t‘(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)‘𝑘)))
7368, 69, 70, 72syl3anc 1370 . . . . . . . . . . 11 ((𝜑𝑘𝐼) → (𝑥 (∏t‘(TopOpen ∘ 𝑅)) ↦ (𝑥𝑘)) ∈ ((∏t‘(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)‘𝑘)))
7467, 73eqeltrd 2839 . . . . . . . . . 10 ((𝜑𝑘𝐼) → (𝑥 ∈ (Base‘𝑌) ↦ (𝑥𝑘)) ∈ ((∏t‘(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)‘𝑘)))
7563eqcomd 2741 . . . . . . . . . . 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 2841 . . . . . . . . 9 ((𝜑𝑘𝐼) → (𝑥 ∈ (Base‘𝑌) ↦ (𝑥𝑘)) ∈ ((TopOpen‘𝑌) Cn (TopOpen‘(𝑅𝑘))))
80 fveq1 6906 . . . . . . . . 9 (𝑥 = 𝑓 → (𝑥𝑘) = (𝑓𝑘))
8160, 60, 61, 60, 79, 80cnmpt21 23695 . . . . . . . 8 ((𝜑𝑘𝐼) → (𝑓 ∈ (Base‘𝑌), 𝑔 ∈ (Base‘𝑌) ↦ (𝑓𝑘)) ∈ (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn (TopOpen‘(𝑅𝑘))))
8260, 60cnmpt2nd 23693 . . . . . . . . 9 ((𝜑𝑘𝐼) → (𝑓 ∈ (Base‘𝑌), 𝑔 ∈ (Base‘𝑌) ↦ 𝑔) ∈ (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn (TopOpen‘𝑌)))
83 fveq1 6906 . . . . . . . . 9 (𝑥 = 𝑔 → (𝑥𝑘) = (𝑔𝑘))
8460, 60, 82, 60, 79, 83cnmpt21 23695 . . . . . . . 8 ((𝜑𝑘𝐼) → (𝑓 ∈ (Base‘𝑌), 𝑔 ∈ (Base‘𝑌) ↦ (𝑔𝑘)) ∈ (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn (TopOpen‘(𝑅𝑘))))
8557, 58, 59, 60, 60, 81, 84cnmpt2plusg 24112 . . . . . . 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 2842 . . . . . 6 ((𝜑𝑘𝐼) → (𝑓 ∈ (Base‘𝑌), 𝑔 ∈ (Base‘𝑌) ↦ ((𝑓𝑘)(+g‘(𝑅𝑘))(𝑔𝑘))) ∈ (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn ((TopOpen ∘ 𝑅)‘𝑘)))
8856, 87eqeltrid 2843 . . . . 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 23651 . . . 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 2839 . . 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 2842 . 2 (𝜑 → (+𝑓𝑌) ∈ (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn (TopOpen‘𝑌)))
9325, 38istmd 24098 . 2 (𝑌 ∈ TopMnd ↔ (𝑌 ∈ Mnd ∧ 𝑌 ∈ TopSp ∧ (+𝑓𝑌) ∈ (((TopOpen‘𝑌) ×t (TopOpen‘𝑌)) Cn (TopOpen‘𝑌))))
949, 14, 92, 93syl3anbrc 1342 1 (𝜑𝑌 ∈ TopMnd)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  Vcvv 3478  wss 3963  cop 4637   cuni 4912  cmpt 5231   × cxp 5687  cres 5691  ccom 5693   Fn wfn 6558  wf 6559  cfv 6563  (class class class)co 7431  cmpo 7433  1st c1st 8011  2nd c2nd 8012  Basecbs 17245  +gcplusg 17298  TopOpenctopn 17468  tcpt 17485  Xscprds 17492  +𝑓cplusf 18663  Mndcmnd 18760  Topctop 22915  TopOnctopon 22932  TopSpctps 22954   Cn ccn 23248   ×t ctx 23584  TopMndctmd 24094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-er 8744  df-map 8867  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fi 9449  df-sup 9480  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-dec 12732  df-uz 12877  df-fz 13545  df-struct 17181  df-slot 17216  df-ndx 17228  df-base 17246  df-plusg 17311  df-mulr 17312  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  df-hom 17322  df-cco 17323  df-rest 17469  df-topn 17470  df-0g 17488  df-topgen 17490  df-pt 17491  df-prds 17494  df-plusf 18665  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-top 22916  df-topon 22933  df-topsp 22955  df-bases 22969  df-cn 23251  df-cnp 23252  df-tx 23586  df-tmd 24096
This theorem is referenced by:  prdstgpd  24149
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