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Theorem pt1hmeo 21823
Description: The canonical homeomorphism from a topological product on a singleton to the topology of the factor. (Contributed by Mario Carneiro, 3-Feb-2015.) (Proof shortened by AV, 18-Apr-2021.)
Hypotheses
Ref Expression
pt1hmeo.j 𝐾 = (∏t‘{⟨𝐴, 𝐽⟩})
pt1hmeo.a (𝜑𝐴𝑉)
pt1hmeo.r (𝜑𝐽 ∈ (TopOn‘𝑋))
Assertion
Ref Expression
pt1hmeo (𝜑 → (𝑥𝑋 ↦ {⟨𝐴, 𝑥⟩}) ∈ (𝐽Homeo𝐾))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐽   𝑥,𝐾   𝜑,𝑥   𝑥,𝑋
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem pt1hmeo
Dummy variables 𝑘 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fconstmpt 5363 . . . . 5 ({𝐴} × {𝑥}) = (𝑘 ∈ {𝐴} ↦ 𝑥)
2 pt1hmeo.a . . . . . . 7 (𝜑𝐴𝑉)
32adantr 468 . . . . . 6 ((𝜑𝑥𝑋) → 𝐴𝑉)
4 sneq 4380 . . . . . . . . 9 (𝑘 = 𝐴 → {𝑘} = {𝐴})
54xpeq1d 5339 . . . . . . . 8 (𝑘 = 𝐴 → ({𝑘} × {𝑥}) = ({𝐴} × {𝑥}))
6 opeq1 4595 . . . . . . . . 9 (𝑘 = 𝐴 → ⟨𝑘, 𝑥⟩ = ⟨𝐴, 𝑥⟩)
76sneqd 4382 . . . . . . . 8 (𝑘 = 𝐴 → {⟨𝑘, 𝑥⟩} = {⟨𝐴, 𝑥⟩})
85, 7eqeq12d 2821 . . . . . . 7 (𝑘 = 𝐴 → (({𝑘} × {𝑥}) = {⟨𝑘, 𝑥⟩} ↔ ({𝐴} × {𝑥}) = {⟨𝐴, 𝑥⟩}))
9 vex 3394 . . . . . . . 8 𝑘 ∈ V
10 vex 3394 . . . . . . . 8 𝑥 ∈ V
119, 10xpsn 6630 . . . . . . 7 ({𝑘} × {𝑥}) = {⟨𝑘, 𝑥⟩}
128, 11vtoclg 3459 . . . . . 6 (𝐴𝑉 → ({𝐴} × {𝑥}) = {⟨𝐴, 𝑥⟩})
133, 12syl 17 . . . . 5 ((𝜑𝑥𝑋) → ({𝐴} × {𝑥}) = {⟨𝐴, 𝑥⟩})
141, 13syl5eqr 2854 . . . 4 ((𝜑𝑥𝑋) → (𝑘 ∈ {𝐴} ↦ 𝑥) = {⟨𝐴, 𝑥⟩})
1514mpteq2dva 4938 . . 3 (𝜑 → (𝑥𝑋 ↦ (𝑘 ∈ {𝐴} ↦ 𝑥)) = (𝑥𝑋 ↦ {⟨𝐴, 𝑥⟩}))
16 pt1hmeo.j . . . 4 𝐾 = (∏t‘{⟨𝐴, 𝐽⟩})
17 pt1hmeo.r . . . 4 (𝜑𝐽 ∈ (TopOn‘𝑋))
18 snex 5098 . . . . 5 {𝐴} ∈ V
1918a1i 11 . . . 4 (𝜑 → {𝐴} ∈ V)
20 topontop 20931 . . . . . 6 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2117, 20syl 17 . . . . 5 (𝜑𝐽 ∈ Top)
222, 21fsnd 6395 . . . 4 (𝜑 → {⟨𝐴, 𝐽⟩}:{𝐴}⟶Top)
2317cnmptid 21678 . . . . . 6 (𝜑 → (𝑥𝑋𝑥) ∈ (𝐽 Cn 𝐽))
2423adantr 468 . . . . 5 ((𝜑𝑘 ∈ {𝐴}) → (𝑥𝑋𝑥) ∈ (𝐽 Cn 𝐽))
25 elsni 4387 . . . . . . . 8 (𝑘 ∈ {𝐴} → 𝑘 = 𝐴)
2625fveq2d 6412 . . . . . . 7 (𝑘 ∈ {𝐴} → ({⟨𝐴, 𝐽⟩}‘𝑘) = ({⟨𝐴, 𝐽⟩}‘𝐴))
27 fvsng 6672 . . . . . . . 8 ((𝐴𝑉𝐽 ∈ (TopOn‘𝑋)) → ({⟨𝐴, 𝐽⟩}‘𝐴) = 𝐽)
282, 17, 27syl2anc 575 . . . . . . 7 (𝜑 → ({⟨𝐴, 𝐽⟩}‘𝐴) = 𝐽)
2926, 28sylan9eqr 2862 . . . . . 6 ((𝜑𝑘 ∈ {𝐴}) → ({⟨𝐴, 𝐽⟩}‘𝑘) = 𝐽)
3029oveq2d 6890 . . . . 5 ((𝜑𝑘 ∈ {𝐴}) → (𝐽 Cn ({⟨𝐴, 𝐽⟩}‘𝑘)) = (𝐽 Cn 𝐽))
3124, 30eleqtrrd 2888 . . . 4 ((𝜑𝑘 ∈ {𝐴}) → (𝑥𝑋𝑥) ∈ (𝐽 Cn ({⟨𝐴, 𝐽⟩}‘𝑘)))
3216, 17, 19, 22, 31ptcn 21644 . . 3 (𝜑 → (𝑥𝑋 ↦ (𝑘 ∈ {𝐴} ↦ 𝑥)) ∈ (𝐽 Cn 𝐾))
3315, 32eqeltrrd 2886 . 2 (𝜑 → (𝑥𝑋 ↦ {⟨𝐴, 𝑥⟩}) ∈ (𝐽 Cn 𝐾))
34 simprr 780 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩})) → 𝑦 = {⟨𝐴, 𝑥⟩})
3514adantrr 699 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩})) → (𝑘 ∈ {𝐴} ↦ 𝑥) = {⟨𝐴, 𝑥⟩})
3634, 35eqtr4d 2843 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩})) → 𝑦 = (𝑘 ∈ {𝐴} ↦ 𝑥))
37 simprl 778 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩})) → 𝑥𝑋)
3837adantr 468 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩})) ∧ 𝑘 ∈ {𝐴}) → 𝑥𝑋)
3938fmpttd 6607 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩})) → (𝑘 ∈ {𝐴} ↦ 𝑥):{𝐴}⟶𝑋)
40 toponmax 20944 . . . . . . . . . . . 12 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
4117, 40syl 17 . . . . . . . . . . 11 (𝜑𝑋𝐽)
4241adantr 468 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩})) → 𝑋𝐽)
43 elmapg 8105 . . . . . . . . . 10 ((𝑋𝐽 ∧ {𝐴} ∈ V) → ((𝑘 ∈ {𝐴} ↦ 𝑥) ∈ (𝑋𝑚 {𝐴}) ↔ (𝑘 ∈ {𝐴} ↦ 𝑥):{𝐴}⟶𝑋))
4442, 18, 43sylancl 576 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩})) → ((𝑘 ∈ {𝐴} ↦ 𝑥) ∈ (𝑋𝑚 {𝐴}) ↔ (𝑘 ∈ {𝐴} ↦ 𝑥):{𝐴}⟶𝑋))
4539, 44mpbird 248 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩})) → (𝑘 ∈ {𝐴} ↦ 𝑥) ∈ (𝑋𝑚 {𝐴}))
4636, 45eqeltrd 2885 . . . . . . 7 ((𝜑 ∧ (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩})) → 𝑦 ∈ (𝑋𝑚 {𝐴}))
4734fveq1d 6410 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩})) → (𝑦𝐴) = ({⟨𝐴, 𝑥⟩}‘𝐴))
482adantr 468 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩})) → 𝐴𝑉)
49 fvsng 6672 . . . . . . . . 9 ((𝐴𝑉𝑥𝑋) → ({⟨𝐴, 𝑥⟩}‘𝐴) = 𝑥)
5048, 37, 49syl2anc 575 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩})) → ({⟨𝐴, 𝑥⟩}‘𝐴) = 𝑥)
5147, 50eqtr2d 2841 . . . . . . 7 ((𝜑 ∧ (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩})) → 𝑥 = (𝑦𝐴))
5246, 51jca 503 . . . . . 6 ((𝜑 ∧ (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩})) → (𝑦 ∈ (𝑋𝑚 {𝐴}) ∧ 𝑥 = (𝑦𝐴)))
53 simprr 780 . . . . . . . 8 ((𝜑 ∧ (𝑦 ∈ (𝑋𝑚 {𝐴}) ∧ 𝑥 = (𝑦𝐴))) → 𝑥 = (𝑦𝐴))
54 simprl 778 . . . . . . . . . 10 ((𝜑 ∧ (𝑦 ∈ (𝑋𝑚 {𝐴}) ∧ 𝑥 = (𝑦𝐴))) → 𝑦 ∈ (𝑋𝑚 {𝐴}))
5541adantr 468 . . . . . . . . . . 11 ((𝜑 ∧ (𝑦 ∈ (𝑋𝑚 {𝐴}) ∧ 𝑥 = (𝑦𝐴))) → 𝑋𝐽)
56 elmapg 8105 . . . . . . . . . . 11 ((𝑋𝐽 ∧ {𝐴} ∈ V) → (𝑦 ∈ (𝑋𝑚 {𝐴}) ↔ 𝑦:{𝐴}⟶𝑋))
5755, 18, 56sylancl 576 . . . . . . . . . 10 ((𝜑 ∧ (𝑦 ∈ (𝑋𝑚 {𝐴}) ∧ 𝑥 = (𝑦𝐴))) → (𝑦 ∈ (𝑋𝑚 {𝐴}) ↔ 𝑦:{𝐴}⟶𝑋))
5854, 57mpbid 223 . . . . . . . . 9 ((𝜑 ∧ (𝑦 ∈ (𝑋𝑚 {𝐴}) ∧ 𝑥 = (𝑦𝐴))) → 𝑦:{𝐴}⟶𝑋)
59 snidg 4400 . . . . . . . . . . 11 (𝐴𝑉𝐴 ∈ {𝐴})
602, 59syl 17 . . . . . . . . . 10 (𝜑𝐴 ∈ {𝐴})
6160adantr 468 . . . . . . . . 9 ((𝜑 ∧ (𝑦 ∈ (𝑋𝑚 {𝐴}) ∧ 𝑥 = (𝑦𝐴))) → 𝐴 ∈ {𝐴})
6258, 61ffvelrnd 6582 . . . . . . . 8 ((𝜑 ∧ (𝑦 ∈ (𝑋𝑚 {𝐴}) ∧ 𝑥 = (𝑦𝐴))) → (𝑦𝐴) ∈ 𝑋)
6353, 62eqeltrd 2885 . . . . . . 7 ((𝜑 ∧ (𝑦 ∈ (𝑋𝑚 {𝐴}) ∧ 𝑥 = (𝑦𝐴))) → 𝑥𝑋)
642adantr 468 . . . . . . . . . . 11 ((𝜑 ∧ (𝑦 ∈ (𝑋𝑚 {𝐴}) ∧ 𝑥 = (𝑦𝐴))) → 𝐴𝑉)
65 fsn2g 6628 . . . . . . . . . . 11 (𝐴𝑉 → (𝑦:{𝐴}⟶𝑋 ↔ ((𝑦𝐴) ∈ 𝑋𝑦 = {⟨𝐴, (𝑦𝐴)⟩})))
6664, 65syl 17 . . . . . . . . . 10 ((𝜑 ∧ (𝑦 ∈ (𝑋𝑚 {𝐴}) ∧ 𝑥 = (𝑦𝐴))) → (𝑦:{𝐴}⟶𝑋 ↔ ((𝑦𝐴) ∈ 𝑋𝑦 = {⟨𝐴, (𝑦𝐴)⟩})))
6758, 66mpbid 223 . . . . . . . . 9 ((𝜑 ∧ (𝑦 ∈ (𝑋𝑚 {𝐴}) ∧ 𝑥 = (𝑦𝐴))) → ((𝑦𝐴) ∈ 𝑋𝑦 = {⟨𝐴, (𝑦𝐴)⟩}))
6867simprd 485 . . . . . . . 8 ((𝜑 ∧ (𝑦 ∈ (𝑋𝑚 {𝐴}) ∧ 𝑥 = (𝑦𝐴))) → 𝑦 = {⟨𝐴, (𝑦𝐴)⟩})
6953opeq2d 4602 . . . . . . . . 9 ((𝜑 ∧ (𝑦 ∈ (𝑋𝑚 {𝐴}) ∧ 𝑥 = (𝑦𝐴))) → ⟨𝐴, 𝑥⟩ = ⟨𝐴, (𝑦𝐴)⟩)
7069sneqd 4382 . . . . . . . 8 ((𝜑 ∧ (𝑦 ∈ (𝑋𝑚 {𝐴}) ∧ 𝑥 = (𝑦𝐴))) → {⟨𝐴, 𝑥⟩} = {⟨𝐴, (𝑦𝐴)⟩})
7168, 70eqtr4d 2843 . . . . . . 7 ((𝜑 ∧ (𝑦 ∈ (𝑋𝑚 {𝐴}) ∧ 𝑥 = (𝑦𝐴))) → 𝑦 = {⟨𝐴, 𝑥⟩})
7263, 71jca 503 . . . . . 6 ((𝜑 ∧ (𝑦 ∈ (𝑋𝑚 {𝐴}) ∧ 𝑥 = (𝑦𝐴))) → (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩}))
7352, 72impbida 826 . . . . 5 (𝜑 → ((𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩}) ↔ (𝑦 ∈ (𝑋𝑚 {𝐴}) ∧ 𝑥 = (𝑦𝐴))))
7473mptcnv 5745 . . . 4 (𝜑(𝑥𝑋 ↦ {⟨𝐴, 𝑥⟩}) = (𝑦 ∈ (𝑋𝑚 {𝐴}) ↦ (𝑦𝐴)))
75 xpsng 6629 . . . . . . . . . . 11 ((𝐴𝑉𝐽 ∈ (TopOn‘𝑋)) → ({𝐴} × {𝐽}) = {⟨𝐴, 𝐽⟩})
762, 17, 75syl2anc 575 . . . . . . . . . 10 (𝜑 → ({𝐴} × {𝐽}) = {⟨𝐴, 𝐽⟩})
7776eqcomd 2812 . . . . . . . . 9 (𝜑 → {⟨𝐴, 𝐽⟩} = ({𝐴} × {𝐽}))
7877fveq2d 6412 . . . . . . . 8 (𝜑 → (∏t‘{⟨𝐴, 𝐽⟩}) = (∏t‘({𝐴} × {𝐽})))
7916, 78syl5eq 2852 . . . . . . 7 (𝜑𝐾 = (∏t‘({𝐴} × {𝐽})))
80 eqid 2806 . . . . . . . . 9 (∏t‘({𝐴} × {𝐽})) = (∏t‘({𝐴} × {𝐽}))
8180pttoponconst 21614 . . . . . . . 8 (({𝐴} ∈ V ∧ 𝐽 ∈ (TopOn‘𝑋)) → (∏t‘({𝐴} × {𝐽})) ∈ (TopOn‘(𝑋𝑚 {𝐴})))
8219, 17, 81syl2anc 575 . . . . . . 7 (𝜑 → (∏t‘({𝐴} × {𝐽})) ∈ (TopOn‘(𝑋𝑚 {𝐴})))
8379, 82eqeltrd 2885 . . . . . 6 (𝜑𝐾 ∈ (TopOn‘(𝑋𝑚 {𝐴})))
84 toponuni 20932 . . . . . 6 (𝐾 ∈ (TopOn‘(𝑋𝑚 {𝐴})) → (𝑋𝑚 {𝐴}) = 𝐾)
8583, 84syl 17 . . . . 5 (𝜑 → (𝑋𝑚 {𝐴}) = 𝐾)
8685mpteq1d 4932 . . . 4 (𝜑 → (𝑦 ∈ (𝑋𝑚 {𝐴}) ↦ (𝑦𝐴)) = (𝑦 𝐾 ↦ (𝑦𝐴)))
8774, 86eqtrd 2840 . . 3 (𝜑(𝑥𝑋 ↦ {⟨𝐴, 𝑥⟩}) = (𝑦 𝐾 ↦ (𝑦𝐴)))
88 eqid 2806 . . . . . 6 𝐾 = 𝐾
8988, 16ptpjcn 21628 . . . . 5 (({𝐴} ∈ V ∧ {⟨𝐴, 𝐽⟩}:{𝐴}⟶Top ∧ 𝐴 ∈ {𝐴}) → (𝑦 𝐾 ↦ (𝑦𝐴)) ∈ (𝐾 Cn ({⟨𝐴, 𝐽⟩}‘𝐴)))
9018, 22, 60, 89mp3an2i 1583 . . . 4 (𝜑 → (𝑦 𝐾 ↦ (𝑦𝐴)) ∈ (𝐾 Cn ({⟨𝐴, 𝐽⟩}‘𝐴)))
9128oveq2d 6890 . . . 4 (𝜑 → (𝐾 Cn ({⟨𝐴, 𝐽⟩}‘𝐴)) = (𝐾 Cn 𝐽))
9290, 91eleqtrd 2887 . . 3 (𝜑 → (𝑦 𝐾 ↦ (𝑦𝐴)) ∈ (𝐾 Cn 𝐽))
9387, 92eqeltrd 2885 . 2 (𝜑(𝑥𝑋 ↦ {⟨𝐴, 𝑥⟩}) ∈ (𝐾 Cn 𝐽))
94 ishmeo 21776 . 2 ((𝑥𝑋 ↦ {⟨𝐴, 𝑥⟩}) ∈ (𝐽Homeo𝐾) ↔ ((𝑥𝑋 ↦ {⟨𝐴, 𝑥⟩}) ∈ (𝐽 Cn 𝐾) ∧ (𝑥𝑋 ↦ {⟨𝐴, 𝑥⟩}) ∈ (𝐾 Cn 𝐽)))
9533, 93, 94sylanbrc 574 1 (𝜑 → (𝑥𝑋 ↦ {⟨𝐴, 𝑥⟩}) ∈ (𝐽Homeo𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1637  wcel 2156  Vcvv 3391  {csn 4370  cop 4376   cuni 4630  cmpt 4923   × cxp 5309  ccnv 5310  wf 6097  cfv 6101  (class class class)co 6874  𝑚 cmap 8092  tcpt 16304  Topctop 20911  TopOnctopon 20928   Cn ccn 21242  Homeochmeo 21770
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7179
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-reu 3103  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-tp 4375  df-op 4377  df-uni 4631  df-int 4670  df-iun 4714  df-iin 4715  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5219  df-eprel 5224  df-po 5232  df-so 5233  df-fr 5270  df-we 5272  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-pred 5893  df-ord 5939  df-on 5940  df-lim 5941  df-suc 5942  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-ov 6877  df-oprab 6878  df-mpt2 6879  df-om 7296  df-1st 7398  df-2nd 7399  df-wrecs 7642  df-recs 7704  df-rdg 7742  df-1o 7796  df-oadd 7800  df-er 7979  df-map 8094  df-ixp 8146  df-en 8193  df-dom 8194  df-fin 8196  df-fi 8556  df-topgen 16309  df-pt 16310  df-top 20912  df-topon 20929  df-bases 20964  df-cn 21245  df-cnp 21246  df-hmeo 21772
This theorem is referenced by:  xpstopnlem1  21826  ptcmpfi  21830
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