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

Theorem pt1hmeo 21803
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 5312 . . . . 5 ({𝐴} × {𝑥}) = (𝑘 ∈ {𝐴} ↦ 𝑥)
2 pt1hmeo.a . . . . . . 7 (𝜑𝐴𝑉)
32adantr 472 . . . . . 6 ((𝜑𝑥𝑋) → 𝐴𝑉)
4 sneq 4323 . . . . . . . . 9 (𝑘 = 𝐴 → {𝑘} = {𝐴})
54xpeq1d 5287 . . . . . . . 8 (𝑘 = 𝐴 → ({𝑘} × {𝑥}) = ({𝐴} × {𝑥}))
6 opeq1 4545 . . . . . . . . 9 (𝑘 = 𝐴 → ⟨𝑘, 𝑥⟩ = ⟨𝐴, 𝑥⟩)
76sneqd 4325 . . . . . . . 8 (𝑘 = 𝐴 → {⟨𝑘, 𝑥⟩} = {⟨𝐴, 𝑥⟩})
85, 7eqeq12d 2767 . . . . . . 7 (𝑘 = 𝐴 → (({𝑘} × {𝑥}) = {⟨𝑘, 𝑥⟩} ↔ ({𝐴} × {𝑥}) = {⟨𝐴, 𝑥⟩}))
9 vex 3335 . . . . . . . 8 𝑘 ∈ V
10 vex 3335 . . . . . . . 8 𝑥 ∈ V
119, 10xpsn 6562 . . . . . . 7 ({𝑘} × {𝑥}) = {⟨𝑘, 𝑥⟩}
128, 11vtoclg 3398 . . . . . 6 (𝐴𝑉 → ({𝐴} × {𝑥}) = {⟨𝐴, 𝑥⟩})
133, 12syl 17 . . . . 5 ((𝜑𝑥𝑋) → ({𝐴} × {𝑥}) = {⟨𝐴, 𝑥⟩})
141, 13syl5eqr 2800 . . . 4 ((𝜑𝑥𝑋) → (𝑘 ∈ {𝐴} ↦ 𝑥) = {⟨𝐴, 𝑥⟩})
1514mpteq2dva 4888 . . 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 5049 . . . . 5 {𝐴} ∈ V
1918a1i 11 . . . 4 (𝜑 → {𝐴} ∈ V)
20 topontop 20912 . . . . . 6 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2117, 20syl 17 . . . . 5 (𝜑𝐽 ∈ Top)
222, 21fsnd 6332 . . . 4 (𝜑 → {⟨𝐴, 𝐽⟩}:{𝐴}⟶Top)
2317cnmptid 21658 . . . . . 6 (𝜑 → (𝑥𝑋𝑥) ∈ (𝐽 Cn 𝐽))
2423adantr 472 . . . . 5 ((𝜑𝑘 ∈ {𝐴}) → (𝑥𝑋𝑥) ∈ (𝐽 Cn 𝐽))
25 elsni 4330 . . . . . . . 8 (𝑘 ∈ {𝐴} → 𝑘 = 𝐴)
2625fveq2d 6348 . . . . . . 7 (𝑘 ∈ {𝐴} → ({⟨𝐴, 𝐽⟩}‘𝑘) = ({⟨𝐴, 𝐽⟩}‘𝐴))
27 fvsng 6603 . . . . . . . 8 ((𝐴𝑉𝐽 ∈ (TopOn‘𝑋)) → ({⟨𝐴, 𝐽⟩}‘𝐴) = 𝐽)
282, 17, 27syl2anc 696 . . . . . . 7 (𝜑 → ({⟨𝐴, 𝐽⟩}‘𝐴) = 𝐽)
2926, 28sylan9eqr 2808 . . . . . 6 ((𝜑𝑘 ∈ {𝐴}) → ({⟨𝐴, 𝐽⟩}‘𝑘) = 𝐽)
3029oveq2d 6821 . . . . 5 ((𝜑𝑘 ∈ {𝐴}) → (𝐽 Cn ({⟨𝐴, 𝐽⟩}‘𝑘)) = (𝐽 Cn 𝐽))
3124, 30eleqtrrd 2834 . . . 4 ((𝜑𝑘 ∈ {𝐴}) → (𝑥𝑋𝑥) ∈ (𝐽 Cn ({⟨𝐴, 𝐽⟩}‘𝑘)))
3216, 17, 19, 22, 31ptcn 21624 . . 3 (𝜑 → (𝑥𝑋 ↦ (𝑘 ∈ {𝐴} ↦ 𝑥)) ∈ (𝐽 Cn 𝐾))
3315, 32eqeltrrd 2832 . 2 (𝜑 → (𝑥𝑋 ↦ {⟨𝐴, 𝑥⟩}) ∈ (𝐽 Cn 𝐾))
34 simprr 813 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩})) → 𝑦 = {⟨𝐴, 𝑥⟩})
3514adantrr 755 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩})) → (𝑘 ∈ {𝐴} ↦ 𝑥) = {⟨𝐴, 𝑥⟩})
3634, 35eqtr4d 2789 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩})) → 𝑦 = (𝑘 ∈ {𝐴} ↦ 𝑥))
37 simprl 811 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩})) → 𝑥𝑋)
3837adantr 472 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩})) ∧ 𝑘 ∈ {𝐴}) → 𝑥𝑋)
39 eqid 2752 . . . . . . . . . 10 (𝑘 ∈ {𝐴} ↦ 𝑥) = (𝑘 ∈ {𝐴} ↦ 𝑥)
4038, 39fmptd 6540 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩})) → (𝑘 ∈ {𝐴} ↦ 𝑥):{𝐴}⟶𝑋)
41 toponmax 20924 . . . . . . . . . . . 12 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
4217, 41syl 17 . . . . . . . . . . 11 (𝜑𝑋𝐽)
4342adantr 472 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩})) → 𝑋𝐽)
44 elmapg 8028 . . . . . . . . . 10 ((𝑋𝐽 ∧ {𝐴} ∈ V) → ((𝑘 ∈ {𝐴} ↦ 𝑥) ∈ (𝑋𝑚 {𝐴}) ↔ (𝑘 ∈ {𝐴} ↦ 𝑥):{𝐴}⟶𝑋))
4543, 18, 44sylancl 697 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩})) → ((𝑘 ∈ {𝐴} ↦ 𝑥) ∈ (𝑋𝑚 {𝐴}) ↔ (𝑘 ∈ {𝐴} ↦ 𝑥):{𝐴}⟶𝑋))
4640, 45mpbird 247 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩})) → (𝑘 ∈ {𝐴} ↦ 𝑥) ∈ (𝑋𝑚 {𝐴}))
4736, 46eqeltrd 2831 . . . . . . 7 ((𝜑 ∧ (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩})) → 𝑦 ∈ (𝑋𝑚 {𝐴}))
4834fveq1d 6346 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩})) → (𝑦𝐴) = ({⟨𝐴, 𝑥⟩}‘𝐴))
492adantr 472 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩})) → 𝐴𝑉)
50 fvsng 6603 . . . . . . . . 9 ((𝐴𝑉𝑥𝑋) → ({⟨𝐴, 𝑥⟩}‘𝐴) = 𝑥)
5149, 37, 50syl2anc 696 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩})) → ({⟨𝐴, 𝑥⟩}‘𝐴) = 𝑥)
5248, 51eqtr2d 2787 . . . . . . 7 ((𝜑 ∧ (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩})) → 𝑥 = (𝑦𝐴))
5347, 52jca 555 . . . . . 6 ((𝜑 ∧ (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩})) → (𝑦 ∈ (𝑋𝑚 {𝐴}) ∧ 𝑥 = (𝑦𝐴)))
54 simprr 813 . . . . . . . 8 ((𝜑 ∧ (𝑦 ∈ (𝑋𝑚 {𝐴}) ∧ 𝑥 = (𝑦𝐴))) → 𝑥 = (𝑦𝐴))
55 simprl 811 . . . . . . . . . 10 ((𝜑 ∧ (𝑦 ∈ (𝑋𝑚 {𝐴}) ∧ 𝑥 = (𝑦𝐴))) → 𝑦 ∈ (𝑋𝑚 {𝐴}))
5642adantr 472 . . . . . . . . . . 11 ((𝜑 ∧ (𝑦 ∈ (𝑋𝑚 {𝐴}) ∧ 𝑥 = (𝑦𝐴))) → 𝑋𝐽)
57 elmapg 8028 . . . . . . . . . . 11 ((𝑋𝐽 ∧ {𝐴} ∈ V) → (𝑦 ∈ (𝑋𝑚 {𝐴}) ↔ 𝑦:{𝐴}⟶𝑋))
5856, 18, 57sylancl 697 . . . . . . . . . 10 ((𝜑 ∧ (𝑦 ∈ (𝑋𝑚 {𝐴}) ∧ 𝑥 = (𝑦𝐴))) → (𝑦 ∈ (𝑋𝑚 {𝐴}) ↔ 𝑦:{𝐴}⟶𝑋))
5955, 58mpbid 222 . . . . . . . . 9 ((𝜑 ∧ (𝑦 ∈ (𝑋𝑚 {𝐴}) ∧ 𝑥 = (𝑦𝐴))) → 𝑦:{𝐴}⟶𝑋)
60 snidg 4343 . . . . . . . . . . 11 (𝐴𝑉𝐴 ∈ {𝐴})
612, 60syl 17 . . . . . . . . . 10 (𝜑𝐴 ∈ {𝐴})
6261adantr 472 . . . . . . . . 9 ((𝜑 ∧ (𝑦 ∈ (𝑋𝑚 {𝐴}) ∧ 𝑥 = (𝑦𝐴))) → 𝐴 ∈ {𝐴})
6359, 62ffvelrnd 6515 . . . . . . . 8 ((𝜑 ∧ (𝑦 ∈ (𝑋𝑚 {𝐴}) ∧ 𝑥 = (𝑦𝐴))) → (𝑦𝐴) ∈ 𝑋)
6454, 63eqeltrd 2831 . . . . . . 7 ((𝜑 ∧ (𝑦 ∈ (𝑋𝑚 {𝐴}) ∧ 𝑥 = (𝑦𝐴))) → 𝑥𝑋)
652adantr 472 . . . . . . . . . . 11 ((𝜑 ∧ (𝑦 ∈ (𝑋𝑚 {𝐴}) ∧ 𝑥 = (𝑦𝐴))) → 𝐴𝑉)
66 fsn2g 6560 . . . . . . . . . . 11 (𝐴𝑉 → (𝑦:{𝐴}⟶𝑋 ↔ ((𝑦𝐴) ∈ 𝑋𝑦 = {⟨𝐴, (𝑦𝐴)⟩})))
6765, 66syl 17 . . . . . . . . . 10 ((𝜑 ∧ (𝑦 ∈ (𝑋𝑚 {𝐴}) ∧ 𝑥 = (𝑦𝐴))) → (𝑦:{𝐴}⟶𝑋 ↔ ((𝑦𝐴) ∈ 𝑋𝑦 = {⟨𝐴, (𝑦𝐴)⟩})))
6859, 67mpbid 222 . . . . . . . . 9 ((𝜑 ∧ (𝑦 ∈ (𝑋𝑚 {𝐴}) ∧ 𝑥 = (𝑦𝐴))) → ((𝑦𝐴) ∈ 𝑋𝑦 = {⟨𝐴, (𝑦𝐴)⟩}))
6968simprd 482 . . . . . . . 8 ((𝜑 ∧ (𝑦 ∈ (𝑋𝑚 {𝐴}) ∧ 𝑥 = (𝑦𝐴))) → 𝑦 = {⟨𝐴, (𝑦𝐴)⟩})
7054opeq2d 4552 . . . . . . . . 9 ((𝜑 ∧ (𝑦 ∈ (𝑋𝑚 {𝐴}) ∧ 𝑥 = (𝑦𝐴))) → ⟨𝐴, 𝑥⟩ = ⟨𝐴, (𝑦𝐴)⟩)
7170sneqd 4325 . . . . . . . 8 ((𝜑 ∧ (𝑦 ∈ (𝑋𝑚 {𝐴}) ∧ 𝑥 = (𝑦𝐴))) → {⟨𝐴, 𝑥⟩} = {⟨𝐴, (𝑦𝐴)⟩})
7269, 71eqtr4d 2789 . . . . . . 7 ((𝜑 ∧ (𝑦 ∈ (𝑋𝑚 {𝐴}) ∧ 𝑥 = (𝑦𝐴))) → 𝑦 = {⟨𝐴, 𝑥⟩})
7364, 72jca 555 . . . . . 6 ((𝜑 ∧ (𝑦 ∈ (𝑋𝑚 {𝐴}) ∧ 𝑥 = (𝑦𝐴))) → (𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩}))
7453, 73impbida 913 . . . . 5 (𝜑 → ((𝑥𝑋𝑦 = {⟨𝐴, 𝑥⟩}) ↔ (𝑦 ∈ (𝑋𝑚 {𝐴}) ∧ 𝑥 = (𝑦𝐴))))
7574mptcnv 5684 . . . 4 (𝜑(𝑥𝑋 ↦ {⟨𝐴, 𝑥⟩}) = (𝑦 ∈ (𝑋𝑚 {𝐴}) ↦ (𝑦𝐴)))
76 xpsng 6561 . . . . . . . . . . 11 ((𝐴𝑉𝐽 ∈ (TopOn‘𝑋)) → ({𝐴} × {𝐽}) = {⟨𝐴, 𝐽⟩})
772, 17, 76syl2anc 696 . . . . . . . . . 10 (𝜑 → ({𝐴} × {𝐽}) = {⟨𝐴, 𝐽⟩})
7877eqcomd 2758 . . . . . . . . 9 (𝜑 → {⟨𝐴, 𝐽⟩} = ({𝐴} × {𝐽}))
7978fveq2d 6348 . . . . . . . 8 (𝜑 → (∏t‘{⟨𝐴, 𝐽⟩}) = (∏t‘({𝐴} × {𝐽})))
8016, 79syl5eq 2798 . . . . . . 7 (𝜑𝐾 = (∏t‘({𝐴} × {𝐽})))
81 eqid 2752 . . . . . . . . 9 (∏t‘({𝐴} × {𝐽})) = (∏t‘({𝐴} × {𝐽}))
8281pttoponconst 21594 . . . . . . . 8 (({𝐴} ∈ V ∧ 𝐽 ∈ (TopOn‘𝑋)) → (∏t‘({𝐴} × {𝐽})) ∈ (TopOn‘(𝑋𝑚 {𝐴})))
8319, 17, 82syl2anc 696 . . . . . . 7 (𝜑 → (∏t‘({𝐴} × {𝐽})) ∈ (TopOn‘(𝑋𝑚 {𝐴})))
8480, 83eqeltrd 2831 . . . . . 6 (𝜑𝐾 ∈ (TopOn‘(𝑋𝑚 {𝐴})))
85 toponuni 20913 . . . . . 6 (𝐾 ∈ (TopOn‘(𝑋𝑚 {𝐴})) → (𝑋𝑚 {𝐴}) = 𝐾)
8684, 85syl 17 . . . . 5 (𝜑 → (𝑋𝑚 {𝐴}) = 𝐾)
8786mpteq1d 4882 . . . 4 (𝜑 → (𝑦 ∈ (𝑋𝑚 {𝐴}) ↦ (𝑦𝐴)) = (𝑦 𝐾 ↦ (𝑦𝐴)))
8875, 87eqtrd 2786 . . 3 (𝜑(𝑥𝑋 ↦ {⟨𝐴, 𝑥⟩}) = (𝑦 𝐾 ↦ (𝑦𝐴)))
89 eqid 2752 . . . . . 6 𝐾 = 𝐾
9089, 16ptpjcn 21608 . . . . 5 (({𝐴} ∈ V ∧ {⟨𝐴, 𝐽⟩}:{𝐴}⟶Top ∧ 𝐴 ∈ {𝐴}) → (𝑦 𝐾 ↦ (𝑦𝐴)) ∈ (𝐾 Cn ({⟨𝐴, 𝐽⟩}‘𝐴)))
9118, 22, 61, 90mp3an2i 1570 . . . 4 (𝜑 → (𝑦 𝐾 ↦ (𝑦𝐴)) ∈ (𝐾 Cn ({⟨𝐴, 𝐽⟩}‘𝐴)))
9228oveq2d 6821 . . . 4 (𝜑 → (𝐾 Cn ({⟨𝐴, 𝐽⟩}‘𝐴)) = (𝐾 Cn 𝐽))
9391, 92eleqtrd 2833 . . 3 (𝜑 → (𝑦 𝐾 ↦ (𝑦𝐴)) ∈ (𝐾 Cn 𝐽))
9488, 93eqeltrd 2831 . 2 (𝜑(𝑥𝑋 ↦ {⟨𝐴, 𝑥⟩}) ∈ (𝐾 Cn 𝐽))
95 ishmeo 21756 . 2 ((𝑥𝑋 ↦ {⟨𝐴, 𝑥⟩}) ∈ (𝐽Homeo𝐾) ↔ ((𝑥𝑋 ↦ {⟨𝐴, 𝑥⟩}) ∈ (𝐽 Cn 𝐾) ∧ (𝑥𝑋 ↦ {⟨𝐴, 𝑥⟩}) ∈ (𝐾 Cn 𝐽)))
9633, 94, 95sylanbrc 701 1 (𝜑 → (𝑥𝑋 ↦ {⟨𝐴, 𝑥⟩}) ∈ (𝐽Homeo𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1624  wcel 2131  Vcvv 3332  {csn 4313  cop 4319   cuni 4580  cmpt 4873   × cxp 5256  ccnv 5257  wf 6037  cfv 6041  (class class class)co 6805  𝑚 cmap 8015  tcpt 16293  Topctop 20892  TopOnctopon 20909   Cn ccn 21222  Homeochmeo 21750
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-reu 3049  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-int 4620  df-iun 4666  df-iin 4667  df-br 4797  df-opab 4857  df-mpt 4874  df-tr 4897  df-id 5166  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-pred 5833  df-ord 5879  df-on 5880  df-lim 5881  df-suc 5882  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-om 7223  df-1st 7325  df-2nd 7326  df-wrecs 7568  df-recs 7629  df-rdg 7667  df-1o 7721  df-oadd 7725  df-er 7903  df-map 8017  df-ixp 8067  df-en 8114  df-dom 8115  df-fin 8117  df-fi 8474  df-topgen 16298  df-pt 16299  df-top 20893  df-topon 20910  df-bases 20944  df-cn 21225  df-cnp 21226  df-hmeo 21752
This theorem is referenced by:  xpstopnlem1  21806  ptcmpfi  21810
  Copyright terms: Public domain W3C validator