Metamath Proof Explorer


Theorem prdstmdd

Description: The product of a family of topological monoids is a topological monoid. (Contributed by Mario Carneiro, 22-Sep-2015)

Ref Expression
Hypotheses prdstmdd.y Y = S 𝑠 R
prdstmdd.i φ I W
prdstmdd.s φ S V
prdstmdd.r φ R : I TopMnd
Assertion prdstmdd φ Y TopMnd

Proof

Step Hyp Ref Expression
1 prdstmdd.y Y = S 𝑠 R
2 prdstmdd.i φ I W
3 prdstmdd.s φ S V
4 prdstmdd.r φ R : I TopMnd
5 tmdmnd x TopMnd x Mnd
6 5 ssriv TopMnd Mnd
7 fss R : I TopMnd TopMnd Mnd R : I Mnd
8 4 6 7 sylancl φ R : I Mnd
9 1 2 3 8 prdsmndd φ Y Mnd
10 tmdtps x TopMnd x TopSp
11 10 ssriv TopMnd TopSp
12 fss R : I TopMnd TopMnd TopSp R : I TopSp
13 4 11 12 sylancl φ R : I TopSp
14 1 3 2 13 prdstps φ Y TopSp
15 eqid Base Y = Base Y
16 3 3ad2ant1 φ f Base Y g Base Y S V
17 2 3ad2ant1 φ f Base Y g Base Y I W
18 4 ffnd φ R Fn I
19 18 3ad2ant1 φ f Base Y g Base Y R Fn I
20 simp2 φ f Base Y g Base Y f Base Y
21 simp3 φ f Base Y g Base Y g Base Y
22 eqid + Y = + Y
23 1 15 16 17 19 20 21 22 prdsplusgval φ f Base Y g Base Y f + Y g = k I f k + R k g k
24 23 mpoeq3dva φ f Base Y , g Base Y f + Y g = f Base Y , g Base Y k I f k + R k g k
25 eqid + 𝑓 Y = + 𝑓 Y
26 15 22 25 plusffval + 𝑓 Y = f Base Y , g Base Y f + Y g
27 vex f V
28 vex g V
29 27 28 op1std z = f g 1 st z = f
30 29 fveq1d z = f g 1 st z k = f k
31 27 28 op2ndd z = f g 2 nd z = g
32 31 fveq1d z = f g 2 nd z k = g k
33 30 32 oveq12d z = f g 1 st z k + R k 2 nd z k = f k + R k g k
34 33 mpteq2dv z = f g k I 1 st z k + R k 2 nd z k = k I f k + R k g k
35 34 mpompt z Base Y × Base Y k I 1 st z k + R k 2 nd z k = f Base Y , g Base Y k I f k + R k g k
36 24 26 35 3eqtr4g φ + 𝑓 Y = z Base Y × Base Y k I 1 st z k + R k 2 nd z k
37 eqid 𝑡 TopOpen R = 𝑡 TopOpen R
38 eqid TopOpen Y = TopOpen Y
39 15 38 istps Y TopSp TopOpen Y TopOn Base Y
40 14 39 sylib φ TopOpen Y TopOn Base Y
41 txtopon TopOpen Y TopOn Base Y TopOpen Y TopOn Base Y TopOpen Y × t TopOpen Y TopOn Base Y × Base Y
42 40 40 41 syl2anc φ TopOpen Y × t TopOpen Y TopOn Base Y × Base Y
43 topnfn TopOpen Fn V
44 ssv TopSp V
45 fnssres TopOpen Fn V TopSp V TopOpen TopSp Fn TopSp
46 43 44 45 mp2an TopOpen TopSp Fn TopSp
47 fvres x TopSp TopOpen TopSp x = TopOpen x
48 eqid TopOpen x = TopOpen x
49 48 tpstop x TopSp TopOpen x Top
50 47 49 eqeltrd x TopSp TopOpen TopSp x Top
51 50 rgen x TopSp TopOpen TopSp x Top
52 ffnfv TopOpen TopSp : TopSp Top TopOpen TopSp Fn TopSp x TopSp TopOpen TopSp x Top
53 46 51 52 mpbir2an TopOpen TopSp : TopSp Top
54 fco2 TopOpen TopSp : TopSp Top R : I TopSp TopOpen R : I Top
55 53 13 54 sylancr φ TopOpen R : I Top
56 33 mpompt z Base Y × Base Y 1 st z k + R k 2 nd z k = f Base Y , g Base Y f k + R k g k
57 eqid TopOpen R k = TopOpen R k
58 eqid + R k = + R k
59 4 ffvelrnda φ k I R k TopMnd
60 40 adantr φ k I TopOpen Y TopOn Base Y
61 60 60 cnmpt1st φ k I f Base Y , g Base Y f TopOpen Y × t TopOpen Y Cn TopOpen Y
62 1 3 2 18 38 prdstopn φ TopOpen Y = 𝑡 TopOpen R
63 62 adantr φ k I TopOpen Y = 𝑡 TopOpen R
64 63 60 eqeltrrd φ k I 𝑡 TopOpen R TopOn Base Y
65 toponuni 𝑡 TopOpen R TopOn Base Y Base Y = 𝑡 TopOpen R
66 64 65 syl φ k I Base Y = 𝑡 TopOpen R
67 66 mpteq1d φ k I x Base Y x k = x 𝑡 TopOpen R x k
68 2 adantr φ k I I W
69 55 adantr φ k I TopOpen R : I Top
70 simpr φ k I k I
71 eqid 𝑡 TopOpen R = 𝑡 TopOpen R
72 71 37 ptpjcn I W TopOpen R : I Top k I x 𝑡 TopOpen R x k 𝑡 TopOpen R Cn TopOpen R k
73 68 69 70 72 syl3anc φ k I x 𝑡 TopOpen R x k 𝑡 TopOpen R Cn TopOpen R k
74 67 73 eqeltrd φ k I x Base Y x k 𝑡 TopOpen R Cn TopOpen R k
75 63 eqcomd φ k I 𝑡 TopOpen R = TopOpen Y
76 fvco3 R : I TopMnd k I TopOpen R k = TopOpen R k
77 4 76 sylan φ k I TopOpen R k = TopOpen R k
78 75 77 oveq12d φ k I 𝑡 TopOpen R Cn TopOpen R k = TopOpen Y Cn TopOpen R k
79 74 78 eleqtrd φ k I x Base Y x k TopOpen Y Cn TopOpen R k
80 fveq1 x = f x k = f k
81 60 60 61 60 79 80 cnmpt21 φ k I f Base Y , g Base Y f k TopOpen Y × t TopOpen Y Cn TopOpen R k
82 60 60 cnmpt2nd φ k I f Base Y , g Base Y g TopOpen Y × t TopOpen Y Cn TopOpen Y
83 fveq1 x = g x k = g k
84 60 60 82 60 79 83 cnmpt21 φ k I f Base Y , g Base Y g k TopOpen Y × t TopOpen Y Cn TopOpen R k
85 57 58 59 60 60 81 84 cnmpt2plusg φ k I f Base Y , g Base Y f k + R k g k TopOpen Y × t TopOpen Y Cn TopOpen R k
86 77 oveq2d φ k I TopOpen Y × t TopOpen Y Cn TopOpen R k = TopOpen Y × t TopOpen Y Cn TopOpen R k
87 85 86 eleqtrrd φ k I f Base Y , g Base Y f k + R k g k TopOpen Y × t TopOpen Y Cn TopOpen R k
88 56 87 eqeltrid φ k I z Base Y × Base Y 1 st z k + R k 2 nd z k TopOpen Y × t TopOpen Y Cn TopOpen R k
89 37 42 2 55 88 ptcn φ z Base Y × Base Y k I 1 st z k + R k 2 nd z k TopOpen Y × t TopOpen Y Cn 𝑡 TopOpen R
90 36 89 eqeltrd φ + 𝑓 Y TopOpen Y × t TopOpen Y Cn 𝑡 TopOpen R
91 62 oveq2d φ TopOpen Y × t TopOpen Y Cn TopOpen Y = TopOpen Y × t TopOpen Y Cn 𝑡 TopOpen R
92 90 91 eleqtrrd φ + 𝑓 Y TopOpen Y × t TopOpen Y Cn TopOpen Y
93 25 38 istmd Y TopMnd Y Mnd Y TopSp + 𝑓 Y TopOpen Y × t TopOpen Y Cn TopOpen Y
94 9 14 92 93 syl3anbrc φ Y TopMnd