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Theorem prdstmdd 24056
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 π‘Œ = (𝑆Xs𝑅)
prdstmdd.i (πœ‘ β†’ 𝐼 ∈ π‘Š)
prdstmdd.s (πœ‘ β†’ 𝑆 ∈ 𝑉)
prdstmdd.r (πœ‘ β†’ 𝑅:𝐼⟢TopMnd)
Assertion
Ref Expression
prdstmdd (πœ‘ β†’ π‘Œ ∈ TopMnd)

Proof of Theorem prdstmdd
Dummy variables 𝑓 𝑔 π‘˜ π‘₯ 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prdstmdd.y . . 3 π‘Œ = (𝑆Xs𝑅)
2 prdstmdd.i . . 3 (πœ‘ β†’ 𝐼 ∈ π‘Š)
3 prdstmdd.s . . 3 (πœ‘ β†’ 𝑆 ∈ 𝑉)
4 prdstmdd.r . . . 4 (πœ‘ β†’ 𝑅:𝐼⟢TopMnd)
5 tmdmnd 24007 . . . . 5 (π‘₯ ∈ TopMnd β†’ π‘₯ ∈ Mnd)
65ssriv 3986 . . . 4 TopMnd βŠ† Mnd
7 fss 6744 . . . 4 ((𝑅:𝐼⟢TopMnd ∧ TopMnd βŠ† Mnd) β†’ 𝑅:𝐼⟢Mnd)
84, 6, 7sylancl 584 . . 3 (πœ‘ β†’ 𝑅:𝐼⟢Mnd)
91, 2, 3, 8prdsmndd 18736 . 2 (πœ‘ β†’ π‘Œ ∈ Mnd)
10 tmdtps 24008 . . . . 5 (π‘₯ ∈ TopMnd β†’ π‘₯ ∈ TopSp)
1110ssriv 3986 . . . 4 TopMnd βŠ† TopSp
12 fss 6744 . . . 4 ((𝑅:𝐼⟢TopMnd ∧ TopMnd βŠ† TopSp) β†’ 𝑅:𝐼⟢TopSp)
134, 11, 12sylancl 584 . . 3 (πœ‘ β†’ 𝑅:𝐼⟢TopSp)
141, 3, 2, 13prdstps 23561 . 2 (πœ‘ β†’ π‘Œ ∈ TopSp)
15 eqid 2728 . . . . . . 7 (Baseβ€˜π‘Œ) = (Baseβ€˜π‘Œ)
1633ad2ant1 1130 . . . . . . 7 ((πœ‘ ∧ 𝑓 ∈ (Baseβ€˜π‘Œ) ∧ 𝑔 ∈ (Baseβ€˜π‘Œ)) β†’ 𝑆 ∈ 𝑉)
1723ad2ant1 1130 . . . . . . 7 ((πœ‘ ∧ 𝑓 ∈ (Baseβ€˜π‘Œ) ∧ 𝑔 ∈ (Baseβ€˜π‘Œ)) β†’ 𝐼 ∈ π‘Š)
184ffnd 6728 . . . . . . . 8 (πœ‘ β†’ 𝑅 Fn 𝐼)
19183ad2ant1 1130 . . . . . . 7 ((πœ‘ ∧ 𝑓 ∈ (Baseβ€˜π‘Œ) ∧ 𝑔 ∈ (Baseβ€˜π‘Œ)) β†’ 𝑅 Fn 𝐼)
20 simp2 1134 . . . . . . 7 ((πœ‘ ∧ 𝑓 ∈ (Baseβ€˜π‘Œ) ∧ 𝑔 ∈ (Baseβ€˜π‘Œ)) β†’ 𝑓 ∈ (Baseβ€˜π‘Œ))
21 simp3 1135 . . . . . . 7 ((πœ‘ ∧ 𝑓 ∈ (Baseβ€˜π‘Œ) ∧ 𝑔 ∈ (Baseβ€˜π‘Œ)) β†’ 𝑔 ∈ (Baseβ€˜π‘Œ))
22 eqid 2728 . . . . . . 7 (+gβ€˜π‘Œ) = (+gβ€˜π‘Œ)
231, 15, 16, 17, 19, 20, 21, 22prdsplusgval 17464 . . . . . 6 ((πœ‘ ∧ 𝑓 ∈ (Baseβ€˜π‘Œ) ∧ 𝑔 ∈ (Baseβ€˜π‘Œ)) β†’ (𝑓(+gβ€˜π‘Œ)𝑔) = (π‘˜ ∈ 𝐼 ↦ ((π‘“β€˜π‘˜)(+gβ€˜(π‘…β€˜π‘˜))(π‘”β€˜π‘˜))))
2423mpoeq3dva 7504 . . . . 5 (πœ‘ β†’ (𝑓 ∈ (Baseβ€˜π‘Œ), 𝑔 ∈ (Baseβ€˜π‘Œ) ↦ (𝑓(+gβ€˜π‘Œ)𝑔)) = (𝑓 ∈ (Baseβ€˜π‘Œ), 𝑔 ∈ (Baseβ€˜π‘Œ) ↦ (π‘˜ ∈ 𝐼 ↦ ((π‘“β€˜π‘˜)(+gβ€˜(π‘…β€˜π‘˜))(π‘”β€˜π‘˜)))))
25 eqid 2728 . . . . . 6 (+π‘“β€˜π‘Œ) = (+π‘“β€˜π‘Œ)
2615, 22, 25plusffval 18615 . . . . 5 (+π‘“β€˜π‘Œ) = (𝑓 ∈ (Baseβ€˜π‘Œ), 𝑔 ∈ (Baseβ€˜π‘Œ) ↦ (𝑓(+gβ€˜π‘Œ)𝑔))
27 vex 3477 . . . . . . . . . 10 𝑓 ∈ V
28 vex 3477 . . . . . . . . . 10 𝑔 ∈ V
2927, 28op1std 8011 . . . . . . . . 9 (𝑧 = βŸ¨π‘“, π‘”βŸ© β†’ (1st β€˜π‘§) = 𝑓)
3029fveq1d 6904 . . . . . . . 8 (𝑧 = βŸ¨π‘“, π‘”βŸ© β†’ ((1st β€˜π‘§)β€˜π‘˜) = (π‘“β€˜π‘˜))
3127, 28op2ndd 8012 . . . . . . . . 9 (𝑧 = βŸ¨π‘“, π‘”βŸ© β†’ (2nd β€˜π‘§) = 𝑔)
3231fveq1d 6904 . . . . . . . 8 (𝑧 = βŸ¨π‘“, π‘”βŸ© β†’ ((2nd β€˜π‘§)β€˜π‘˜) = (π‘”β€˜π‘˜))
3330, 32oveq12d 7444 . . . . . . 7 (𝑧 = βŸ¨π‘“, π‘”βŸ© β†’ (((1st β€˜π‘§)β€˜π‘˜)(+gβ€˜(π‘…β€˜π‘˜))((2nd β€˜π‘§)β€˜π‘˜)) = ((π‘“β€˜π‘˜)(+gβ€˜(π‘…β€˜π‘˜))(π‘”β€˜π‘˜)))
3433mpteq2dv 5254 . . . . . 6 (𝑧 = βŸ¨π‘“, π‘”βŸ© β†’ (π‘˜ ∈ 𝐼 ↦ (((1st β€˜π‘§)β€˜π‘˜)(+gβ€˜(π‘…β€˜π‘˜))((2nd β€˜π‘§)β€˜π‘˜))) = (π‘˜ ∈ 𝐼 ↦ ((π‘“β€˜π‘˜)(+gβ€˜(π‘…β€˜π‘˜))(π‘”β€˜π‘˜))))
3534mpompt 7541 . . . . 5 (𝑧 ∈ ((Baseβ€˜π‘Œ) Γ— (Baseβ€˜π‘Œ)) ↦ (π‘˜ ∈ 𝐼 ↦ (((1st β€˜π‘§)β€˜π‘˜)(+gβ€˜(π‘…β€˜π‘˜))((2nd β€˜π‘§)β€˜π‘˜)))) = (𝑓 ∈ (Baseβ€˜π‘Œ), 𝑔 ∈ (Baseβ€˜π‘Œ) ↦ (π‘˜ ∈ 𝐼 ↦ ((π‘“β€˜π‘˜)(+gβ€˜(π‘…β€˜π‘˜))(π‘”β€˜π‘˜))))
3624, 26, 353eqtr4g 2793 . . . 4 (πœ‘ β†’ (+π‘“β€˜π‘Œ) = (𝑧 ∈ ((Baseβ€˜π‘Œ) Γ— (Baseβ€˜π‘Œ)) ↦ (π‘˜ ∈ 𝐼 ↦ (((1st β€˜π‘§)β€˜π‘˜)(+gβ€˜(π‘…β€˜π‘˜))((2nd β€˜π‘§)β€˜π‘˜)))))
37 eqid 2728 . . . . 5 (∏tβ€˜(TopOpen ∘ 𝑅)) = (∏tβ€˜(TopOpen ∘ 𝑅))
38 eqid 2728 . . . . . . . 8 (TopOpenβ€˜π‘Œ) = (TopOpenβ€˜π‘Œ)
3915, 38istps 22864 . . . . . . 7 (π‘Œ ∈ TopSp ↔ (TopOpenβ€˜π‘Œ) ∈ (TopOnβ€˜(Baseβ€˜π‘Œ)))
4014, 39sylib 217 . . . . . 6 (πœ‘ β†’ (TopOpenβ€˜π‘Œ) ∈ (TopOnβ€˜(Baseβ€˜π‘Œ)))
41 txtopon 23523 . . . . . 6 (((TopOpenβ€˜π‘Œ) ∈ (TopOnβ€˜(Baseβ€˜π‘Œ)) ∧ (TopOpenβ€˜π‘Œ) ∈ (TopOnβ€˜(Baseβ€˜π‘Œ))) β†’ ((TopOpenβ€˜π‘Œ) Γ—t (TopOpenβ€˜π‘Œ)) ∈ (TopOnβ€˜((Baseβ€˜π‘Œ) Γ— (Baseβ€˜π‘Œ))))
4240, 40, 41syl2anc 582 . . . . 5 (πœ‘ β†’ ((TopOpenβ€˜π‘Œ) Γ—t (TopOpenβ€˜π‘Œ)) ∈ (TopOnβ€˜((Baseβ€˜π‘Œ) Γ— (Baseβ€˜π‘Œ))))
43 topnfn 17416 . . . . . . . 8 TopOpen Fn V
44 ssv 4006 . . . . . . . 8 TopSp βŠ† V
45 fnssres 6683 . . . . . . . 8 ((TopOpen Fn V ∧ TopSp βŠ† V) β†’ (TopOpen β†Ύ TopSp) Fn TopSp)
4643, 44, 45mp2an 690 . . . . . . 7 (TopOpen β†Ύ TopSp) Fn TopSp
47 fvres 6921 . . . . . . . . 9 (π‘₯ ∈ TopSp β†’ ((TopOpen β†Ύ TopSp)β€˜π‘₯) = (TopOpenβ€˜π‘₯))
48 eqid 2728 . . . . . . . . . 10 (TopOpenβ€˜π‘₯) = (TopOpenβ€˜π‘₯)
4948tpstop 22867 . . . . . . . . 9 (π‘₯ ∈ TopSp β†’ (TopOpenβ€˜π‘₯) ∈ Top)
5047, 49eqeltrd 2829 . . . . . . . 8 (π‘₯ ∈ TopSp β†’ ((TopOpen β†Ύ TopSp)β€˜π‘₯) ∈ Top)
5150rgen 3060 . . . . . . 7 βˆ€π‘₯ ∈ TopSp ((TopOpen β†Ύ TopSp)β€˜π‘₯) ∈ Top
52 ffnfv 7134 . . . . . . 7 ((TopOpen β†Ύ TopSp):TopSp⟢Top ↔ ((TopOpen β†Ύ TopSp) Fn TopSp ∧ βˆ€π‘₯ ∈ TopSp ((TopOpen β†Ύ TopSp)β€˜π‘₯) ∈ Top))
5346, 51, 52mpbir2an 709 . . . . . 6 (TopOpen β†Ύ TopSp):TopSp⟢Top
54 fco2 6755 . . . . . 6 (((TopOpen β†Ύ TopSp):TopSp⟢Top ∧ 𝑅:𝐼⟢TopSp) β†’ (TopOpen ∘ 𝑅):𝐼⟢Top)
5553, 13, 54sylancr 585 . . . . 5 (πœ‘ β†’ (TopOpen ∘ 𝑅):𝐼⟢Top)
5633mpompt 7541 . . . . . 6 (𝑧 ∈ ((Baseβ€˜π‘Œ) Γ— (Baseβ€˜π‘Œ)) ↦ (((1st β€˜π‘§)β€˜π‘˜)(+gβ€˜(π‘…β€˜π‘˜))((2nd β€˜π‘§)β€˜π‘˜))) = (𝑓 ∈ (Baseβ€˜π‘Œ), 𝑔 ∈ (Baseβ€˜π‘Œ) ↦ ((π‘“β€˜π‘˜)(+gβ€˜(π‘…β€˜π‘˜))(π‘”β€˜π‘˜)))
57 eqid 2728 . . . . . . . 8 (TopOpenβ€˜(π‘…β€˜π‘˜)) = (TopOpenβ€˜(π‘…β€˜π‘˜))
58 eqid 2728 . . . . . . . 8 (+gβ€˜(π‘…β€˜π‘˜)) = (+gβ€˜(π‘…β€˜π‘˜))
594ffvelcdmda 7099 . . . . . . . 8 ((πœ‘ ∧ π‘˜ ∈ 𝐼) β†’ (π‘…β€˜π‘˜) ∈ TopMnd)
6040adantr 479 . . . . . . . 8 ((πœ‘ ∧ π‘˜ ∈ 𝐼) β†’ (TopOpenβ€˜π‘Œ) ∈ (TopOnβ€˜(Baseβ€˜π‘Œ)))
6160, 60cnmpt1st 23600 . . . . . . . . 9 ((πœ‘ ∧ π‘˜ ∈ 𝐼) β†’ (𝑓 ∈ (Baseβ€˜π‘Œ), 𝑔 ∈ (Baseβ€˜π‘Œ) ↦ 𝑓) ∈ (((TopOpenβ€˜π‘Œ) Γ—t (TopOpenβ€˜π‘Œ)) Cn (TopOpenβ€˜π‘Œ)))
621, 3, 2, 18, 38prdstopn 23560 . . . . . . . . . . . . . . 15 (πœ‘ β†’ (TopOpenβ€˜π‘Œ) = (∏tβ€˜(TopOpen ∘ 𝑅)))
6362adantr 479 . . . . . . . . . . . . . 14 ((πœ‘ ∧ π‘˜ ∈ 𝐼) β†’ (TopOpenβ€˜π‘Œ) = (∏tβ€˜(TopOpen ∘ 𝑅)))
6463, 60eqeltrrd 2830 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘˜ ∈ 𝐼) β†’ (∏tβ€˜(TopOpen ∘ 𝑅)) ∈ (TopOnβ€˜(Baseβ€˜π‘Œ)))
65 toponuni 22844 . . . . . . . . . . . . 13 ((∏tβ€˜(TopOpen ∘ 𝑅)) ∈ (TopOnβ€˜(Baseβ€˜π‘Œ)) β†’ (Baseβ€˜π‘Œ) = βˆͺ (∏tβ€˜(TopOpen ∘ 𝑅)))
6664, 65syl 17 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘˜ ∈ 𝐼) β†’ (Baseβ€˜π‘Œ) = βˆͺ (∏tβ€˜(TopOpen ∘ 𝑅)))
6766mpteq1d 5247 . . . . . . . . . . 11 ((πœ‘ ∧ π‘˜ ∈ 𝐼) β†’ (π‘₯ ∈ (Baseβ€˜π‘Œ) ↦ (π‘₯β€˜π‘˜)) = (π‘₯ ∈ βˆͺ (∏tβ€˜(TopOpen ∘ 𝑅)) ↦ (π‘₯β€˜π‘˜)))
682adantr 479 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘˜ ∈ 𝐼) β†’ 𝐼 ∈ π‘Š)
6955adantr 479 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘˜ ∈ 𝐼) β†’ (TopOpen ∘ 𝑅):𝐼⟢Top)
70 simpr 483 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘˜ ∈ 𝐼) β†’ π‘˜ ∈ 𝐼)
71 eqid 2728 . . . . . . . . . . . . 13 βˆͺ (∏tβ€˜(TopOpen ∘ 𝑅)) = βˆͺ (∏tβ€˜(TopOpen ∘ 𝑅))
7271, 37ptpjcn 23543 . . . . . . . . . . . 12 ((𝐼 ∈ π‘Š ∧ (TopOpen ∘ 𝑅):𝐼⟢Top ∧ π‘˜ ∈ 𝐼) β†’ (π‘₯ ∈ βˆͺ (∏tβ€˜(TopOpen ∘ 𝑅)) ↦ (π‘₯β€˜π‘˜)) ∈ ((∏tβ€˜(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)β€˜π‘˜)))
7368, 69, 70, 72syl3anc 1368 . . . . . . . . . . 11 ((πœ‘ ∧ π‘˜ ∈ 𝐼) β†’ (π‘₯ ∈ βˆͺ (∏tβ€˜(TopOpen ∘ 𝑅)) ↦ (π‘₯β€˜π‘˜)) ∈ ((∏tβ€˜(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)β€˜π‘˜)))
7467, 73eqeltrd 2829 . . . . . . . . . 10 ((πœ‘ ∧ π‘˜ ∈ 𝐼) β†’ (π‘₯ ∈ (Baseβ€˜π‘Œ) ↦ (π‘₯β€˜π‘˜)) ∈ ((∏tβ€˜(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)β€˜π‘˜)))
7563eqcomd 2734 . . . . . . . . . . 11 ((πœ‘ ∧ π‘˜ ∈ 𝐼) β†’ (∏tβ€˜(TopOpen ∘ 𝑅)) = (TopOpenβ€˜π‘Œ))
76 fvco3 7002 . . . . . . . . . . . 12 ((𝑅:𝐼⟢TopMnd ∧ π‘˜ ∈ 𝐼) β†’ ((TopOpen ∘ 𝑅)β€˜π‘˜) = (TopOpenβ€˜(π‘…β€˜π‘˜)))
774, 76sylan 578 . . . . . . . . . . 11 ((πœ‘ ∧ π‘˜ ∈ 𝐼) β†’ ((TopOpen ∘ 𝑅)β€˜π‘˜) = (TopOpenβ€˜(π‘…β€˜π‘˜)))
7875, 77oveq12d 7444 . . . . . . . . . 10 ((πœ‘ ∧ π‘˜ ∈ 𝐼) β†’ ((∏tβ€˜(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)β€˜π‘˜)) = ((TopOpenβ€˜π‘Œ) Cn (TopOpenβ€˜(π‘…β€˜π‘˜))))
7974, 78eleqtrd 2831 . . . . . . . . 9 ((πœ‘ ∧ π‘˜ ∈ 𝐼) β†’ (π‘₯ ∈ (Baseβ€˜π‘Œ) ↦ (π‘₯β€˜π‘˜)) ∈ ((TopOpenβ€˜π‘Œ) Cn (TopOpenβ€˜(π‘…β€˜π‘˜))))
80 fveq1 6901 . . . . . . . . 9 (π‘₯ = 𝑓 β†’ (π‘₯β€˜π‘˜) = (π‘“β€˜π‘˜))
8160, 60, 61, 60, 79, 80cnmpt21 23603 . . . . . . . 8 ((πœ‘ ∧ π‘˜ ∈ 𝐼) β†’ (𝑓 ∈ (Baseβ€˜π‘Œ), 𝑔 ∈ (Baseβ€˜π‘Œ) ↦ (π‘“β€˜π‘˜)) ∈ (((TopOpenβ€˜π‘Œ) Γ—t (TopOpenβ€˜π‘Œ)) Cn (TopOpenβ€˜(π‘…β€˜π‘˜))))
8260, 60cnmpt2nd 23601 . . . . . . . . 9 ((πœ‘ ∧ π‘˜ ∈ 𝐼) β†’ (𝑓 ∈ (Baseβ€˜π‘Œ), 𝑔 ∈ (Baseβ€˜π‘Œ) ↦ 𝑔) ∈ (((TopOpenβ€˜π‘Œ) Γ—t (TopOpenβ€˜π‘Œ)) Cn (TopOpenβ€˜π‘Œ)))
83 fveq1 6901 . . . . . . . . 9 (π‘₯ = 𝑔 β†’ (π‘₯β€˜π‘˜) = (π‘”β€˜π‘˜))
8460, 60, 82, 60, 79, 83cnmpt21 23603 . . . . . . . 8 ((πœ‘ ∧ π‘˜ ∈ 𝐼) β†’ (𝑓 ∈ (Baseβ€˜π‘Œ), 𝑔 ∈ (Baseβ€˜π‘Œ) ↦ (π‘”β€˜π‘˜)) ∈ (((TopOpenβ€˜π‘Œ) Γ—t (TopOpenβ€˜π‘Œ)) Cn (TopOpenβ€˜(π‘…β€˜π‘˜))))
8557, 58, 59, 60, 60, 81, 84cnmpt2plusg 24020 . . . . . . 7 ((πœ‘ ∧ π‘˜ ∈ 𝐼) β†’ (𝑓 ∈ (Baseβ€˜π‘Œ), 𝑔 ∈ (Baseβ€˜π‘Œ) ↦ ((π‘“β€˜π‘˜)(+gβ€˜(π‘…β€˜π‘˜))(π‘”β€˜π‘˜))) ∈ (((TopOpenβ€˜π‘Œ) Γ—t (TopOpenβ€˜π‘Œ)) Cn (TopOpenβ€˜(π‘…β€˜π‘˜))))
8677oveq2d 7442 . . . . . . 7 ((πœ‘ ∧ π‘˜ ∈ 𝐼) β†’ (((TopOpenβ€˜π‘Œ) Γ—t (TopOpenβ€˜π‘Œ)) Cn ((TopOpen ∘ 𝑅)β€˜π‘˜)) = (((TopOpenβ€˜π‘Œ) Γ—t (TopOpenβ€˜π‘Œ)) Cn (TopOpenβ€˜(π‘…β€˜π‘˜))))
8785, 86eleqtrrd 2832 . . . . . 6 ((πœ‘ ∧ π‘˜ ∈ 𝐼) β†’ (𝑓 ∈ (Baseβ€˜π‘Œ), 𝑔 ∈ (Baseβ€˜π‘Œ) ↦ ((π‘“β€˜π‘˜)(+gβ€˜(π‘…β€˜π‘˜))(π‘”β€˜π‘˜))) ∈ (((TopOpenβ€˜π‘Œ) Γ—t (TopOpenβ€˜π‘Œ)) Cn ((TopOpen ∘ 𝑅)β€˜π‘˜)))
8856, 87eqeltrid 2833 . . . . 5 ((πœ‘ ∧ π‘˜ ∈ 𝐼) β†’ (𝑧 ∈ ((Baseβ€˜π‘Œ) Γ— (Baseβ€˜π‘Œ)) ↦ (((1st β€˜π‘§)β€˜π‘˜)(+gβ€˜(π‘…β€˜π‘˜))((2nd β€˜π‘§)β€˜π‘˜))) ∈ (((TopOpenβ€˜π‘Œ) Γ—t (TopOpenβ€˜π‘Œ)) Cn ((TopOpen ∘ 𝑅)β€˜π‘˜)))
8937, 42, 2, 55, 88ptcn 23559 . . . 4 (πœ‘ β†’ (𝑧 ∈ ((Baseβ€˜π‘Œ) Γ— (Baseβ€˜π‘Œ)) ↦ (π‘˜ ∈ 𝐼 ↦ (((1st β€˜π‘§)β€˜π‘˜)(+gβ€˜(π‘…β€˜π‘˜))((2nd β€˜π‘§)β€˜π‘˜)))) ∈ (((TopOpenβ€˜π‘Œ) Γ—t (TopOpenβ€˜π‘Œ)) Cn (∏tβ€˜(TopOpen ∘ 𝑅))))
9036, 89eqeltrd 2829 . . 3 (πœ‘ β†’ (+π‘“β€˜π‘Œ) ∈ (((TopOpenβ€˜π‘Œ) Γ—t (TopOpenβ€˜π‘Œ)) Cn (∏tβ€˜(TopOpen ∘ 𝑅))))
9162oveq2d 7442 . . 3 (πœ‘ β†’ (((TopOpenβ€˜π‘Œ) Γ—t (TopOpenβ€˜π‘Œ)) Cn (TopOpenβ€˜π‘Œ)) = (((TopOpenβ€˜π‘Œ) Γ—t (TopOpenβ€˜π‘Œ)) Cn (∏tβ€˜(TopOpen ∘ 𝑅))))
9290, 91eleqtrrd 2832 . 2 (πœ‘ β†’ (+π‘“β€˜π‘Œ) ∈ (((TopOpenβ€˜π‘Œ) Γ—t (TopOpenβ€˜π‘Œ)) Cn (TopOpenβ€˜π‘Œ)))
9325, 38istmd 24006 . 2 (π‘Œ ∈ TopMnd ↔ (π‘Œ ∈ Mnd ∧ π‘Œ ∈ TopSp ∧ (+π‘“β€˜π‘Œ) ∈ (((TopOpenβ€˜π‘Œ) Γ—t (TopOpenβ€˜π‘Œ)) Cn (TopOpenβ€˜π‘Œ))))
949, 14, 92, 93syl3anbrc 1340 1 (πœ‘ β†’ π‘Œ ∈ TopMnd)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3058  Vcvv 3473   βŠ† wss 3949  βŸ¨cop 4638  βˆͺ cuni 4912   ↦ cmpt 5235   Γ— cxp 5680   β†Ύ cres 5684   ∘ ccom 5686   Fn wfn 6548  βŸΆwf 6549  β€˜cfv 6553  (class class class)co 7426   ∈ cmpo 7428  1st c1st 7999  2nd c2nd 8000  Basecbs 17189  +gcplusg 17242  TopOpenctopn 17412  βˆtcpt 17429  Xscprds 17436  +𝑓cplusf 18606  Mndcmnd 18703  Topctop 22823  TopOnctopon 22840  TopSpctps 22862   Cn ccn 23156   Γ—t ctx 23492  TopMndctmd 24002
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7748  ax-cnex 11204  ax-resscn 11205  ax-1cn 11206  ax-icn 11207  ax-addcl 11208  ax-addrcl 11209  ax-mulcl 11210  ax-mulrcl 11211  ax-mulcom 11212  ax-addass 11213  ax-mulass 11214  ax-distr 11215  ax-i2m1 11216  ax-1ne0 11217  ax-1rid 11218  ax-rnegex 11219  ax-rrecex 11220  ax-cnre 11221  ax-pre-lttri 11222  ax-pre-lttrn 11223  ax-pre-ltadd 11224  ax-pre-mulgt0 11225
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-tp 4637  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-iin 5003  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7879  df-1st 8001  df-2nd 8002  df-frecs 8295  df-wrecs 8326  df-recs 8400  df-rdg 8439  df-1o 8495  df-er 8733  df-map 8855  df-ixp 8925  df-en 8973  df-dom 8974  df-sdom 8975  df-fin 8976  df-fi 9444  df-sup 9475  df-pnf 11290  df-mnf 11291  df-xr 11292  df-ltxr 11293  df-le 11294  df-sub 11486  df-neg 11487  df-nn 12253  df-2 12315  df-3 12316  df-4 12317  df-5 12318  df-6 12319  df-7 12320  df-8 12321  df-9 12322  df-n0 12513  df-z 12599  df-dec 12718  df-uz 12863  df-fz 13527  df-struct 17125  df-slot 17160  df-ndx 17172  df-base 17190  df-plusg 17255  df-mulr 17256  df-sca 17258  df-vsca 17259  df-ip 17260  df-tset 17261  df-ple 17262  df-ds 17264  df-hom 17266  df-cco 17267  df-rest 17413  df-topn 17414  df-0g 17432  df-topgen 17434  df-pt 17435  df-prds 17438  df-plusf 18608  df-mgm 18609  df-sgrp 18688  df-mnd 18704  df-top 22824  df-topon 22841  df-topsp 22863  df-bases 22877  df-cn 23159  df-cnp 23160  df-tx 23494  df-tmd 24004
This theorem is referenced by:  prdstgpd  24057
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