MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pt1hmeo Structured version   Visualization version   GIF version

Theorem pt1hmeo 22727
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 5625 . . . . 5 ({𝐴} × {𝑥}) = (𝑘 ∈ {𝐴} ↦ 𝑥)
2 pt1hmeo.a . . . . . . 7 (𝜑𝐴𝑉)
32adantr 484 . . . . . 6 ((𝜑𝑥𝑋) → 𝐴𝑉)
4 sneq 4565 . . . . . . . . 9 (𝑘 = 𝐴 → {𝑘} = {𝐴})
54xpeq1d 5594 . . . . . . . 8 (𝑘 = 𝐴 → ({𝑘} × {𝑥}) = ({𝐴} × {𝑥}))
6 opeq1 4798 . . . . . . . . 9 (𝑘 = 𝐴 → ⟨𝑘, 𝑥⟩ = ⟨𝐴, 𝑥⟩)
76sneqd 4567 . . . . . . . 8 (𝑘 = 𝐴 → {⟨𝑘, 𝑥⟩} = {⟨𝐴, 𝑥⟩})
85, 7eqeq12d 2754 . . . . . . 7 (𝑘 = 𝐴 → (({𝑘} × {𝑥}) = {⟨𝑘, 𝑥⟩} ↔ ({𝐴} × {𝑥}) = {⟨𝐴, 𝑥⟩}))
9 vex 3424 . . . . . . . 8 𝑘 ∈ V
10 vex 3424 . . . . . . . 8 𝑥 ∈ V
119, 10xpsn 6974 . . . . . . 7 ({𝑘} × {𝑥}) = {⟨𝑘, 𝑥⟩}
128, 11vtoclg 3493 . . . . . 6 (𝐴𝑉 → ({𝐴} × {𝑥}) = {⟨𝐴, 𝑥⟩})
133, 12syl 17 . . . . 5 ((𝜑𝑥𝑋) → ({𝐴} × {𝑥}) = {⟨𝐴, 𝑥⟩})
141, 13eqtr3id 2793 . . . 4 ((𝜑𝑥𝑋) → (𝑘 ∈ {𝐴} ↦ 𝑥) = {⟨𝐴, 𝑥⟩})
1514mpteq2dva 5164 . . 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 5338 . . . . 5 {𝐴} ∈ V
1918a1i 11 . . . 4 (𝜑 → {𝐴} ∈ V)
20 topontop 21834 . . . . . 6 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2117, 20syl 17 . . . . 5 (𝜑𝐽 ∈ Top)
222, 21fsnd 6721 . . . 4 (𝜑 → {⟨𝐴, 𝐽⟩}:{𝐴}⟶Top)
2317cnmptid 22582 . . . . . 6 (𝜑 → (𝑥𝑋𝑥) ∈ (𝐽 Cn 𝐽))
2423adantr 484 . . . . 5 ((𝜑𝑘 ∈ {𝐴}) → (𝑥𝑋𝑥) ∈ (𝐽 Cn 𝐽))
25 elsni 4572 . . . . . . . 8 (𝑘 ∈ {𝐴} → 𝑘 = 𝐴)
2625fveq2d 6739 . . . . . . 7 (𝑘 ∈ {𝐴} → ({⟨𝐴, 𝐽⟩}‘𝑘) = ({⟨𝐴, 𝐽⟩}‘𝐴))
27 fvsng 7013 . . . . . . . 8 ((𝐴𝑉𝐽 ∈ (TopOn‘𝑋)) → ({⟨𝐴, 𝐽⟩}‘𝐴) = 𝐽)
282, 17, 27syl2anc 587 . . . . . . 7 (𝜑 → ({⟨𝐴, 𝐽⟩}‘𝐴) = 𝐽)
2926, 28sylan9eqr 2801 . . . . . 6 ((𝜑𝑘 ∈ {𝐴}) → ({⟨𝐴, 𝐽⟩}‘𝑘) = 𝐽)
3029oveq2d 7247 . . . . 5 ((𝜑𝑘 ∈ {𝐴}) → (𝐽 Cn ({⟨𝐴, 𝐽⟩}‘𝑘)) = (𝐽 Cn 𝐽))
3124, 30eleqtrrd 2842 . . . 4 ((𝜑𝑘 ∈ {𝐴}) → (𝑥𝑋𝑥) ∈ (𝐽 Cn ({⟨𝐴, 𝐽⟩}‘𝑘)))
3216, 17, 19, 22, 31ptcn 22548 . . 3 (𝜑 → (𝑥𝑋 ↦ (𝑘 ∈ {𝐴} ↦ 𝑥)) ∈ (𝐽 Cn 𝐾))
3315, 32eqeltrrd 2840 . 2 (𝜑 → (𝑥𝑋 ↦ {⟨𝐴, 𝑥⟩}) ∈ (𝐽 Cn 𝐾))
34 simprr 773 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩})) → 𝑦 = {⟨𝐴, 𝑥⟩})
3514adantrr 717 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩})) → (𝑘 ∈ {𝐴} ↦ 𝑥) = {⟨𝐴, 𝑥⟩})
3634, 35eqtr4d 2781 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩})) → 𝑦 = (𝑘 ∈ {𝐴} ↦ 𝑥))
37 simprl 771 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩})) → 𝑥𝑋)
3837adantr 484 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩})) ∧ 𝑘 ∈ {𝐴}) → 𝑥𝑋)
3938fmpttd 6950 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩})) → (𝑘 ∈ {𝐴} ↦ 𝑥):{𝐴}⟶𝑋)
40 toponmax 21847 . . . . . . . . . . . 12 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
4117, 40syl 17 . . . . . . . . . . 11 (𝜑𝑋𝐽)
4241adantr 484 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩})) → 𝑋𝐽)
43 elmapg 8541 . . . . . . . . . 10 ((𝑋𝐽 ∧ {𝐴} ∈ V) → ((𝑘 ∈ {𝐴} ↦ 𝑥) ∈ (𝑋m {𝐴}) ↔ (𝑘 ∈ {𝐴} ↦ 𝑥):{𝐴}⟶𝑋))
4442, 18, 43sylancl 589 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩})) → ((𝑘 ∈ {𝐴} ↦ 𝑥) ∈ (𝑋m {𝐴}) ↔ (𝑘 ∈ {𝐴} ↦ 𝑥):{𝐴}⟶𝑋))
4539, 44mpbird 260 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩})) → (𝑘 ∈ {𝐴} ↦ 𝑥) ∈ (𝑋m {𝐴}))
4636, 45eqeltrd 2839 . . . . . . 7 ((𝜑 ∧ (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩})) → 𝑦 ∈ (𝑋m {𝐴}))
4734fveq1d 6737 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩})) → (𝑦𝐴) = ({⟨𝐴, 𝑥⟩}‘𝐴))
482adantr 484 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩})) → 𝐴𝑉)
49 fvsng 7013 . . . . . . . . 9 ((𝐴𝑉𝑥𝑋) → ({⟨𝐴, 𝑥⟩}‘𝐴) = 𝑥)
5048, 37, 49syl2anc 587 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩})) → ({⟨𝐴, 𝑥⟩}‘𝐴) = 𝑥)
5147, 50eqtr2d 2779 . . . . . . 7 ((𝜑 ∧ (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩})) → 𝑥 = (𝑦𝐴))
5246, 51jca 515 . . . . . 6 ((𝜑 ∧ (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩})) → (𝑦 ∈ (𝑋m {𝐴}) ∧ 𝑥 = (𝑦𝐴)))
53 simprr 773 . . . . . . . 8 ((𝜑 ∧ (𝑦 ∈ (𝑋m {𝐴}) ∧ 𝑥 = (𝑦𝐴))) → 𝑥 = (𝑦𝐴))
54 simprl 771 . . . . . . . . . 10 ((𝜑 ∧ (𝑦 ∈ (𝑋m {𝐴}) ∧ 𝑥 = (𝑦𝐴))) → 𝑦 ∈ (𝑋m {𝐴}))
5541adantr 484 . . . . . . . . . . 11 ((𝜑 ∧ (𝑦 ∈ (𝑋m {𝐴}) ∧ 𝑥 = (𝑦𝐴))) → 𝑋𝐽)
56 elmapg 8541 . . . . . . . . . . 11 ((𝑋𝐽 ∧ {𝐴} ∈ V) → (𝑦 ∈ (𝑋m {𝐴}) ↔ 𝑦:{𝐴}⟶𝑋))
5755, 18, 56sylancl 589 . . . . . . . . . 10 ((𝜑 ∧ (𝑦 ∈ (𝑋m {𝐴}) ∧ 𝑥 = (𝑦𝐴))) → (𝑦 ∈ (𝑋m {𝐴}) ↔ 𝑦:{𝐴}⟶𝑋))
5854, 57mpbid 235 . . . . . . . . 9 ((𝜑 ∧ (𝑦 ∈ (𝑋m {𝐴}) ∧ 𝑥 = (𝑦𝐴))) → 𝑦:{𝐴}⟶𝑋)
59 snidg 4589 . . . . . . . . . . 11 (𝐴𝑉𝐴 ∈ {𝐴})
602, 59syl 17 . . . . . . . . . 10 (𝜑𝐴 ∈ {𝐴})
6160adantr 484 . . . . . . . . 9 ((𝜑 ∧ (𝑦 ∈ (𝑋m {𝐴}) ∧ 𝑥 = (𝑦𝐴))) → 𝐴 ∈ {𝐴})
6258, 61ffvelrnd 6923 . . . . . . . 8 ((𝜑 ∧ (𝑦 ∈ (𝑋m {𝐴}) ∧ 𝑥 = (𝑦𝐴))) → (𝑦𝐴) ∈ 𝑋)
6353, 62eqeltrd 2839 . . . . . . 7 ((𝜑 ∧ (𝑦 ∈ (𝑋m {𝐴}) ∧ 𝑥 = (𝑦𝐴))) → 𝑥𝑋)
642adantr 484 . . . . . . . . . . 11 ((𝜑 ∧ (𝑦 ∈ (𝑋m {𝐴}) ∧ 𝑥 = (𝑦𝐴))) → 𝐴𝑉)
65 fsn2g 6971 . . . . . . . . . . 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 235 . . . . . . . . 9 ((𝜑 ∧ (𝑦 ∈ (𝑋m {𝐴}) ∧ 𝑥 = (𝑦𝐴))) → ((𝑦𝐴) ∈ 𝑋𝑦 = {⟨𝐴, (𝑦𝐴)⟩}))
6867simprd 499 . . . . . . . 8 ((𝜑 ∧ (𝑦 ∈ (𝑋m {𝐴}) ∧ 𝑥 = (𝑦𝐴))) → 𝑦 = {⟨𝐴, (𝑦𝐴)⟩})
6953opeq2d 4805 . . . . . . . . 9 ((𝜑 ∧ (𝑦 ∈ (𝑋m {𝐴}) ∧ 𝑥 = (𝑦𝐴))) → ⟨𝐴, 𝑥⟩ = ⟨𝐴, (𝑦𝐴)⟩)
7069sneqd 4567 . . . . . . . 8 ((𝜑 ∧ (𝑦 ∈ (𝑋m {𝐴}) ∧ 𝑥 = (𝑦𝐴))) → {⟨𝐴, 𝑥⟩} = {⟨𝐴, (𝑦𝐴)⟩})
7168, 70eqtr4d 2781 . . . . . . 7 ((𝜑 ∧ (𝑦 ∈ (𝑋m {𝐴}) ∧ 𝑥 = (𝑦𝐴))) → 𝑦 = {⟨𝐴, 𝑥⟩})
7263, 71jca 515 . . . . . 6 ((𝜑 ∧ (𝑦 ∈ (𝑋m {𝐴}) ∧ 𝑥 = (𝑦𝐴))) → (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩}))
7352, 72impbida 801 . . . . 5 (𝜑 → ((𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩}) ↔ (𝑦 ∈ (𝑋m {𝐴}) ∧ 𝑥 = (𝑦𝐴))))
7473mptcnv 6017 . . . 4 (𝜑(𝑥𝑋 ↦ {⟨𝐴, 𝑥⟩}) = (𝑦 ∈ (𝑋m {𝐴}) ↦ (𝑦𝐴)))
75 xpsng 6972 . . . . . . . . . . 11 ((𝐴𝑉𝐽 ∈ (TopOn‘𝑋)) → ({𝐴} × {𝐽}) = {⟨𝐴, 𝐽⟩})
762, 17, 75syl2anc 587 . . . . . . . . . 10 (𝜑 → ({𝐴} × {𝐽}) = {⟨𝐴, 𝐽⟩})
7776eqcomd 2744 . . . . . . . . 9 (𝜑 → {⟨𝐴, 𝐽⟩} = ({𝐴} × {𝐽}))
7877fveq2d 6739 . . . . . . . 8 (𝜑 → (∏t‘{⟨𝐴, 𝐽⟩}) = (∏t‘({𝐴} × {𝐽})))
7916, 78syl5eq 2791 . . . . . . 7 (𝜑𝐾 = (∏t‘({𝐴} × {𝐽})))
80 eqid 2738 . . . . . . . . 9 (∏t‘({𝐴} × {𝐽})) = (∏t‘({𝐴} × {𝐽}))
8180pttoponconst 22518 . . . . . . . 8 (({𝐴} ∈ V ∧ 𝐽 ∈ (TopOn‘𝑋)) → (∏t‘({𝐴} × {𝐽})) ∈ (TopOn‘(𝑋m {𝐴})))
8219, 17, 81syl2anc 587 . . . . . . 7 (𝜑 → (∏t‘({𝐴} × {𝐽})) ∈ (TopOn‘(𝑋m {𝐴})))
8379, 82eqeltrd 2839 . . . . . 6 (𝜑𝐾 ∈ (TopOn‘(𝑋m {𝐴})))
84 toponuni 21835 . . . . . 6 (𝐾 ∈ (TopOn‘(𝑋m {𝐴})) → (𝑋m {𝐴}) = 𝐾)
8583, 84syl 17 . . . . 5 (𝜑 → (𝑋m {𝐴}) = 𝐾)
8685mpteq1d 5158 . . . 4 (𝜑 → (𝑦 ∈ (𝑋m {𝐴}) ↦ (𝑦𝐴)) = (𝑦 𝐾 ↦ (𝑦𝐴)))
8774, 86eqtrd 2778 . . 3 (𝜑(𝑥𝑋 ↦ {⟨𝐴, 𝑥⟩}) = (𝑦 𝐾 ↦ (𝑦𝐴)))
88 eqid 2738 . . . . . 6 𝐾 = 𝐾
8988, 16ptpjcn 22532 . . . . 5 (({𝐴} ∈ V ∧ {⟨𝐴, 𝐽⟩}:{𝐴}⟶Top ∧ 𝐴 ∈ {𝐴}) → (𝑦 𝐾 ↦ (𝑦𝐴)) ∈ (𝐾 Cn ({⟨𝐴, 𝐽⟩}‘𝐴)))
9018, 22, 60, 89mp3an2i 1468 . . . 4 (𝜑 → (𝑦 𝐾 ↦ (𝑦𝐴)) ∈ (𝐾 Cn ({⟨𝐴, 𝐽⟩}‘𝐴)))
9128oveq2d 7247 . . . 4 (𝜑 → (𝐾 Cn ({⟨𝐴, 𝐽⟩}‘𝐴)) = (𝐾 Cn 𝐽))
9290, 91eleqtrd 2841 . . 3 (𝜑 → (𝑦 𝐾 ↦ (𝑦𝐴)) ∈ (𝐾 Cn 𝐽))
9387, 92eqeltrd 2839 . 2 (𝜑(𝑥𝑋 ↦ {⟨𝐴, 𝑥⟩}) ∈ (𝐾 Cn 𝐽))
94 ishmeo 22680 . 2 ((𝑥𝑋 ↦ {⟨𝐴, 𝑥⟩}) ∈ (𝐽Homeo𝐾) ↔ ((𝑥𝑋 ↦ {⟨𝐴, 𝑥⟩}) ∈ (𝐽 Cn 𝐾) ∧ (𝑥𝑋 ↦ {⟨𝐴, 𝑥⟩}) ∈ (𝐾 Cn 𝐽)))
9533, 93, 94sylanbrc 586 1 (𝜑 → (𝑥𝑋 ↦ {⟨𝐴, 𝑥⟩}) ∈ (𝐽Homeo𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1543  wcel 2111  Vcvv 3420  {csn 4555  cop 4561   cuni 4833  cmpt 5149   × cxp 5563  ccnv 5564  wf 6393  cfv 6397  (class class class)co 7231  m cmap 8528  tcpt 16967  Topctop 21814  TopOnctopon 21831   Cn ccn 22145  Homeochmeo 22674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2159  ax-12 2176  ax-ext 2709  ax-rep 5193  ax-sep 5206  ax-nul 5213  ax-pow 5272  ax-pr 5336  ax-un 7541
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2072  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2887  df-ne 2942  df-ral 3067  df-rex 3068  df-reu 3069  df-rab 3071  df-v 3422  df-sbc 3709  df-csb 3826  df-dif 3883  df-un 3885  df-in 3887  df-ss 3897  df-pss 3899  df-nul 4252  df-if 4454  df-pw 4529  df-sn 4556  df-pr 4558  df-tp 4560  df-op 4562  df-uni 4834  df-int 4874  df-iun 4920  df-iin 4921  df-br 5068  df-opab 5130  df-mpt 5150  df-tr 5176  df-id 5469  df-eprel 5474  df-po 5482  df-so 5483  df-fr 5523  df-we 5525  df-xp 5571  df-rel 5572  df-cnv 5573  df-co 5574  df-dm 5575  df-rn 5576  df-res 5577  df-ima 5578  df-ord 6233  df-on 6234  df-lim 6235  df-suc 6236  df-iota 6355  df-fun 6399  df-fn 6400  df-f 6401  df-f1 6402  df-fo 6403  df-f1o 6404  df-fv 6405  df-ov 7234  df-oprab 7235  df-mpo 7236  df-om 7663  df-1st 7779  df-2nd 7780  df-1o 8222  df-er 8411  df-map 8530  df-ixp 8599  df-en 8647  df-dom 8648  df-fin 8650  df-fi 9051  df-topgen 16972  df-pt 16973  df-top 21815  df-topon 21832  df-bases 21867  df-cn 22148  df-cnp 22149  df-hmeo 22676
This theorem is referenced by:  xpstopnlem1  22730  ptcmpfi  22734
  Copyright terms: Public domain W3C validator